1. Introduction
In the present paper, given a convex planar domain, we consider a variant of the usual billiard map to model some dissipative phenomena, which result in the existence of a global attractor. For such dissipative maps, Birkhoff [Reference BirkhoffBir32] introduced an invariant subset of the attractor, the so-called Birkhoff attractor; as we shall see, it is minimal in some sense among all invariant sets which separate phase-space, and it is essentially the place where interesting dynamics occurs. We investigate the properties of the Birkhoff attractor, in particular, how they change as the dissipation parameter is varied.
Loosely speaking, like for conservative billiards, we consider a massless particle moving with unit velocity inside the billiard table
$\Omega \subset {\mathbb R}^2$
according to the usual law except at collisions with the boundary
$\partial \Omega $
, which we now assume to be inelastic. More precisely (see Figure 1):
-
• the motion happens along straight lines between two collisions;
-
• at each orthogonal collision, the velocity vector is replaced with its opposite, while at a non-orthogonal collision, it is changed in such a way that the (unoriented) outgoing angle of reflection is strictly smaller than the incoming angle of incidence, both being measured with respect to the normal to
$\partial \Omega $
.
In other words, the reflected angle bends toward the inner normal at the incidence point. We refer to Definition A here below for more details, and to §3.1 for further properties of these billiard maps.
The standard billiard map and its dissipative counterpart.
Billiards exhibiting some form of dissipation have already been considered in previous works. To the best of the authors’ knowledge, for outer billiards, dissipation was first introduced in [Reference DayDay47, Lemma 2.2]; see also [Reference TabachnikovTab93, p. 83]. Subsequently, dynamical properties of dissipative polygonal outer billiards have been studied in [Reference Del Magno, Gaivão and GutkinDMGaG15]. Regarding standard billiards, the paper [Reference Markarian, Pujals and SambarinoMPS10] by Markarian, Pujals, and Sambarino is dedicated to the study of limit sets of dissipative billiards (called here pinball billiards) for various types of tables (close to a circle, with semi-dispersing walls, which possess some hyperbolicity…), through the existence of a dominated splitting. Motivated by these rigorous results, the paper [Reference Arroyo, Markarian and SandersAMS09] numerically investigates and characterizes the bifurcations of the resulting attractors as the contraction parameter is varied. In [Reference Markarian, Oliffson Kamphorst and Pinto-de CarvalhoMOKPdC12], the authors construct simple examples of non-elastic convex billiards with dominated splitting and attractors supporting a rational or irrational rotation. Let us conclude this brief overview by mentioning some works about dissipative billiards for tables with flat walls. The paper [Reference Arroyo, Markarian and SandersAMS12] concentrates on inelastic billiard dynamics in an equilateral triangular table and studies, mainly numerically, the structure of fractal strange attractors and their evolution as the contraction parameter changes. Finally, in a series of works [Reference Duarte, Gaivão and SoufiDGaS17, Reference Del Magno, Lopes Dias, Duarte, Gaivão and PinheiroDMLDD+12, Reference Del Magno, Lopes Dias, Duarte, Gaivão and PinheiroDMLDD+14, Reference Del Magno, Lopes Dias, Duarte and GaivãoDMLDDGa18], Del Magno et al investigate dissipative billiards within various types of polygonal tables; in particular, they study the structure of the non-wandering sets of such billiards, the existence of hyperbolic attractors, and prove the existence of countably many Sinai–Ruelle–Bowen (SRB) measures on these attractors under suitable conditions.
Let us now move on to the formal definition of dissipative billiard maps considered in the present work. Let
$\Omega \subset \mathbb R^2$
be a strictly convex domain with
$C^k$
boundary
$\partial \Omega $
,
$k\geq 2$
. We say that
$\Omega \subset \mathbb R^2$
is strongly convex if, additionally, its curvature never vanishes. We assume that the perimeter of
$\partial \Omega $
is normalized to one. We fix an orientation of
$\partial \Omega $
and parameterize
$\partial \Omega $
in arclength by some map
$\Upsilon \colon {\mathbb T} \to \mathbb R^2$
, where
${\mathbb T}:=\mathbb R/\mathbb Z$
. The phase-space is the set of pairs
$(x,v)$
consisting of a point x on
$\partial \Omega $
, and a unit vector
$v\in T_x \Omega $
pointing inward or tangent to
$\partial \Omega $
. It is naturally identified with the cylinder
${\mathbb A}:={\mathbb T} \times [-1,1]$
; indeed, any point
$(x,v)$
in phase-space corresponds to a pair
$(s,r)\in {\mathbb T} \times [-1,1]$
, where
$x=\Upsilon (s)\in \partial \Omega $
, and
$r = \sin \varphi $
is the sine of the oriented angle
$\varphi \in [-{\pi }/{2},{\pi }/{2}]$
from the vector v to the inward normal to
$\partial \Omega $
at x. The usual conservative billiard map is then defined as
$$ \begin{align} f=f_1\colon \begin{cases} \mathbb{A} \ \to \mathbb{A}, \\ (s,r) \ \mapsto f(s,r)=(s',r_1^{\prime}), \end{cases} \end{align} $$
where
$\Upsilon (s')$
represents the point where the trajectory, starting at
$\Upsilon (s)$
along the direction making an angle
$\arcsin r$
with the normal at
$\Upsilon (s)$
, hits the boundary again, and
$r_1^{\prime }$
is the sine of the reflected angle at
$\Upsilon (s')$
, according to the standard reflection law (angle of incidence = angle of reflection). Let us now fix a dissipation parameter
$\unicode{x3bb} \in (0,1)$
.
Definition A. Given a domain
$\Omega $
as above, let us fix a
$C^{k-1}$
function
$\unicode{x3bb} \colon {\mathbb A}\to (0,1)$
such that
$$ \begin{align} 0<\partial_r \unicode{x3bb}(s,r)r+\unicode{x3bb}(s,r)<1\quad \text{for all } (s,r)\in \mathrm{int}({\mathbb A}), \end{align} $$
and let
$\mathcal {H}_{\unicode{x3bb} }\colon (s,r)\mapsto (s,\unicode{x3bb} (s,r) r)$
. The dissipative billiard map
$f_{\unicode{x3bb} }$
associated to
$\unicode{x3bb} $
is then defined as the map
$$ \begin{align*} f_{\unicode{x3bb}}:=\mathcal{H}_{\unicode{x3bb}} \circ f\colon \begin{cases} \mathbb{A}\ \to \mathbb{A}, \\ (s,r) \ \mapsto f_{\unicode{x3bb}}(s,r)=(s',r_{\unicode{x3bb}}'), \end{cases} \end{align*} $$
where
$$ \begin{align*} r_{\unicode{x3bb}}'=r_{\unicode{x3bb}}'(s,r):=\unicode{x3bb}(s',r_1^{\prime})r_1^{\prime} \end{align*} $$
for
$r_1^{\prime }=r_1^{\prime }(s,r)$
as in equation (1.1). Note that for any
$(s,r)\in \mathrm {int}({\mathbb A})$
, we have
$\det D\mathcal {H}_{\unicode{x3bb} }(f_1(s,r))=\partial _r \unicode{x3bb} (s',r_1^{\prime })r_1^{\prime }+\unicode{x3bb} (s',r_1^{\prime })$
. In particular, since
$\det Df_{1}(s,r)=1$
and by equation (1.2), we obtain
$$ \begin{align} 0 < \det Df_{\unicode{x3bb}}(s,r) =\det D\mathcal{H}_{\unicode{x3bb}}(f_1(s,r))< 1\quad \text{for all } (s,r) \in \mathrm{int}(\mathbb{A}). \end{align} $$
By equation (1.3), the resulting billiard map
$f_{\unicode{x3bb} }$
is no longer conservative; actually, it turns out to be a dissipative map in the sense of [Reference Le CalvezLC88] (see Definition 2.1). In particular,
$f_{\unicode{x3bb} }$
contracts the standard area form
$\omega = d r \wedge ds$
. We refer to §3 for a few general facts about dissipative billiards.
Remark 1.1. For the purpose of this article, it is sufficient to consider
$C^2$
billiard tables. Indeed, due to dissipation, trajectories which are close to the boundary drift further away from the boundary and pathological phenomena, such as in [Reference HalpernHal77], do not occur.
Remark 1.2. For simplicity, in most of what follows, we will restrict ourselves to the case where
$\unicode{x3bb} $
is actually a constant function, that is,
$\unicode{x3bb} \equiv \unicode{x3bb} _*\in (0,1)$
. In that case, we will say that
$f_{\unicode{x3bb} }$
has constant dissipation. Then, the dissipative billiard map associated to
$\unicode{x3bb} $
simply becomes
$$ \begin{align*} f_{\unicode{x3bb}}\colon \begin{cases} \mathbb{A}\ \to \mathbb{A}, \\ (s,r) \ \mapsto f_{\unicode{x3bb}}(s,r)=(s',r_{\unicode{x3bb}}'), \end{cases} \end{align*} $$
where
$$ \begin{align*} r_{\unicode{x3bb}}'=r_{\unicode{x3bb}}'(s,r):= \unicode{x3bb} r_1^{\prime}. \end{align*} $$
In the following, when it is clear from the context, we will abbreviate
$r_{\unicode{x3bb} }'=r'$
.
For constant dissipation, there is a natural one-parameter family of dissipative billiard maps
$\{f_{\unicode{x3bb} }\}_{\unicode{x3bb} \in (0,1)}$
; in particular, we will study transitions in the behavior of the Birkhoff attractor as
$\unicode{x3bb} $
changes. However, the simplifying hypothesis that
$\unicode{x3bb} $
is constant is not essential. Indeed, as we will explain, most results shown in the present work hold under the more general assumption that
$\unicode{x3bb} \colon {\mathbb A} \to (0,1)$
is a
$C^1$
function as in Definition A that is close enough to being constant, namely
$\|D\unicode{x3bb} \|\ll 1$
.
Due to the dissipative character of
$f_{\unicode{x3bb} }$
, there is a contraction of the phase-space which results in the existence of attractors. Indeed, as
$f_{\unicode{x3bb} }(\mathbb {A})\subset \mathrm {int}({\mathbb A})$
, there exists a global attractor
$$ \begin{align} \Lambda_{\unicode{x3bb}}^0:= \bigcap_{k \ge 0} f_{\unicode{x3bb}}^k({\mathbb A}). \end{align} $$
The attractor
$\Lambda _{\unicode{x3bb} }^0$
is
$f_{\unicode{x3bb} }$
-invariant, non-empty, compact, and connected. Moreover,
$\Lambda _{\unicode{x3bb} }^0$
separates
${\mathbb A}$
, that is,
${\mathbb A}\setminus \Lambda _{\unicode{x3bb} }^0$
is the disjoint union of two connected open sets
$U_{\unicode{x3bb} },V_{\unicode{x3bb} }$
. However, we can find a smaller invariant set—the so-called Birkhoff attractor—by ‘removing the hairs’ from
$\Lambda _{\unicode{x3bb} }^0$
(see e.g. [Reference Le CalvezLC90, p. 91]). The Birkhoff attractor, here denoted
$\Lambda _{\unicode{x3bb} }$
, is then defined as
$$ \begin{align} \Lambda_{\unicode{x3bb}}:= \overline{U}_{\unicode{x3bb}}\cap \overline{V}_{\unicode{x3bb}}. \end{align} $$
We remark that, even if
$\Lambda _{\unicode{x3bb} }$
is compact and
$f_{\unicode{x3bb} }$
-invariant, it may no longer be an attractor in the usual sense. Actually,
$\Lambda _{\unicode{x3bb} }$
can also be characterized as the minimal element (with respect to inclusion) among all sets which are compact, connected,
$f_{\unicode{x3bb} }$
-invariant, and separate
${\mathbb A}$
. We refer to §2 for more details about the Birkhoff attractor and its properties.
The notion of Birkhoff attractor was first introduced by Birkhoff in [Reference BirkhoffBir32]. In the framework of dissipative twist maps of the annulus, further properties of the Birkhoff attractor have been investigated by the works of Charpentier [Reference CharpentierCha34] and of Le Calvez [Reference Le CalvezLC88]. The Birkhoff attractor of the thickened Arnol’d family has been studied by Crovisier in [Reference CrovisierCro02]. Different authors have derived criteria to guarantee the existence of chaotic behaviors for invariant annular continua, see [Reference Barge and GilletteBG91a, Reference Barge and GilletteBG91b, Reference CasdagliCas88, Reference Hockett and HolmesHH86, Reference KoropeckiKor17, Reference Passeggi, Potrie and SambarinoPPS18, Reference Passeggi and TalPT23]. Recently, the notion of Birkhoff attractor has been generalized to higher dimensions for conformally symplectic maps of some symplectic manifolds by Arnaud, Humilière, and Viterbo, see [Reference Arnaud, Humilière and ViterboAHV23, Reference ViterboVit22].
Notation 1.3. Fix some dissipative map
$f\colon {\mathbb A}\to {\mathbb A}$
, with a hyperbolic periodic point
$p\in {\mathbb A}$
, of period
$q \geq 1$
. If p is of saddle type, we will denote its one-dimensional stable, respectively unstable manifold, as
$$ \begin{align*} \mathcal{W}^{s}(p;f^q) &:= \Big\{x\in{\mathbb A} :\lim_{n \to +\infty} d(f^{qn}(x),p)=0\Big\},\\ \mathcal{W}^{u}(p;f^q) &:= \Big\{x\in{\mathbb A} :\lim_{n \to +\infty} d(f^{-qn}(x),p)=0\Big\}. \end{align*} $$
If p is a sink, we will denote its two-dimensional stable manifold as
$$ \begin{align*} \mathcal{W}^{s}(p;f^q) := \Big\{x\in{\mathbb A} :\lim_{n \to +\infty} d(f^{qn}(x),p)=0\Big\}. \end{align*} $$
In either case, for
$*={\kern-1pt}s/u$
, we will sometimes abbreviate
$\mathcal {W}^{*}(\mathcal {O}_f(p){\kern-1pt}){\kern-2pt}:={\kern-2pt}\bigcup _{i=0}^{q-1} \mathcal {W}^{*}(f^i(p); f^q)$
, or simply
$\mathcal {W}^{*}(\mathcal {O}_{\unicode{x3bb} }(p))$
, when
$f=f_{\unicode{x3bb} }$
is some dissipative billiard map.
Considering the crucial role of elliptic tables in the conservative case, it is natural to start our study with dissipative billiard maps within ellipses. The detailed study of the corresponding dynamics is contained in §4, whose main result is the next theorem.
Theorem B. Given an ellipse
$\mathcal {E}$
of eccentricity
$e\in (0,1)$
, let
$f_{\unicode{x3bb} } \colon \mathbb {A} \to f_{\unicode{x3bb} }(\mathbb {A})\subset \mathrm {int}({\mathbb A})$
be a dissipative billiard map within
$\mathcal {E}$
in the sense of Definition A (we allow non-constant dissipation). Then, the
$2$
-periodic orbits
$\{H,f_{\unicode{x3bb} }(H)\}$
and
$\{E,f_{\unicode{x3bb} }(E)\}$
, corresponding to the trajectories along the major and minor axes, are hyperbolic of saddle and sink type, respectively, and the Birkhoff attractor satisfies
$$ \begin{align*} \Lambda_{\unicode{x3bb}}^0=\Lambda_{\unicode{x3bb}}=\mathcal{W}^u(\mathcal{O}_{\unicode{x3bb}}(H)) \cup \{E,f_{\unicode{x3bb}}(E)\} = \overline{\mathcal{W}^u(\mathcal{O}_{\unicode{x3bb}}(H))}. \end{align*} $$
Moreover, for
$i=0,1$
,
$\mathcal {W}^u(f_{\unicode{x3bb} }^i(H);f_{\unicode{x3bb} }^2)\setminus \{f_{\unicode{x3bb} }^i(H)\}$
is the disjoint union of two branches
$\mathscr {C}_i^1,\mathscr {C}_i^2$
, with
$\mathscr {C}_i^j\subset \mathcal {W}^s(f_{\unicode{x3bb} }^j(E);f_{\unicode{x3bb} }^2)$
,
$j=0,1$
.
At the end of §4, we prove that the conclusion of Theorem B remains true also for strictly convex domains whose boundary is sufficiently
$C^2$
-close to an ellipse, as stated in the next corollary. For simplicity, we state it in the case where the dissipation function
$\unicode{x3bb} $
is a constant in
$(0,1)$
.
Corollary C. Let
$\mathcal {E}$
be an ellipse of eccentricity
$e\in (0,1)$
. Let
$\unicode{x3bb} \in (0,1)$
. There exists
$\epsilon =\epsilon (\mathcal {E},\unicode{x3bb} )>0$
such that for any
$C^k$
(
$k\geq 2$
) domain
$\Omega \subset \mathbb R^2$
satisfying
$d_{C^2}(\partial \Omega ,\mathcal {E})<\epsilon $
, the following holds. Denoting by
$f_{\unicode{x3bb} }^\Omega $
the dissipative billiard map within
$\Omega $
, there exist
$2$
-periodic orbits
$\mathcal {O}_{f_{\unicode{x3bb} }^\Omega }(H)=\{H,f_{\unicode{x3bb} }^\Omega (H)\}$
and
$\mathcal {O}_{f_{\unicode{x3bb} }^\Omega }(E)=\{E,f_{\unicode{x3bb} }^\Omega (E)\}$
of saddle and sink type, respectively, and the Birkhoff attractor is equal to
$$ \begin{align*} \Lambda_{\unicode{x3bb}}= \mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}^\Omega}(H)) \cup \mathcal{O}_{f_{\unicode{x3bb}}^\Omega}(E). \end{align*} $$
Moreover, the function
$(\mathcal {E},\unicode{x3bb} )\mapsto \epsilon (\mathcal {E},\unicode{x3bb} )$
can be chosen to be continuous.
The first examples of Birkhoff attractors for a dissipative billiard map
$f_{\unicode{x3bb} }$
within a circle or an ellipse (see Figure 2 illustrating the Birkhoff attractor in the case of an ellipse when the dissipation is mild, that is,
$\unicode{x3bb} $
close to
$1$
) naturally lead us to consider topological properties of Birkhoff attractors, in particular, to investigate when
$\Lambda _{\unicode{x3bb} }$
is topologically as simple as it can be, namely, a graph. The main results in this direction are contained in §5. Through the following definition, we introduce the class of billiards for which we can guarantee such a simple behavior of the Birkhoff attractor.
Birkhoff attractor of a dissipative billiard map
$f_{\unicode{x3bb} }$
within an ellipse of non-zero eccentricity when the dissipation is mild, that is,
$\unicode{x3bb} $
is close to
$1$
.
Definition D. For any
$k \geq 2$
, let
$\mathcal {D}^k$
be the set of strongly convex domains
$\Omega $
with
$C^k$
boundary
$\partial \Omega $
such that, given a parameterization
$\Upsilon \colon \mathbb {T} \to \mathbb R^2$
of
$\partial \Omega $
, the following geometric condition holds (see Figure 3):
$$ \begin{align} \max_{s \in {\mathbb T}} \tau(s)\mathcal{K}(s)<-1, \end{align} $$
where
$\mathcal {K}(s)<0$
denotes the curvature of
$\partial \Omega $
at the point
$\Upsilon (s)$
, and
$\tau (s)>0$
is the length of the first segment of the
$f_1$
-orbit starting at
$\Upsilon (s)$
perpendicularly to
$\partial \Omega $
. Alternatively, equation (1.6) amounts to asking that the centers of the osculating circles at the points of
$\partial \Omega $
remain in
$\Omega $
.
The geometric condition
$\max _{s \in {\mathbb T}} \tau (s)\mathcal {K}(s)<-1$
in Definition D. Here,
$R(s):=-{1}/{\mathcal {K}(s)}$
is the radius of curvature, and
$\mathscr {C}(O(s),R(s))$
is the osculating circle at
$\Upsilon (s)$
.
Clearly, the set
$\mathcal {D}^k$
is
$C^k$
-open. More precisely, for any
$\Omega \in \mathcal {D}^k$
, there exists a
$C^2$
-open neighborhood
$\mathcal {U}$
of
$\Omega $
such that for any
$C^k$
domain
$\Omega '\in \mathcal {U}$
, we have
$\Omega '\in \mathcal {D}^k$
.
The main result of §5 is proving that the geometric condition contained in Definition D together with strong dissipation (
$\unicode{x3bb} $
close to
$0$
) suffice to guarantee that the corresponding Birkhoff attractor is a graph. Our result is also about the dynamics on the attractor and the graph’s regularity. We use the notions of dominated splitting for an invariant set and of normally contracted manifold. We refer to §5 for more details about such definitions.
Theorem E. Let
$f_{\unicode{x3bb} } \colon \mathbb {A} \to f_{\unicode{x3bb} }(\mathbb {A})\subset \mathrm {int}({\mathbb A})$
be a dissipative billiard map with constant dissipation
$\unicode{x3bb} \in (0,1)$
within some domain
$\Omega \in \mathcal {D}^k$
,
$k\geq 2$
. Then the following assertions hold.
-
(1) There exists
$\unicode{x3bb} (\Omega )\in (0,1)$
such that, for any
$\unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
coincides with
$\Lambda _{\unicode{x3bb} }^0$
and has a dominated splitting
$E^s \oplus E^c$
, where
$E^s$
is uniformly contracted by
$Df_{\unicode{x3bb} }$
. Moreover,
$\Lambda _{\unicode{x3bb} }$
is a normally contracted
$C^{1}$
graph over
$\mathbb {T}\times \{0\}$
which is tangent to
$E^c$
. -
(2) There exists
$\unicode{x3bb} '(\Omega )\in (0,\unicode{x3bb} (\Omega ))$
such that for any
$\unicode{x3bb} \in (0,\unicode{x3bb} '(\Omega ))$
,
$\Lambda _{\unicode{x3bb} }$
is actually a
$C^{k-1}$
graph. -
(3) There exists an open and dense set
$\mathscr {U}$
of
$C^k$
domains such that if, moreover,
$\Omega \in \mathcal {D}^k\cap \mathscr {U}$
, then there exists
$\unicode{x3bb} "(\Omega )\in (0,\unicode{x3bb} '(\Omega ))$
such that, for any
$\unicode{x3bb} \in (0,\unicode{x3bb} "(\Omega ))$
,
$\Lambda _{\unicode{x3bb} }$
is a
$C^{k-1}$
normally contracted graph of rotation number
$\frac 12$
. Moreover, for some finite collection
$$ \begin{align*} \Lambda_{\unicode{x3bb}}=\bigcup_{i=1}^\ell \overline{\mathcal{W}^u(\mathcal{O}_{\unicode{x3bb}}(H_i))}, \end{align*} $$
$\{\mathcal {O}_{\unicode{x3bb} }(H_i)\}_{i=1,\ldots ,\ell }=\{H_i,f_{\unicode{x3bb} }(H_i)\}_{i=1,\ldots ,\ell }$
of
$2$
-periodic orbits of saddle type (we refer the reader e.g. to [Reference Dias Carneiro, Oliffson Kamphorst and Pinto-de CarvalhoDCOKPdC07, §2] for more details on the topology on the space of
$C^k$
convex billiards).
Remark 1.4. Given
$k \geq 2$
and a domain
$\Omega \in \mathcal {D}^k\cap \mathscr {U}$
as in the above statement, the conclusion of Theorem E holds for general dissipative billiard maps
$f_{\unicode{x3bb} }$
in the sense of Definition A, provided that the dissipation function
$\unicode{x3bb} \colon {\mathbb A}\to (0,1)$
satisfies
$\|\unicode{x3bb} \|_{C^1}\ll 1$
. See e.g. Remark 5.17 for more details.
Remark 1.5. A consequence of Theorem E is that if
$\partial \Omega $
is an ellipse
$\mathcal {E}$
of eccentricity
$e\in (0,{\sqrt 2}/{2})$
, then
$\mathcal {E} \in \mathcal {D}^\infty $
and, for any
$\unicode{x3bb} \in (0,\unicode{x3bb} (\mathcal {E}))$
, the corresponding Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
is a normally contracted
$C^1$
graph, which is actually
$C^\infty $
except possibly at the
$2$
-periodic sink
$\{E,f_{\unicode{x3bb} }(E)\}$
, where
$\Lambda _{\unicode{x3bb} }$
is tangent to the weak stable space of the sink, see Corollary 5.11. We will also see that, when the eccentricity e is larger than
${\sqrt 2}/{2}$
, then for
$\unicode{x3bb} \in (0,1)$
small, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
is no longer a graph (see Proposition 5.16).
We may wonder if Birkhoff attractors of dissipative billiards may exhibit more complex topological properties than the examples described in §§4 and 5. In fact, following a result by Charpentier [Reference CharpentierCha34], a Birkhoff attractor for a dissipative diffeomorphism can be an ‘indecomposable continuum’, and a sufficient condition for this to occur is that the Birkhoff attractor contains points with different rotation numbers. The aim of §6 is to explore this direction and discuss some topological and dynamical implications of such a phenomenon.
To state the main results, we need to premise the notion of upper and lower rotation number for
$\Lambda _{\unicode{x3bb} }$
. Denote by
$V_{\Lambda }$
(respectively
$U_{\Lambda }$
) the connected component of
${\mathbb A}\setminus \Lambda _{\unicode{x3bb} }$
containing
$\{(s,1)\in {\mathbb A} :s\in {\mathbb T}\}$
(respectively
$\{(s,-1))\in {\mathbb A} :s\in {\mathbb T}\}$
). For any
$(s,r)\in {\mathbb A}$
, the upper and lower vertical lines are respectively
$$ \begin{align*} V^+(s,r):=\{(s,y)\in{\mathbb A} :y\geq r\}\quad \text{and}\quad V^-(s,r):=\{(s,y)\in{\mathbb A} :y\leq r\}. \end{align*} $$
Let us now define
$$ \begin{align*} \Lambda_{\unicode{x3bb}}^+:=\{ x\in\Lambda_{\unicode{x3bb}} :V^+(x)\setminus\{x\}\subset V_{\Lambda}\}\quad\text{and}\quad\Lambda_{\unicode{x3bb}}^-:=\{ x\in\Lambda_{\unicode{x3bb}} :V^-(x)\setminus\{x\}\subset U_{\Lambda}\}. \end{align*} $$
Given a covering
$\pi \colon \mathbb R\times [-1,1]\to {\mathbb T}\times [-1,1]$
of
${\mathbb A}$
, let
$\tilde \Lambda _{\unicode{x3bb} }^\pm :=\pi ^{-1}(\Lambda _{\unicode{x3bb} }^\pm )$
. Moreover, let
$\tilde \pi _1\colon \mathbb R\times [-1,1]\to \mathbb R$
be the first coordinate projection and
$F_{\unicode{x3bb} }\colon \mathbb R\times [-1,1]\to \mathbb R\times [-1,1]$
a continuous lift of
$f_{\unicode{x3bb} }$
. Then, by a result due to Birkhoff [Reference BirkhoffBir32] and rephrased in all details by Le Calvez [Reference Le CalvezLC88], the sequence
$$ \begin{align*}\bigg( \frac{\tilde \pi_1\circ F_{\unicode{x3bb}}^{-n}-\tilde \pi_1}{n} \bigg)_{n\in\mathbb N}\end{align*} $$
converges uniformly on
$\tilde \Lambda _{\unicode{x3bb} }^+$
$(\text {respectively} \ \tilde \Lambda _{\unicode{x3bb} }^-)$
to a constant
$\rho _{\unicode{x3bb} }^+$
$(\text {respectively} \ \rho _{\unicode{x3bb} }^-)$
. The constants
$\rho _{\unicode{x3bb} }^+$
and
$\rho _{\unicode{x3bb} }^-$
—called upper and lower rotation numbers—do depend on the chosen lift, but not their difference. We refer the reader to §6.2 for more details.
For the conservative billiard map
$f=f_1$
, let us denote by
$\mathscr {V}(f)$
the union of all f-invariant essential curves in
${\mathbb A}$
, that is, f-invariant homotopically non-trivial curves. We recall that an instability region for f is an open bounded connected component of
${\mathbb A} \setminus \mathscr {V}(f)$
that contains in its interior an essential curve. The main result of §6 is the next theorem, whose proof is mainly based on an adaptation of some arguments of the work [Reference Le CalvezLC88]. Let us recall that a continuum is a compact connected topological space.
Theorem F. Let
$\Omega \subset \mathbb R^2$
be a strongly convex domain with
$C^k$
boundary,
$k\geq 2$
. Let
$f=f_1$
be the associated conservative billiard map. If f admits an instability region that contains the zero section
$\mathbb {T}\times \{0\}$
, then there exists
$\unicode{x3bb} _0(\Omega )\in (0,1)$
such that, for any
$\unicode{x3bb} \in [\unicode{x3bb} _0(\Omega ),1)$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
of the dissipative billiard map
$f_{\unicode{x3bb} }$
with constant dissipation
$\unicode{x3bb} $
has
$\rho _{\unicode{x3bb} }^+-\rho ^-_{\unicode{x3bb} }>0$
, with
$\frac 12 \in (\rho _{\unicode{x3bb} }^-,\rho ^+_{\unicode{x3bb} })\,\mod \mathbb Z$
.
Remark 1.6. Theorem F was stated for a dissipative map
$f_{\unicode{x3bb} }$
with constant dissipation. However, the result holds for general dissipative billiard maps in the sense of Definition A, as long as the dissipation function
$\unicode{x3bb} \colon {\mathbb A} \to (0,1)$
is sufficiently close to the constant function
$1$
in the
$C^1$
-topology, that is,
$\|1-\unicode{x3bb} \|_{C^1}\ll 1$
. See e.g. Proposition 6.10 for more details in this direction.
The above theorem has several interesting consequences for
$\Lambda _{\unicode{x3bb} }$
. In fact, in the case where
$\rho _{\unicode{x3bb} }^+-\rho ^-_{\unicode{x3bb} }>0$
, the corresponding Birkhoff attractor turns out to be complicated both topologically and dynamically. In particular:
-
•
$\Lambda _{\unicode{x3bb} }$
is an indecomposable continuum, that is, it cannot be written as the union of two proper continua (directly from the work [Reference CharpentierCha34] of Charpentier; see also [Reference Barge and GilletteBG91a]); -
• each rational
${p}/{q}\in (\rho _{\unicode{x3bb} }^-,\rho ^+_{\unicode{x3bb} })$
is the rotation number of a periodic orbit in
$\Lambda _{\unicode{x3bb} }$
(as a straightforward application of [Reference Barge and GilletteBG91b]); -
• if x is a saddle periodic point of type
$(p,q)$
, with
${p}/{q}\in (\rho _{\unicode{x3bb} }^-,\rho ^+_{\unicode{x3bb} })$
, then its unstable manifold
$\mathcal {W}^u(x;f^q)$
satisfies
$\overline {\mathcal {W}^u(x;f^q)}\subset \Lambda _{\unicode{x3bb} }$
(by [Reference Le CalvezLC88, Proposition 14.3]); -
• there exists
$n_0\in \mathbb {N}$
so that
$f_{\unicode{x3bb} }^{n_0}$
has a rotational horseshoe (by [Reference Passeggi, Potrie and SambarinoPPS18, Theorem A]). In particular,
$f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }}$
has positive topological entropy.
Applying essentially [Reference Dias Carneiro, Oliffson Kamphorst and Pinto-de CarvalhoDCOKPdC07], we prove that the conclusions of Theorem F hold generically for
$C^k$
strongly convex domains,
$k \ge 3$
, as explained in the next corollary.
Corollary G. For
$k \geq 3$
, there exists an open and dense subset
$\mathscr {U}$
of the set of
$C^k$
strongly convex domains such that for every
$\Omega \in \mathscr {U}$
, the following assertions hold.
-
(1) There exists
$\unicode{x3bb} _0(\Omega )\in (0,1)$
such that, for any
$\unicode{x3bb} \in [\unicode{x3bb} _0(\Omega ),1)$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
of the corresponding dissipative billiard map
$f_{\unicode{x3bb} }$
has
$\rho _{\unicode{x3bb} }^+-\rho ^-_{\unicode{x3bb} }>0$
, with
$\tfrac 12 \in (\rho _{\unicode{x3bb} }^-,\rho ^+_{\unicode{x3bb} })\,\mod \mathbb Z$
. -
(2) There exists
$\unicode{x3bb} _1(\Omega )\in [\unicode{x3bb} _0(\Omega ),1)$
such that, for any
$\unicode{x3bb} \in [\unicode{x3bb} _1(\Omega ),1)$
and any
$2$
-periodic point p of saddle type (e.g. when the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
corresponds to a diameter of the table), there exists a horseshoe
$K_{\unicode{x3bb} }(p)\subset \Lambda _{\unicode{x3bb} }$
in the homoclinic class of the
$2$
-periodic point p.
Finally, as a consequence of [Reference MatherMat82], we emphasize that the conclusion of point (1) in Corollary G also holds for any convex domain
$\Omega $
whose boundary is
$C^2$
and contains some point at which the curvature vanishes. In this case (see Corollary 6.16), for any
$\epsilon>0$
, there exists
$\unicode{x3bb} _0=\unicode{x3bb} _0(\Omega ,\epsilon )\in (0,1)$
such that for any
$\unicode{x3bb} \in [\unicode{x3bb} _0,1)$
, the corresponding Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
has
$\rho ^+_{\unicode{x3bb} }-\rho ^-_{\unicode{x3bb} }\in (1-\epsilon ,1)$
.
From the results presented above, it is possible to highlight a phase transition for Birkhof attractors of dissipative billiards when the parameter
$\unicode{x3bb} $
varies. We would like to emphasize how the topological and dynamical properties of the Birkhoff attractor change in terms of the dissipative parameter. From Corollaries C and G, we obtain the following conclusion.
Corollary H. Let
$\mathcal {E}$
be an ellipse of eccentricity
$e\in (0,1)$
. Fix
$k \ge 3$
. There exists an open and dense set
$\mathscr {G}$
of
$C^k$
domains such that the following holds. For any
${0<\unicode{x3bb} _1<\unicode{x3bb} _2 <1}$
, there exists
$\delta>0$
so that if
$\Omega \in \mathscr {G}$
and
$d_{C^2}(\partial \Omega ,\mathcal {E})<\delta $
, then:
-
(1) there are
$2$
-periodic orbits
$\{H,f_{\unicode{x3bb} }(H)\}$
and
$\{E,f_{\unicode{x3bb} }(E)\}$
of saddle and sink type, respectively; -
(2) for any
$\unicode{x3bb} \in [\unicode{x3bb} _1,\unicode{x3bb} _2]$
, (1.7)In particular,
$$ \begin{align}\Lambda_{\unicode{x3bb}}=\overline{\mathcal{W}^u(\mathcal{O}_{\unicode{x3bb}}(H))}=\mathcal{W}^u(\mathcal{O}_{\unicode{x3bb}}(H)) \cup \{E,f_{\unicode{x3bb}}(E)\}. \end{align} $$
$\Lambda _{\unicode{x3bb} }$
has rotation number
$\frac 12$
;
-
(3) there exists
$\unicode{x3bb} _0 (\Omega )>\unicode{x3bb} _2$
such that, if
$\unicode{x3bb} \in [\unicode{x3bb} _0(\Omega ),1)$
, then
$\rho _{\unicode{x3bb} }^+-\rho ^-_{\unicode{x3bb} }>0$
, with
$\frac 12 \in (\rho _{\unicode{x3bb} }^-,\rho ^+_{\unicode{x3bb} })\,\mod \mathbb Z$
. In particular,
$\Lambda _{\unicode{x3bb} }$
is an indecomposable continuum that contains a horseshoe.
Proof. Let
$\unicode{x3bb} \in (0,1)\mapsto \epsilon (\mathcal {E},\unicode{x3bb} )>0$
be the continuous function given by Corollary C; let
Fix
$k \geq 2$
. Then, for any
$C^k$
domain
$\Omega $
with
$d_{C^2}(\partial \Omega ,\mathcal {E})<\delta $
, and for any
$\unicode{x3bb} \in [\unicode{x3bb} _1,\unicode{x3bb} _2]$
, we have
$d_{C^2}(\partial \Omega ,\mathcal {E})<\epsilon (\mathcal {E},\unicode{x3bb} )$
, and hence equation (1.7) holds for
$2$
-periodic orbits
$\{H,f_{\unicode{x3bb} }(H)\}$
and
$\{E,f_{\unicode{x3bb} }(E)\}$
of saddle and sink type, respectively. Now, by Corollary G, we obtain point (3) of the corollary if, moreover,
$\Omega $
is chosen within an open and dense set of
$C^k$
domains.
We refer to Figure 4. The phase transition described for perturbations of elliptic tables also holds for domains in
$\mathcal {D}^k$
,
$k \geq 3$
. The following corollary is a consequence of Theorem E and Corollary G.
Phase transition for a
$C^k$
-generic domain near an ellipse of non-zero eccentricity,
$k \geq 3$
.
Corollary I. For any
$k\geq 3$
, there exists an open and dense set
$\mathscr {U}$
of
$C^k$
domains such that for every
$\Omega \in \mathcal {D}^k\cap \mathscr {U}$
, the following holds. There exist
$0<\unicode{x3bb} "(\Omega )<\unicode{x3bb} _0(\Omega )<1$
such that:
-
(1) if
$\unicode{x3bb} \in (0,\unicode{x3bb} "(\Omega ))$
, then
$\Lambda _{\unicode{x3bb} }$
is equal to the attractor
$\Lambda _{\unicode{x3bb} }^0$
and it is a
$C^{k-1}$
normally contracted graph of rotation number
$\tfrac 12$
satisfying for some finite collection
$$ \begin{align*} \Lambda_{\unicode{x3bb}}=\bigcup_{i=1}^\ell \overline{\mathcal{W}^u(\mathcal{O}_{\unicode{x3bb}}(H_i))} \end{align*} $$
$\{\mathcal {O}_{\unicode{x3bb} }(H_i)\}_{i=1,\ldots ,\ell }=\{H_i,f_{\unicode{x3bb} }(H_i)\}_{i=1,\ldots ,\ell }$
of
$2$
-periodic orbits of saddle type;
-
(2) if
$\unicode{x3bb} \in [\unicode{x3bb} _0(\Omega ),1)$
, then
$\rho _{\unicode{x3bb} }^+-\rho ^-_{\unicode{x3bb} }>0$
, with
$\tfrac 12 \in (\rho _{\unicode{x3bb} }^-,\rho ^+_{\unicode{x3bb} })\,\mod \mathbb Z$
. In particular,
$\Lambda _{\unicode{x3bb} }$
is an indecomposable continuum that contains a horseshoe.
2. Dissipative maps and Birkhoff attractors
Let
${\mathbb T}:=\mathbb R/\mathbb Z$
and
${\mathbb A}:={\mathbb T}\times [-1,1]$
, with coordinates
$(s,r)\in {\mathbb A}$
. Endow
${\mathbb A}$
with the standard 2-form
$\omega =dr\wedge ds= d\alpha $
, where
$\alpha = r\,ds$
is the standard Liouville 1-form. The form
$\omega $
induces then an orientation on
${\mathbb A}$
and the Lebesgue measure denoted by m. For the following definition of a dissipative map, we refer to [Reference Le CalvezLC88, p. 245].
Definition 2.1. For two continuous maps
$\phi ^-,\phi ^+\colon {\mathbb T}\to \mathbb R$
such that
$\phi ^-<\phi ^+$
, let us denote
$$ \begin{align*}C := \{(s,r)\in{\mathbb A} :\phi^-(s)\leq r\leq \phi^+(s)\}.\end{align*} $$
A map
$f\colon C\to \mathrm {int}(C)$
is a dissipative map if:
-
(1) f is a homeomorphism of C into its image, homotopic to the identity;
-
(2) f is a
$C^1$
diffeomorphism of
$\mathrm {int}(C)$
into its image; -
(3) there exists
$\unicode{x3bb} \in (0,1)$
such that for any Borel set
$Y\subset C$
, it holds that
$m(f(Y))\leq \unicode{x3bb} m(Y)$
.
Observe that the following condition is equivalent to condition (3) in Definition 2.1:
-
(3’) there exists
$\unicode{x3bb} \in (0,1)$
such that for every
$(s,r)\in \mathrm {int}(C)$
, it holds that
$0<\det Df (s,r)\leq \unicode{x3bb} $
.
When considering dynamical systems with dissipation, it is natural to mention the notion of conformally symplectic maps, which is stated here in the more general framework of symplectic manifolds. See [Reference Arnaud and FéjozAF24, Reference Arnaud, Florio and RoosAFR24, Reference Marò and SorrentinoMS17].
Definition 2.2. Let
$(\mathcal {M},\omega )$
be a symplectic manifold. A diffeomorphism f from
$\mathcal {M}$
into its image (contained in
$\mathcal {M}$
) is conformally symplectic if there exists a smooth function
$a\colon \mathcal {M}\to \mathbb R$
such that
$f^*\omega =a\omega $
.
As shown by Libermann in [Reference LibermannLib59, p. 210], if the dimension of
$\mathcal {M}$
is greater than or equal to
$4$
, then the smooth function a is a constant, called the conformality ratio. In our case, that is, for
$\dim \mathcal {M}=2$
, the function a is not a priori constant. This motivates the next definition.
Definition 2.3. Let
$(\mathcal {M},\omega )$
be a symplectic manifold with
$\dim \mathcal {M}=2$
. A diffeomorphism f from
$\mathcal {M}$
into its image (contained in
$\mathcal {M}$
) is constant conformally symplectic if there exists a constant
$a>0$
such that
$f^*\omega =a\omega $
.
Observe that conformally symplectic maps are stable under symplectic changes of coordinates. This is not true anymore for constant conformally symplectic maps.
Let
$\mathcal {X}$
be the set of compact sets of
$\mathrm {int}(C)$
, endowed with the Hausdorff distance. We say that an element
$X \in \mathcal {X}$
separates C if its complement has two connected components: a lower one
$U_{X}{\kern-1pt}\supset{\kern-1pt} \{(s,\phi ^-(s)){\kern-1pt}\in{\kern-1pt} {\mathbb A} {\kern-1pt}:{\kern-1pt}s{\kern-1pt}\in{\kern-1pt} {\mathbb T}\}$
and an upper one
$V_X{\kern-1pt}\supset{\kern-1pt} \{(s,\phi ^+(s)){\kern-1pt}\in{\kern-1pt} {\mathbb A}{\kern-1pt}:{\kern-1pt}s{\kern-1pt}\in{\kern-1pt} {\mathbb T}\}$
. We denote by
$\mathcal {X}(f)\subset \mathcal {X}$
the subset of
$\mathcal {X}$
consisting of the sets which are compact, connected, f-invariant, and separate C.
Let f be a dissipative map according to Definition 2.1. We can define the Birkhoff attractor of f as follows, see [Reference BirkhoffBir32], [Reference Le CalvezLC88, §2], and [Reference Le CalvezLC90, Ch. 6] for an exhaustive treatment of the argument. First, observe that, by the dissipative character of f and as
$f(C)\subset \mathrm {int}(C)$
, there exists an attractor
$$ \begin{align} \Lambda^0:= \bigcap_{k \ge 0} f^k(C), \end{align} $$
which is f-invariant, non-empty, compact, and connected. Moreover, it separates C, that is,
$C\setminus \Lambda ^0$
is the disjoint union of two connected open sets
$U_{\Lambda ^0},V_{\Lambda ^0}$
as above. In other words,
$\Lambda ^0 \in \mathcal {X}(f)$
.
Definition 2.4. Let f be a dissipative map and let
$\Lambda ^0$
be its corresponding attractor, see equation (2.1). Let
$U_{\Lambda ^0},V_{\Lambda ^0}$
be the two connected components of
$C\setminus \Lambda ^0$
. Then, the Birkhoff attractor
$\Lambda $
is defined as
$$ \begin{align} \Lambda:= \overline{U}_{\Lambda^0}\cap \overline{V}_{\Lambda^0}. \end{align} $$
The Birkhoff attractor of a dissipative map can also be described as the minimal set, with respect to the inclusion, among the elements of
$\mathcal {X}(f)$
, as we will state immediately.
Proposition 2.5. [Reference Le CalvezLC90, Proposition 6.1]
-
(1) The set
$\mathcal {X}(f)$
contains a minimal element with respect to the inclusion, which is the Birkhoff attractor
$\Lambda $
for f. -
(2) If
$X \in \mathcal {X}(f)$
, then
$\Lambda =\overline {U}_X \cap \overline {V}_X$
; in particular,
$\Lambda = \mathrm {Fr}\, U_{\Lambda } = \mathrm {Fr} \, V_{\Lambda }$
.
From the properties of the Birkhoff attractor, we can deduce the following two lemmas.
Lemma 2.6. Let
$X\in \mathcal {X}(f)$
. Let
$x\in X$
be such that
$C\setminus (X\setminus \{x\})$
is connected. Then,
$x\in \Lambda $
.
Proof. By Proposition 2.5(1), it holds that
$\Lambda \subset X$
; in particular,
$U_X\subset U_{\Lambda }$
and
$V_X\subset V_{\Lambda }$
. By hypothesis,
$U_X\cup V_X\cup \{x\}$
is connected. Assume by contradiction that
$x\in X\setminus \Lambda \subset C\setminus \Lambda =U_{\Lambda }\cup V_{\Lambda }$
. Without loss of generality, we can assume that
$x\in U_{\Lambda }$
. Therefore,
$$ \begin{align*} U_X\cup\{x\}\cup V_X\subset U_{\Lambda}\cup V_{\Lambda}. \end{align*} $$
The set on the left-hand side is connected, while the set on the right-hand side is the disjoint union of two open sets. We then conclude that the left term is contained in one of the two open sets. Without loss of generality, we suppose that
$U_X\cup \{x\}\cup V_X\subset U_{\Lambda }$
. In particular,
$V_X\subset U_{\Lambda }\cap V_{\Lambda }=\emptyset $
, which is the required contradiction.
Lemma 2.7. Let
$X\subset \mathrm {int}({\mathbb A})$
be f-invariant and such that
$\Lambda \subset X$
. Then, X separates the annulus.
Proof. Observe that, since
$X\subset \mathrm {int}({\mathbb A})$
, clearly both
${\mathbb T}\times \{1\}$
and
${\mathbb T}\times \{-1\}$
are contained in
${\mathbb A}\setminus X$
. By contradiction, assume that X does not separate the annulus, that is,
${\mathbb A}\setminus X$
is connected. Indeed, it cannot disconnect
${\mathbb A}$
into more than two connected components, by the dissipative character of f. Moreover, since
$\Lambda \subset X$
, it holds that
${\mathbb A}\setminus X$
is contained in either
$U_{\Lambda }$
or
$V_{\Lambda }$
, the two connected components of
${\mathbb A}\setminus \Lambda $
. Without loss of generality, assume that
${\mathbb A}\setminus X\subset U_{\Lambda }$
. Thus, since
${\mathbb T}\times \{1\}\subset {\mathbb A}\setminus X$
, we have
${{\mathbb T}\times \{1\}\subset U_{\Lambda }\cap V_{\Lambda }=\emptyset} $
. This provides the required contradiction and concludes the proof.
Given a dissipative map
$f \colon C \to \mathrm {int} (C)$
and a hyperbolic periodic point
$p\in C$
of f of period
$q \geq 1$
, we will denote by
$\mathcal {W}^s(p;f^q)$
its stable manifold and by
$\mathcal {W}^u(p;f^q)$
its unstable manifold for the iterate
$f^q$
. Note that p cannot be a source by the dissipative character of f. We will also sometimes abbreviate
$\mathcal {W}^*(p; f^q)$
simply as
$\mathcal {W}^*(p)$
when the information about the map and the period are clear from the context. Observe that, for any hyperbolic point in the attractor
$\Lambda ^0$
, its unstable manifold is also contained in
$\Lambda ^0$
. Similarly, the next proposition guarantees that for any periodic point of saddle type in the Birkhoff attractor
$\Lambda $
, certain branches of its stable/unstable manifold have to belong to the Birkhoff attractor. See also [Reference Le CalvezLC88, §14.3] for related results.
Proposition 2.8. Let
$f\colon C \to \mathrm {int} (C)$
be a dissipative map. Assume that
$p \in \Lambda $
is a hyperbolic periodic point of saddle type, with period
$q \geq 1$
. Then at least one the two branches of
$\mathcal {W}^u(p;f^q)\setminus \{p\}$
is contained in
$\Lambda $
, unless
$\Lambda $
locally coincides near p with the local stable manifold
$\mathcal {W}_{\mathrm {loc}}^s(p;f^q)$
of p; in the latter case,
$\mathcal {W}^s(p;f^q)\subset \Lambda $
. In particular, at least one of the following non-exclusive properties holds:
-
•
$\overline {\mathcal {W}^s(p;f^q)}\subset \Lambda $
; -
•
$\overline {\mathcal {W}^u(p;f^q)}\subset \Lambda $
; -
•
$\text {there exists }\,\delta>0$
such that
$\Lambda \cap B(p,\delta )\setminus \{p\}=\mathcal {B}^u \cup \mathcal {B}^s$
, where
$\mathcal {B}^*$
is a branch of
$\mathcal {W}_{\mathrm {loc}}^*(p;f^q)\setminus \{p\}$
,
$*=s,u$
.
Proof. Let
$p \in \Lambda $
be a hyperbolic periodic point of saddle type, with period
$q \geq 1$
. We denote by
$0<\mu _1<1<\mu _2$
the eigenvalues of
$Df^q(p)$
. By Hartman–Grobman theorem, the dynamics can be linearized near p. Let then U be an open neighborhood of p, and
$\Psi \colon U\to \mathbb R^2$
be a homeomorphism such that
$\Psi (p)=0$
and
$\Psi \circ f^q|_U= A\circ \Psi |_U$
, where
$$ \begin{align*} A=\begin{bmatrix} \mu_1 & 0 \\ 0 & \mu_2 \end{bmatrix},\quad 0<\mu_1<1<\mu_2. \end{align*} $$
Let us assume that
$\Lambda $
does not locally coincide with the local stable manifold of p, that is, for any neighborhood V of p, there exists a point in
$\Lambda \setminus \mathcal {W}_{\mathrm {loc}}^s(p;f^q)$
. Let
$\epsilon>0$
be such that the ball of radius
$\epsilon $
centered at the origin
$B(0,\epsilon )$
is contained in
$\Psi (U)$
. Then there exists
$n_0\in \mathbb N$
such that for every
$n\geq n_0$
, there exists a point
$q_n\in \Lambda \cap U$
such that:
-
•
$\Psi (q_n)=(x_n,y_n)\in B(0,\epsilon )$
; -
•
$|y_n|\in (({\epsilon}/ {2})\mu _2^{-n},\epsilon \mu _2^{-n})$
.
Observe that each
$q_n$
does not belong to the local stable manifold of p and that
${f^{qk}(q_n)\in U}$
for all
$0\leq k\leq n$
. Thus, we can consider the sequence
$(f^{qn}(q_n))_n$
of points in
$\Lambda $
. Denote
$\Psi \circ f^{qn}(q_n)=:(X_n,Y_n)$
; in particular, for every
$n \in \mathbb {N}$
, it holds that
${|X_n|<\epsilon \mu _1^n}$
and
$|Y_n|\in ({\epsilon}/ {2},\epsilon)$
. Therefore, up to passing to a subsequence, the sequence
$(f^{qn}(q_n))_n$
converges to a point
$Q\neq p$
belonging to the local unstable manifold of p. Since
$\Lambda $
is closed and invariant, we conclude that
$$ \begin{align} f^{qn}(Q)\in(\Lambda\cap \mathcal{W}^u(p;f^q))\setminus \{p\} \quad \text{for all } n \in \mathbb{Z}. \end{align} $$
Let us now prove that the branch
$\mathcal {B}^u(p,Q)$
of
$\mathcal {W}^u_{\mathrm {loc}}(p;f^q)\setminus \{p\}$
containing Q is contained in
$\Lambda $
, and therefore, by the invariance of
$\Lambda $
, that the whole branch of the unstable manifold of p also does. Assume this is not the case: since
$\Lambda $
is closed and since, by equation (2.3), in any neighborhood of p, there exists a point in
$\Lambda \cap \mathcal {W}^u_{\mathrm {loc}}(p;f^q)$
, for every neighborhood
$V\subset U$
of p, we can find two points
$q_1=\Psi ^{-1}(0,a_1),q_2=\Psi ^{-1}(0,a_2)\in (\Lambda \cap U)\setminus \{p\}$
belonging to the local unstable manifold of p such that
$$ \begin{align*} \Gamma_0:=\Psi^{-1}(\{ 0 \}\times (a_1,a_2))\subset V,\quad \text{and}\quad \Gamma_0\cap\Lambda=\emptyset. \end{align*} $$
Let
$\mathcal {U}$
be a bounded neighborhood of
$\Lambda $
such that
$f^{-1}|_{\mathcal {U}}$
is a diffeomorphism onto its image. Up to considering V small enough, we have that
$$ \begin{align*} \bigcup_{n \in \mathbb{N}}f^{-qn}(\Lambda\cup \Gamma_0)\subset \mathcal{U}. \end{align*} $$
Let
$O_0\subset \mathcal {U}$
be a bounded open set among the connected components of
$C\setminus (\Lambda \cup \Gamma _0)$
, with
$\Gamma _0 \subset \partial O_0$
. Then, the family
$(f^{-qn}(O_0))_{n \in \mathbb {N}}$
is uniformly bounded in measure, which contradicts the dissipative character of f.
With the notation of Proposition 2.8, we can show that, in some cases,
$\Lambda $
contains both the stable manifold
$\mathcal {W}^s(p;f^q)$
and the unstable manifold
$\mathcal {W}^u(p;f^q)$
of the point p. In the following, for
$*=s,u$
, we denote
$\mathcal {W}^*(\mathcal {O}_f(p))=\bigcup _{k=0}^{q-1}\mathcal {W}^*(f^k(p);f^q)$
.
Lemma 2.9. Let p be a periodic hyperbolic point belonging to
$\Lambda $
. Let us assume that one connected component
$\mathcal {B}$
of
$\mathcal {W}^s(p;f^q)\setminus \{p\}$
belongs to
$\Lambda $
, and that
$\mathcal {B}$
intersects
$\mathcal {W}^u(p;f^q)$
transversally. Then,
$$ \begin{align*}\overline{\mathcal{W}^u(\mathcal{O}_f(p))\cup\mathcal{W}^s(\mathcal{O}_f(p))}\subset\Lambda. \end{align*} $$
Proof. Let z be a point of transverse intersection between
$\mathcal {B}$
and
$\mathcal {W}^u(p;f^q)$
. The standard
$\unicode{x3bb} $
-lemma (see e.g. [Reference Palis and de MeloPdM82, Ch. 2.7]) guarantees that there exists a one-dimensional disk
$D\subset \mathcal {B}$
whose past iterates
$(f^{-qn}(D))_{n \geq 0}$
accumulate
$\mathcal {W}^s(p;f^q)$
. As
$f^{-qn}(D)\subset \Lambda $
for every
$n \geq 0$
and since
$\Lambda $
is closed, we deduce that
$\overline {\mathcal {W}^s(p;f^q)}\subset \Lambda $
. Let us denote by
$\mathcal {B}'$
the branch of
$\mathcal {W}^u(p;f^q)\setminus \{p\}$
containing z. By invariance of
$\Lambda $
, all the points
$(f^{-qn}(z))_{n \geq 0}$
belong to
$\mathcal {W}^u(p;f^q)\cap \Lambda $
; in particular, for any
$\delta>0$
,
$\Lambda \cap B(p,\delta )$
contains points of the branch
$\mathcal {B}'\subset \mathcal {W}^u(p;f^q)\setminus \{p\}$
. Thus, arguing as in the proof of Proposition 2.8, we deduce that
$\mathcal {B}'\subset \Lambda $
. Another application of the
$\unicode{x3bb} $
-lemma (now for a small disk
$D'\subset \mathcal {B}'$
) gives that, in fact,
$\overline {\mathcal {W}^u(p;f^q)}\subset \Lambda $
. By f-invariance of
$\Lambda $
, we conclude that
$\overline {\mathcal {W}^u(\mathcal {O}_f(p))\cup \mathcal {W}^s(\mathcal {O}_f(p))}\subset \Lambda $
.
Corollary 2.10. Let p be a periodic hyperbolic point belonging to
$\Lambda $
. Let us assume that for each pair of branches
$\mathcal {B}^u\subset \mathcal {W}^u(p;f^q)\setminus \{p\}$
and
$\mathcal {B}^s\subset \mathcal {W}^s(p;f^q)\setminus \{p\}$
,
$\mathcal {B}^u$
and
$\mathcal {B}^s$
intersect transversally. Then,
$\Lambda $
contains a horseshoe
$K(p)$
; more precisely, it holds that
$$ \begin{align*} K(p)\subset H(p):=\overline{\mathcal{W}^s(\mathcal{O}_f(p))\pitchfork \mathcal{W}^u(\mathcal{O}_f(p))}\subset \overline{ \mathcal{W}^u(\mathcal{O}_f(p))}\subset \Lambda. \end{align*} $$
Proof. The fact that
$\mathcal {W}^u(p;f^q)\setminus \{p\}$
and
$\mathcal {W}^s(p;f^q)\setminus \{p\}$
intersect transversally guarantees the existence of a horseshoe
$K(p)$
which is contained in the homoclinic class
$H(p):=\overline {\mathcal {W}^s(\mathcal {O}_f(p))\pitchfork \mathcal {W}^u(\mathcal {O}_f(p))}$
of p (see [Reference SmaleSma65, Reference SmaleSma67] and [Reference NewhouseNew72, §2]). If
$\Lambda $
contains one branch
$\mathcal {B}^s\subset \mathcal {W}^s(p;f^q)\setminus \{p\}$
, then by Lemma 2.9,
$\overline {\mathcal {W}^s(\mathcal {O}_f(p))\cup \mathcal {W}^u(\mathcal {O}_f(p))}\subset \Lambda $
; otherwise, by Proposition 2.8, it holds that
$\overline {\mathcal {W}^u(p;f^q)}\subset \Lambda $
. In either case,
$K(p)\subset H(p)\subset \overline { \mathcal {W}^u(\mathcal {O}_f(p))}\subset \Lambda $
.
3. General facts about dissipative billiards
3.1. Dissipative billiard map
Examples of dissipative maps are given by dissipative billiard maps within convex domains of the plane. Let
$\Omega $
be a convex domain of the plane
$\mathbb R^2$
with
$C^k$
boundary,
$k \geq 2$
. Let
$\Upsilon \colon {\mathbb T}\to \mathbb R^2$
be an arclength parameterization of
$\partial \Omega $
. We denote by
$f_1$
the conservative billiard map associated to
$\Omega $
. Now, as in Definition A, we take a
$C^{k-1}$
function
$\unicode{x3bb} \colon {\mathbb A}\to (0,1)$
satisfying the condition in equation (1.2). As in Definition A, the dissipative billiard map within
$\Omega $
associated to the dissipation function
$\unicode{x3bb} $
is then given by
$$ \begin{align*} f_{\unicode{x3bb}}\colon \begin{cases} \mathbb{A} \ \to \mathbb{A},\\ (s,r) \ \mapsto f_{\unicode{x3bb}}(s,r)=(s',r'), \end{cases} \end{align*} $$
where
$\Upsilon (s')$
is the point where the half-line, starting at
$\Upsilon (s)$
and making an angle
$\varphi =\arcsin r$
with the normal, hits the boundary
$\partial \Omega $
again, and
$r'= \unicode{x3bb} (s',r_1^{\prime }) r_1^{\prime }$
,
$r_1^{\prime }$
being the sine of the outgoing angle of reflection in the case of an elastic collision. Letting
$\mathcal {H}_{\unicode{x3bb} } \colon {\mathbb A}\ni (s,r)\mapsto (s,\unicode{x3bb} (s,r) r)\in {\mathbb A}$
, we observe that
$$ \begin{align} f_{\unicode{x3bb}}=\mathcal{H}_{\unicode{x3bb}}\circ f_1. \end{align} $$
Here are some basic properties of the dissipative billiard map.
-
(1) The equality
$r' = r_1^{\prime }$
happens if and only if
$r' = r_1^{\prime }= 0$
, that is, the bounce at
$x'$
is perpendicular. -
(2) We recall that the standard billiard map
$f_1$
preserves the area form
$\omega = d r \wedge ds = d\alpha $
, where
$\alpha $
denotes the 1-form
$r\,ds$
. Thus, by equation (3.1),
$f_{\unicode{x3bb} }$
is a conformally symplectic map; indeed, In particular, if
$$ \begin{align*} f_{\unicode{x3bb}}^*\omega=(\partial_r \unicode{x3bb}(s',r_1^{\prime})r_1^{\prime}+\unicode{x3bb}(s',r_1^{\prime}))\omega. \end{align*} $$
$\unicode{x3bb} $
is a constant function, then
$f_{\unicode{x3bb} }$
is a conformally symplectic map with constant factor
$\unicode{x3bb} $
, as
$f_{\unicode{x3bb} }^*\omega = f_1^*(\mathcal {H}_{\unicode{x3bb} }^*\omega ) = \unicode{x3bb} \omega $
.
-
(3) The map
$f_{\unicode{x3bb} }$
is a dissipative map of
${\mathbb A}$
, according to Definition 2.1. In particular,
$f_{\unicode{x3bb} }$
verifies:-
(i) for any
$(s,r) \in \mathrm {int}(\mathbb {A})$
, it holds that (3.2)in particular, if
$$ \begin{align} 0 < \det Df_{\unicode{x3bb}}(s,r) =\partial_r \unicode{x3bb}(s',r_1^{\prime})r_1^{\prime}+\unicode{x3bb}(s',r_1^{\prime}) < 1\,, \end{align} $$
$\unicode{x3bb} $
is constant, then for any
$(s,r) \in \mathrm {int}(\mathbb {A})$
, it holds that
$0 < \det Df_{\unicode{x3bb} }(s,r) =\unicode{x3bb} < 1$
;
-
(ii) it holds that
$f_{\unicode{x3bb} }(\mathbb {T} \times [-1,1])\subset \mathbb {T} \times (-1,1)$
, in particular, if
$\unicode{x3bb} $
is constant, then (3.3)
$$ \begin{align} f_{\unicode{x3bb}}(\mathbb{T} \times [-1,1]) = \mathbb{T}\times[-\unicode{x3bb},\unicode{x3bb}]\subset \mathbb{T} \times (-1,1). \end{align} $$
Moreover, we will see in §6 that the map
$f_{\unicode{x3bb} }$
is a positive twist map. -
-
(4) Again from equation (3.1), for every
$(s,r)\in \mathrm {int}({\mathbb A})$
and
$(s',r'):=f_{\unicode{x3bb} }(s,r)$
, we have (3.4)where
$$ \begin{align} &Df_{\unicode{x3bb}}(s,r) =D\mathcal{H}_{\unicode{x3bb}}(f_1(s,r))\, Df_1(s,r) \nonumber\\\kern1pt&= \begin{bmatrix} 1 & 0\\ \partial_s \unicode{x3bb}(s',r_1^{\prime})r_1^{\prime} & \partial_r \unicode{x3bb}(s',r_1^{\prime})r_1^{\prime}+\unicode{x3bb}(s',r_1^{\prime}) \end{bmatrix} \begin{bmatrix} -\dfrac{\tau \mathcal{K} + \nu}{\nu'} & \dfrac{\tau}{\nu \nu'} \\ \\ \tau \mathcal{K} \mathcal{K}' + \mathcal{K} \nu'+ \mathcal{K}' \nu & -\dfrac{\tau \mathcal{K}'+ \nu'}{\nu} \end{bmatrix}\!, \end{align} $$
$\tau =\ell (s,s'):=\|\Upsilon (s)-\Upsilon (s')\|$
is the Euclidean distance between the points
$x=\Upsilon (s)$
,
$x'=\Upsilon (s')$
,
$\mathcal {K}$
,
$\mathcal {K}'$
denote the curvatures at
$\Upsilon (s),\Upsilon (s')$
, respectively, and
$\nu = \sqrt {1 - r^2}$
,
$\nu ' = \sqrt {1 - (r_1^{\prime })^2}=\sqrt {1 - ({r'}/{\unicode{x3bb} })^2}$
. Equation (3.4) can be deduced from [Reference Chernov and MarkarianCM06, §2.11], by applying the change of coordinates
$(s,\varphi )\mapsto (s,r=\sin \varphi )$
. If the dissipation is constant, equal to
$\unicode{x3bb} \in (0,1)$
, then with the same notation as above, for
$(s,r)\in \mathrm {int}({\mathbb A})$
,
$$ \begin{align*} Df_{\unicode{x3bb}}(s,r) = \begin{bmatrix} -\dfrac{\tau \mathcal{K} + \nu}{\nu'} & \dfrac{\tau}{\nu \nu'} \\ \\ \unicode{x3bb}(\tau \mathcal{K} \mathcal{K}' + \mathcal{K} \nu'+ \mathcal{K}' \nu ) & -\unicode{x3bb} \dfrac{\tau \mathcal{K}'+ \nu'}{\nu} \end{bmatrix}\!. \end{align*} $$
Notation 3.1. In the following, while considering the dissipative billiard map
$f_{\unicode{x3bb} }$
, we denote its attractor by
$\Lambda _{\unicode{x3bb} }^0$
and its Birkhoff attractor by
$\Lambda _{\unicode{x3bb} }$
.
3.2. Properties of the Birkhoff attractor for axially symmetric billiards
This subsection is devoted to proving some properties of the Birkhoff attractor for a dissipative billiard map under some symmetric assumptions on the domain
$\Omega $
. In particular, they can be applied to the case of elliptic billiards considered in §4. In this subsection, we assume that the billiard map
$f_{\unicode{x3bb} }$
has constant dissipation
$\unicode{x3bb} \in (0,1)$
. In fact, this would be more generally as long as the dissipation function
$\unicode{x3bb} \colon {\mathbb A}\to (0,1)$
respects the symmetries of
$\Omega $
.
Definition 3.2. (Axially symmetric billiard table)
Let
$\Omega \subset \mathbb R^2$
be a strictly convex domain with
$C^2$
boundary. We say that
$\Omega $
is axially symmetric with respect to some line
$\Delta \subset \mathbb R^2$
if
$\Omega $
is invariant under the reflection along the line
$\Delta $
.
Lemma 3.3. Let
$\Omega \subset \mathbb R^2$
be a strictly convex domain with
$C^2$
boundary, with perimeter
$2L>0$
, which is axially symmetric with respect to some line
$\Delta $
. Let
$s_0$
and
$s_0+L$
be the arclength parameters of the points in
$\partial \Omega \cap \Delta $
. For
$\unicode{x3bb} \in (0,1)$
, let
$f_{\unicode{x3bb} }$
be the associated dissipative billiard map. Then:
-
(1) the pair
$\{(s_0,0),(s_0+L,0)\}$
corresponds to a
$2$
-periodic orbit; -
(2) let us denote by
$\mathcal {I}_{\Delta }$
the involution
$\mathcal {I}_{\Delta }\colon (s_0+s,\varphi )\mapsto (s_0-s,-\varphi )$
, then, (3.5)
$$ \begin{align} \mathcal{I}_{\Delta} \circ f_{\unicode{x3bb}} = f_{\unicode{x3bb}} \circ \mathcal{I}_{\Delta}. \end{align} $$
Proof. Item (1) follows by noticing that the osculating circles at
$s_0$
and
$s_0+L$
are invariant under the reflection through the axis
$\Delta $
, and hence the line segment connecting the points
$s_0,s_0+L$
(which is collinear to
$\Delta $
) is perpendicular to the boundary of
$\Omega $
at the points
$s_0,s_0+L$
. The second point is immediate once we have observed that
$f_{\unicode{x3bb} }=\mathcal {H}_{\unicode{x3bb} }\circ f_1$
, as
$\mathcal {H}_{\unicode{x3bb} }$
commutes with
$\mathcal {I}_{\Delta }$
and, under the assumption of axial symmetry,
$f_1$
and
$\mathcal {I}_{\Delta }$
also commute.
Corollary 3.4. Let
$\Omega \subset \mathbb R^2$
be a strictly convex domain with
$C^2$
boundary with perimeter
$2L>0$
, axially symmetric with respect to some line
$\Delta $
. Let
$s_0$
and
$s_0+L$
be the arclength parameters of the points in
$\partial \Omega \cap \Delta $
. For
$\unicode{x3bb} \in (0,1)$
, let
$f_{\unicode{x3bb} }$
be the associated dissipative billiard map and
$\Lambda _{\unicode{x3bb} }$
the corresponding Birkhoff attractor. Then:
-
(1)
$\mathcal {I}_{\Delta }(\Lambda _{\unicode{x3bb} })=\Lambda _{\unicode{x3bb} }$
; -
(2)
$\{(s_0,0),(s_0+L,0)\}\subset \Lambda _{\unicode{x3bb} }$
.
Proof. Item (1) follows from the fact that
$\Lambda _{\unicode{x3bb} }$
is the smallest, with respect to inclusion, compact connected
$f_{\unicode{x3bb} }$
-invariant subset which separates the annulus. Indeed,
$\mathcal {I}_{\Delta }(\Lambda _{\unicode{x3bb} })$
is compact, connected; it is
$f_{\unicode{x3bb} }$
-invariant, by the fact that
$\Lambda _{\unicode{x3bb} }$
is
$f_{\unicode{x3bb} }$
-invariant and by equation (3.5); moreover,
$\mathcal {I}_{\Delta }(\Lambda _{\unicode{x3bb} })$
separates the annulus. Thus,
$\Lambda _{\unicode{x3bb} }\subset \mathcal {I}_{\Delta }(\Lambda _{\unicode{x3bb} })$
. We conclude because
$\mathcal {I}^2_{\Delta }=\mathrm {Id}$
and so
$\mathcal {I}_{\Delta }(\Lambda _{\unicode{x3bb} })\subset \mathcal {I}^2_{\Delta }(\Lambda _{\unicode{x3bb} })=\Lambda _{\unicode{x3bb} }$
.
To show point (2), let us argue by contradiction, assuming that
$(s_0,0)\notin \Lambda _{\unicode{x3bb} }$
. By compactness of
$\Lambda _{\unicode{x3bb} }$
, there exists a connected open neighborhood U of
$(s_0,0)$
that is disjoint from
$\Lambda _{\unicode{x3bb} }$
. Since
$\mathcal {I}_{\Delta }(s_0,0)=(s_0,0)$
, the set
$U':=U\cap \mathcal {I}_{\Delta }(U)$
is a connected open neighborhood of
$(s_0,0)$
that is
$\mathcal {I}_{\Delta }$
-invariant and disjoint from
$\Lambda _{\unicode{x3bb} }$
. Recall that
$\Lambda _{\unicode{x3bb} }$
separates the annulus
$\mathbb {A}$
, that is,
$\mathbb {A}\setminus \Lambda _{\unicode{x3bb} }$
is the disjoint union of two open connected components, denoted by
$U_{\Lambda _{\unicode{x3bb} }}$
and
$ V_{\Lambda _{\unicode{x3bb} }}$
. Since
$\mathcal {I}_{\Delta }$
maps the top boundary
${\mathbb T}\times \{1\}$
to the bottom boundary
${\mathbb T}\times \{-1\}$
and vice versa, we have
$\mathcal {I}_{\Delta }(U_{\Lambda _{\unicode{x3bb} }})=V_{\Lambda _{\unicode{x3bb} }}$
. As
$U'\cap \Lambda _{\unicode{x3bb} }=\emptyset $
, we can assume without loss of generality that
$U' {\kern-1pt}\subset{\kern-1pt} U_{\Lambda _{\unicode{x3bb} }}$
. However, then,
$U'{\kern-1pt}={\kern-1pt}\mathcal {I}_{\Delta }(U'){\kern-1pt}\subset{\kern-1pt} \mathcal {I}_{\Delta }(U_{\Lambda _{\unicode{x3bb} }}){\kern-1pt}={\kern-1pt} V_{\Lambda _{\unicode{x3bb} }}$
, which is a contradiction. Thus,
$(s_0,0)\in \Lambda _{\unicode{x3bb} }$
and
$(s_0+L,0)=f_{\unicode{x3bb} }(s_0,0)\in \Lambda _{\unicode{x3bb} }$
.
3.3. Bifurcations of
$2$
-periodic points
As we are going to see,
$2$
-periodic points play a special role for dissipative billiards; this is partly due to the fact that by point (1) in §3.1, the usual reflection law and the dissipative one have the same effect at an orthogonal collision. In the previous section, we already saw that for convex billiards with symmetries, symmetric
$2$
-periodic orbits have to belong to the Birkhoff attractor. Here we investigate the eigenvalues of
$2$
-periodic points and their bifurcations as the dissipation parameter changes. In fact, as we will see here, although the set of
$2$
-periodic orbits is independent of the value of the dissipation
$\unicode{x3bb} \colon {\mathbb A}\to (0,1)$
, their type will depend on the strength of the perturbation. Throughout the rest of §3.3, except in Lemma 3.7 and in the last point of Corollary 3.9, we will assume for simplicity that the dissipation is constant, equal to some
$\unicode{x3bb} \in (0,1)$
. Yet, the result of Lemma 3.5 could be fully adapted to the case of a non-constant dissipation
$\unicode{x3bb} $
, but the proof would be even more computational; the version we give in Lemma 3.7 is a little less precise but suffices for our purpose. The proofs of the main technical lemma of this subsection, namely Lemmas 3.5 and 3.7, are mainly computational: for this reason, we postpone them to Appendix A.
Let us denote by
$\mathrm {II}$
the set of
$2$
-periodic points for
$\{f_{\unicode{x3bb} }\}_{\unicode{x3bb} \in [0,1]}$
. In the following, for
$p=(s,0) \in \mathrm {II}$
, let
$(s',0):=f_{\unicode{x3bb} }(p)$
(the point
$(s',0)$
is independent of the value of
$\unicode{x3bb} $
as observed above), and denote by
$\tau =\ell (s,s'):=\|\Upsilon (s)-\Upsilon (s')\|$
the Euclidean distance between the points
$\Upsilon (s),\Upsilon (s')$
. We also denote by
$\mathcal {K}_1$
,
$\mathcal {K}_2$
the respective curvatures at
$\Upsilon (s),\Upsilon (s')$
and let
$$ \begin{align} k_{1,2}=k_{1,2}(p):=(\tau \mathcal{K}_1 + 1)(\tau \mathcal{K}_2 + 1). \end{align} $$
Lemma 3.5. Let
$p\in \mathrm {II}$
and let
$\tau ,\mathcal {K}_1,\mathcal {K}_2,k_{1,2}$
be as above. Fix
$\unicode{x3bb} \in (0,1)$
and denote by
$\{\mu _1=\mu _1(\unicode{x3bb} ),\mu _2=\mu _2(\unicode{x3bb} )\}$
the eigenvalues of
$Df_{\unicode{x3bb} }^2(p)$
, with
$|\mu _1|\leq |\mu _2|$
.
-
(a) If
$k_{1,2}>1$
, then
$0<\mu _1<\unicode{x3bb} ^2<1<\mu _2$
and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle. -
(b) If
$k_{1,2}=1$
, then
$\mu _1=\unicode{x3bb} ^2$
,
$\mu _2=1$
, and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is parabolic. -
(c) If
$k_{1,2}\in (0,1)$
, then the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink; moreover, let (3.7)The following holds:-
(i) if
$\unicode{x3bb} \in (0,\unicode{x3bb} _-)$
, then
$\mu _1,\mu _2$
are real, with
$\unicode{x3bb} ^2<\mu _1< \mu _2 <1$
; -
(ii) if
$\unicode{x3bb} =\unicode{x3bb} _-$
, then
$\mu _1=\mu _2=\unicode{x3bb} \in (0,1)$
; -
(iii) if
$\unicode{x3bb} \in (\unicode{x3bb} _-,1)$
, then
$\mu _1,\mu _2$
are complex conjugate of modulus
$\unicode{x3bb} $
.
-
-
(d) If
$k_{1,2}=0$
, then the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink, with
$\mu _1=\mu _2=-\unicode{x3bb} $
, and
$Df_{\unicode{x3bb} }^2(p)=-\unicode{x3bb} \, \mathrm {id}$
. -
(e) If
$k_{1,2}\in (-1,0)$
, let The following holds:-
(i) if
, then
$-1<\mu _2<-\unicode{x3bb} <\mu _1<0$
and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink; -
(ii) if
, then
$\mu _1=-\unicode{x3bb} ^2$
,
$\mu _2=-1$
, and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is parabolic; -
(iii) if
,
$\mu _2<-1<-\unicode{x3bb} ^2<\mu _1<0$
and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle.
-
-
(f) If
$k_{1,2}\leq -1$
, then
$\mu _2<-1<-\unicode{x3bb} ^2<\mu _1<0$
and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle.
Moreover, for
$\unicode{x3bb} =0$
, the eigenvalues of
$Df_0^2(p)$
are
$\mu _1=0$
and
$\mu _2=k_{1,2}$
, and the respective eigenspaces are vertical and horizontal.
Proof. See Appendix A.
Remark 3.6. We use the same notation as above. In the special case of a point
$p \in \mathrm {II}$
with
$\mathcal {K}_1=\mathcal {K}_2=:\mathcal {K}\leq 0$
, we have
$k_{1,2}=(1+\tau \mathcal {K})^2\geq 0$
and
$\tau \mathcal {K}\leq 0$
, and hence the previous result gives the following outcome.
-
(a) If
$\tau \mathcal {K}<-2$
, then the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle. -
(b) If
$\tau \mathcal {K}\in \{-2,0\}$
, then
$\mu _1=\unicode{x3bb} ^2$
,
$\mu _2=1$
and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is parabolic. -
(c) If
$-2<\tau \mathcal {K}< 0$
, then the 2-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink; for
$\tau \mathcal {K}=-1$
, it holds that
$Df_{\unicode{x3bb} }^2(p)=-\unicode{x3bb} \,\mathrm {id}$
.
In the case of dissipative billiard maps associated to a non-constant dissipation
$\unicode{x3bb} \in (0,1)$
, we also have the following lemma.
Lemma 3.7. Let
$\unicode{x3bb} \colon {\mathbb A}\to (0,1)$
be a
$C^{k-1}$
function such that
$f_{\unicode{x3bb} }:=\mathcal {H}_{\unicode{x3bb} } \circ f$
is a dissipative billiard map in the sense of Definition 1.2, where
$\mathcal {H}_{\unicode{x3bb} }\colon (s,r)\mapsto (s,\unicode{x3bb} (s,r) r)$
. In particular,
$f_{\unicode{x3bb} }$
has the same set
$\mathrm {II}$
of
$2$
-periodic points as f. Fix a
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
and assume that
$k_{1,2}\geq 0$
, with
$k_{1,2}=k_{1,2}(p):=(\tau \mathcal {K}_1 + 1)(\tau \mathcal {K}_2 + 1)$
as in equation (3.6). Then
$\{p,f_{\unicode{x3bb} }(p)\}$
is parabolic for
$f_{\unicode{x3bb} }$
if and only if it is for the conservative billiard map
$f=f_1$
, and this happens if and only if
$k_{1,2}=1$
.
Otherwise, the orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is either a sink or a saddle for
$f_{\unicode{x3bb} }$
; more precisely, it is a saddle if and only if
$k_{1,2}>1$
, and it is a sink if and only if
$k_{1,2}<1$
.
Proof. See Appendix A.
In the next statement, we summarize some results obtained by Dias Carneiro, Oliffson Kamphorst, and Pinto-de-Carvalho, see [Reference Dias Carneiro, Oliffson Kamphorst and Pinto-de CarvalhoDCOKPdC07, Theorem 1] and [Reference Dias Carneiro, Oliffson Kamphorst and Pinto-de CarvalhoDCOKPdC03], and Xia and Zhang [Reference Xia and ZhangXZ14, Theorem 1.1–Corollary 4.4].
Theorem 3.8. Fix
$k \geq 2$
.
-
(1) For every
$q\geq 2$
, there exists an open and dense set
$\mathscr {U}^q$
of strongly convex domains with
$C^k$
boundary such that the number of q-periodic points (for the usual conservative billiard map) is finite; moreover, all the q-periodic points are either elliptic or hyperbolic. -
(2) There exists a
$G_{\delta }$
-dense set
$\mathcal {G}^k$
of strongly convex domains with
$C^k$
boundary such that, for each
$q \ge 2$
, the number of q-periodic points is finite. Moreover, all the periodic points are either hyperbolic or elliptic. -
(3) There exists an open and dense set
$\mathscr {U}$
of strongly convex domains with
$C^k$
boundary,
$k\geq 3$
, such that for each
$2$
-periodic point
$p \in \mathrm {II}$
of saddle type, each branch of
$\mathcal {W}^s(p;f^2)\setminus \{p\}$
and
$\mathcal {W}^u(p;f^2)\setminus \{p\}$
contains a transverse homoclinic point.
Corollary 3.9. Let
$k \geq 3$
. There exists an open and dense set
$\mathscr {U}$
of strongly convex domains with
$C^k$
boundary such that, for every
$\Omega \in \mathscr {U}$
, the following assertions hold.
If
$f_{\unicode{x3bb} }$
is a dissipative billiard maps with constant dissipation
$\unicode{x3bb} \in (0,1)$
, then:
-
(1) for any
$\unicode{x3bb} \in [0,1]$
, the set
$\mathrm {II}$
of
$2$
-periodic points of
$f_{\unicode{x3bb} }$
is finite; -
(2) for all but at most finitely many
$\unicode{x3bb} \in (0,1)$
, all the
$2$
-periodic points are non-degenerate, that is, they are either saddles or sinks; -
(3) there exists
$\unicode{x3bb} _*(\Omega )\in (0,1)$
such that for any
$\unicode{x3bb} \in [\unicode{x3bb} _*(\Omega ),1)$
and for any point
$p \in \mathrm {II}$
of saddle type, each branch of
$\mathcal {W}^s(p;f_{\unicode{x3bb} }^2)\setminus \{p\}$
and
$\mathcal {W}^u(p;f_{\unicode{x3bb} }^2)\setminus \{p\}$
contains a transverse homoclinic point.
If, moreover,
$\Omega \in \mathcal {D}^k \cap \mathscr {U}$
, where
$\mathcal {D}^k$
is the subset of strongly convex domains with
$C^k$
boundary as in Definition D, then for a general dissipative billiard map
$f_{\unicode{x3bb} }$
in the sense of Definition A (with possibly non-constant dissipation), all the
$2$
-periodic points are either saddles or sinks. It is also true for the (degenerate) map
$f_0$
, namely, when the dissipation
$\unicode{x3bb} $
vanishes.
Proof. By the definition of the (dissipative) billiard law given in Definition A, given a convex domain, the set
$\mathrm {II}$
of
$2$
-periodic points is common to all the maps
$\{f_{\unicode{x3bb} }\}_{\unicode{x3bb} \in [0,1]}$
. By Theorem 3.8, we deduce that there exists an open and dense set
$\mathscr {U}$
of strongly convex
$C^k$
domains,
$k\geq 3$
, such that, for every
$\Omega \in \mathscr {U}$
and for any
$\unicode{x3bb} \in [0,1]$
, the billiard map
$f_{\unicode{x3bb} }$
has finitely many
$2$
-periodic points. Fix
$\Omega \in \mathscr {U}$
and denote by
$\mathrm {II}$
its finite set of
$2$
-periodic points. For any
$p \in \mathrm {II}$
, let
$\tau ,\mathcal {K}_1,\mathcal {K}_2$
be as in Lemma 3.5 and let
$k_{1,2}=k_{1,2}(p):= (\tau \mathcal {K}_1+1)(\tau \mathcal {K}_2+1)$
. For any
$\unicode{x3bb} \in (0,1)$
, the
$2$
-periodic
$f_{\unicode{x3bb} }$
-orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle or a sink, unless
$k_{1,2}=1$
(see Lemma 3.5(b)), or
$k_{1,2}\in (-1,0)$
and
(see Lemma 3.5(e)(ii)). On the one hand, let us examine the case where
$k_{1,2}=1$
for the
$2$
-periodic orbit
$\{p,f_1(p)\}$
of the conservative billiard map
$f_1$
. By equation (A.2), for
$\unicode{x3bb} =1$
,
$k_{1,2}=1$
if and only if
$\mathrm {tr} Df_1^2(p)=2$
, that is, the
$2$
-periodic
$f_1$
-orbit
$\{p,f_1(p)\}$
is parabolic, which does not occur for the domain
$\Omega $
, since it is in
$\mathscr {U}$
. On the other hand, again since
$\Omega \in \mathscr {U}$
,
$\mathrm {II}$
is finite, and hence so is the set
By the above discussion and by Lemma 3.5, we conclude that for any
$p\in \mathrm {II}$
and for any
$\unicode{x3bb} \in (0,1)\setminus \mathcal {F}$
, the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is non-degenerate, that is, it is either a saddle or a sink.
Point (3) follows immediately from Theorem 3.9; indeed, for the conservative billiard map
$f_1$
, there exist transverse homoclinic points on each of the branches of any
$2$
-periodic point of saddle type; the existence of transverse homoclinic points is stable under
$C^1$
-small perturbations of the dynamics, and hence the same property holds true for any
$f_{\unicode{x3bb} }$
with
$\unicode{x3bb} \in (0,1)$
close enough to
$1$
.
Finally, assume that
$\Omega $
belongs to the set
$\mathcal {D}^k\cap \mathscr {U}$
. Let
$f_{\unicode{x3bb} }$
be a general dissipative billiard map for
$\Omega $
in the sense of Definition 1.2. It has the same set
$\mathrm {II}$
of
$2$
-periodic points as
$f_1$
. By the condition in equation (1.6), for any
$p \in \mathrm {II}$
, we have
$k_{1,2}(p)> 0$
. Moreover,
$k_{1,2}(p)\neq 1$
, since
$\Omega \in \mathscr {U}$
, and hence by Lemma 3.7, the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is either a saddle or a sink. In addition, for
$\unicode{x3bb} =0$
, Lemma 3.5 says that the eigenvalues of p are
$\mu _1=0$
and
$\mu _2=k_{1,2}(p)$
; as
$k_{1,2}(p)>0$
and
$k_{1,2}(p)\neq 1$
, we deduce that the
$2$
-periodic
$\{p,f_0(p)\}$
of
$f_0$
is either a saddle or a sink.
Let us recall some classical definitions and results for conservative billiard maps. Let
$\Omega $
be a strongly convex domain with
$C^2$
boundary. Then, the billiard map expressed in coordinates
$(s,\varphi )\in {\mathbb T}\times [-{\pi }/{2},{\pi }/{2}]$
is a
$C^1$
diffeomorphism, see [Reference DouadyDou82, Proposition I.3.2] and [Reference Le CalvezLC90, p. 11]. Let
$F_1\colon \mathbb R\times [-1,1]\to \mathbb R\times [-1,1]$
be a lift of the conservative billiard map. Let
$\tilde \pi _1\colon \mathbb R\times [-1,1]\to \mathbb R$
be the projection onto the first coordinate. Then the
$F_1$
-orbit of a point
$(S,r)$
is completely determined by the bi-infinite sequence
$(S_i)_{i\in \mathbb Z}:=(\tilde \pi _1\circ F_1^i(S,r))_{i\in \mathbb Z}$
.
Definition 3.10. Let
$(S,r)\in \mathbb R\times (-1,1)$
. The rotation number of
$(S,r)$
is
$$ \begin{align*} \rho(S,r):=\lim_{n\to \infty}\dfrac{\tilde\pi_1\circ F_1^n(S,r)}{n} , \end{align*} $$
whenever the limit exists.
Observe that the rotation number depends on the chosen lift
$F_1$
. Up to the choice of the lift, that is, a lift such that
$F_1$
is the identity on the lower boundary
$\mathbb R\times \{-1\}$
, the rotation number of any point belongs to the interval
$[0,1]$
. Observe that, for such a lift, the rotation number of a periodic (conservative) billiard trajectory corresponds to
${\text {winding number}}/{\text {number of reflections}}$
.
Let
$\ell \colon \mathbb R^2\to \mathbb R$
be the generating function of the conservative billiard map. From the geometric point of view, the quantity
$\ell (S_i,S_{i+1})$
corresponds to the Euclidean distance on
$\mathbb R^2$
between
$\Upsilon (s_i)$
and
$\Upsilon (s_{i+1})$
, where
$\pi \colon \mathbb R\to {\mathbb T}$
is a covering and
$s_i=\pi (S_i)$
. This is also the quantity previously denoted as
$\tau (s_i,s_{i+1})$
. In the next proposition, with an abuse of notation, since
$\ell $
is invariant under the action of
$\mathbb Z$
, we also denote by
$\ell $
the function induced on
${\mathbb T}^2$
.
By a standard construction due to Birkhoff, see e.g. [Reference SiburgSib04, Theorem 1.2.4], it is well known that there exist at least two periodic orbits for every rational rotation number. They are obtained by considering the length functional, given by
$\sum _{i \in \mathbb {Z}} \ell (S_i,S_{i+1})$
. In particular, the first orbit is given by maximizing the functional, while the other one is given by a min-max procedure (sometimes referred to as the ‘Mountain Pass lemma’). In particular, for the rotation number
$\frac 1 2$
, we obtain two
$2$
-periodic orbits for the conservative billiard map. Then, as remarked at the beginning of the section, the dissipative billiard map
$f_{\unicode{x3bb} }$
has two 2-periodic orbits for any
$\unicode{x3bb} \in [0,1]$
. More precisely, for every
$\unicode{x3bb} \in (0,1)$
, the set of 2-periodic points is non-empty and it contains at least two different orbits.
Proposition 3.11. Let
$f_{\unicode{x3bb} }$
be the dissipative billiard map of a strongly convex domain
$\Omega $
with
$C^k$
boundary,
$k \geq 2$
, that belongs to the open and dense set
$\mathscr {U}$
of Theorem 3.8. Assume that
$\{p=(s_1,0),f_{\unicode{x3bb} }(p)=(s_2,0)\}$
is a
$2$
-periodic orbit. We denote by
$\mathcal {K}_1,\mathcal {K}_2<0$
the respective curvatures at the points
$\Upsilon (s_1)$
and
$\Upsilon (s_2)$
, where
$\Upsilon \colon {\mathbb T}\to \mathbb R^2$
is an arclength parameterization of the boundary. Let
$\tau :=\ell (s_1,s_2)$
and
$k_{1,2}:=(\tau \mathcal {K}_1 + 1)(\tau \mathcal {K}_2 + 1)$
. Denote by
$\{\mu _1,\mu _2\}$
the eigenvalues of
$Df_{\unicode{x3bb} }^2(p)$
, with
$|\mu _1|\leq |\mu _2|$
. Then:
-
(a) if
$(s_1,s_2)$
corresponds to a local maximum of
$\ell $
(e.g. when
$[\Upsilon (s_1),\Upsilon (s_2)]$
is a diameter), then
$k_{1,2}>1$
and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle; -
(b) if
$(s_1,s_2)$
corresponds to a critical point of saddle type of
$\ell $
, then
$k_{1,2}<1$
and it holds that:-
(i) if
$k_{1,2}\geq 0$
(note that it is always the case when
$\mathcal {K}_1=\mathcal {K}_2$
or when
$\Omega \in \mathcal {D}^k$
), then the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink; -
(ii) if
$k_{1,2}\in (-1,0)$
, let
; then, we have:-
• for any
, the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink; -
• for
, the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is parabolic; -
• for any
, the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle;
-
-
(iii) if
$k_{1,2}\leq -1$
, then for any
$\unicode{x3bb} \in (0,1)$
, the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle.
-
Proof. Fix a
$2$
-periodic orbit
$\{p=(s_1,0),f_{\unicode{x3bb} }(p)=(s_2,0)\}$
. Since
$\partial _1 \ell (s,s')=-\sin \varphi $
and
$\partial _2 \ell (s,s')=\sin \varphi '$
, the point
$(s_1,s_2)$
is a critical point of
$\ell $
. Moreover, we have (see e.g. [Reference Chen, Kaloshin and ZhangCKZ23, Lemma 2.1])
$$ \begin{align*} &\ell(s_1+\delta s,s_2+\delta s')-\ell(s_1,s_2)\\&\quad=\frac 12\begin{bmatrix} \delta s & \delta s' \end{bmatrix}\underbrace{\begin{bmatrix} \mathcal{K}_1+\dfrac{1}{\tau} & \dfrac{1}{\tau}\\ \dfrac{1}{\tau} & \mathcal{K}_2+\dfrac{1}{\tau} \end{bmatrix}}_{=:A}\begin{bmatrix} \delta s \\ \delta s' \end{bmatrix}+o((\delta s)^2+(\delta s')^2). \end{align*} $$
Let us distinguish between two cases, namely, when the pair
$(s_1,s_2)$
corresponds to a local maximum or a critical point of saddle type of the length functional.
-
(a) In the first case, when
$\ell $
is locally maximal at
$(s_1,s_2)$
, the Hessian matrix A of
$\ell $
is negative semi-definite, that is,
$\mathrm {tr}A=\mathcal {K}_1+\mathcal {K}_1+{2}/{\tau }\leq 0$
, and
$\det A={1}/{\tau ^2}(k_{1,2}-1)\geq 0$
with
$k_{1,2}:=(1+\tau \mathcal {K}_1)(1+\tau \mathcal {K}_2)$
. Since
$\mathcal {K}_1,\mathcal {K}_2<0$
, from the inequality for
$\det A$
, we deduce that
$ k_{1,2}\geq 1 $
. Therefore, by Lemma 3.5(a),(b), the real eigenvalues
$\mu _1\leq \mu _2$
of
$Df_{\unicode{x3bb} }^2(p)$
satisfy
$0<\mu _1\leq \unicode{x3bb} ^2<1\leq \mu _2$
. In particular, if the local maximum is non-degenerate, then
$\det A>0$
, and hence
$k_{1,2}>1$
, and by Lemma 3.5(a), the
$2$
-periodic point is a saddle, with
$0<\mu _1<\unicode{x3bb} ^2<1<\mu _2$
. Note that local maxima of
$\ell $
are always non-degenerate if
$\Omega \in \mathscr {U}$
; indeed, as in the proof of Corollary 3.9, we see that in that case,
$k_{1,2}\neq 1$
. -
(b) In the second case, the matrix A satisfies
$\det A={1}/{\tau ^2}(k_{1,2}-1)\leq 0$
, and hence
$ k_{1,2}\leq 1 $
. When the critical point is non-degenerate (in particular, when
$\Omega \in \mathscr {U}$
), it holds that
$\det A<0$
, and hence
$k_{1,2}<1$
. Then, the conclusion of point (b) in the above statement follows respectively from Lemma 3.5(c),(d), when
$k_{1,2}\in [0,1)$
, from Lemma 3.5(e), when
$k_{1,2}\in (-1,0)$
, and from Lemma 3.5(f), when
$k_{1,2}\leq -1$
.
4. Birkhoff attractor for circular and elliptic billiards
This section is devoted to the study of the Birkhoff attractor for the dissipative billiard map of (circles and) ellipses. To fix ideas, we can imagine that in what follows, the billiard maps have constant dissipation
$\unicode{x3bb} $
, but in fact, all the results presented in this section hold for a general dissipative billiard map as in Definition A. A useful tool through the whole section is the notion of Lyapunov function.
Definition 4.1. Let
$(X,d)$
be a metric space and let
$f\colon X\to X$
be a continuous map. A continuous function
$L\colon X\to \mathbb R$
is a Lyapunov function for f if
$L\circ f(x)\leq L(x)$
for every
$x\in X$
. If L is a Lyapunov function for f, the corresponding neutral set is defined as
$\mathcal {N}(L):=\{x\in X :L\circ f(x)=L(x)\}$
.
As in the Birkhoff case, the simplest example of a dissipative billiard is when the boundary of the billiard table is a circle
$$ \begin{align*} \mathcal{C}:=\{x=(x_1,x_2)\in \mathbb R^2: x_1^2 + x_2^2 = R^2 \}. \end{align*} $$
The proof of the next result is straightforward.
Proposition 4.2. Let
$f_{\unicode{x3bb} } \colon \mathbb {A} \to \mathbb {A}$
be a dissipative billiard map within a circle
$\mathcal {C}$
. The corresponding Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
is equal to the attractor
$\Lambda _{\unicode{x3bb} }^0$
, and
$$ \begin{align*} \Lambda_{\unicode{x3bb}} = \Lambda_{\unicode{x3bb}}^0 = \mathbb{T} \times \{0\}. \end{align*} $$
Proof. We notice that, since for any
$M \in ( 0,1 ]$
,
$$ \begin{align*} f_{\unicode{x3bb}}(\mathbb{T} \times [-M,M]) \subset \mathbb{T} \times [-\max_{\mathbb A}\unicode{x3bb} \, M,\max_{\mathbb A}\unicode{x3bb}\, M] \subset \mathbb{T} \times (-M,M), \end{align*} $$
the attractor (see equation (2.1)) corresponds to
$\Lambda _{\unicode{x3bb} }^0= \mathbb {T} \times \{0\}$
. Since
$\mathbb {T} \times \{0\}$
is the minimal element, with respect to the inclusion, in
$\mathcal {X}(f_{\unicode{x3bb} })$
, this concludes the proof.
Remark 4.3. The following are easy observations about dissipative maps inside a circular billiard. For this remark, we assume that the dissipation
$\unicode{x3bb} $
is constant.
-
(1) Since
$\mathcal {D}$
is axially symmetric with respect to every line passing through its center, the fact that
$\mathbb {T} \times \{0\} \subset \Lambda _{\unicode{x3bb} }$
is a direct application of Corollary 3.4. -
(2) It is worth noting that, in the case of the map
$f_1$
on the disc, the angle
$\varphi =\arcsin r$
stays constant along every orbit and it represents an integral of motion; as a consequence, in the dissipative case,
$L(s,r) = r$
is a Lyapunov function for
$f_{\unicode{x3bb} }$
and
$\Lambda _{\unicode{x3bb} }$
corresponds to the neutral set
$\mathcal {N}(L)$
of L (see Definition 4.1). -
(3) The foliation
$\{{\mathbb T}\times \{r\} :r\in [-1,1]\}$
is
$f_{\unicode{x3bb} }$
-invariant, that is, for every
$r\in [-1,1]$
, there exists
$r'\in [-1,1]$
such that
$f_{\unicode{x3bb} }({\mathbb T}\times \{r\})={\mathbb T}\times \{r'\}$
. In particular, r and
$r'$
have the same sign and
$\vert r'\vert \leq \vert r\vert $
.
In the following, we investigate the dynamics of the dissipative billiard map within an ellipse
$\mathcal {E}$
of non-zero eccentricity e. As the dynamics is unchanged under rigid motion of the table (affine maps of
$\mathbb R^2$
), without loss of generality, we assume that the major axis is horizontal, and the minor axis is vertical, that is, for some parameters
$a_1>a_2>0$
, we have
$$ \begin{align*} \mathcal{E}:=\bigg\{x=(x_1,x_2)\in \mathbb R^2: \frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}=1\bigg\},\quad e:=\frac{\sqrt{a_1^2-a_2^2}}{a_1}\in (0,1). \end{align*} $$
Denoting by
$\cdot $
the Euclidean scalar product and by B the diagonal matrix
$$ \begin{align*}B:=\begin{bmatrix}\dfrac{1}{a_1^2} & 0 \\ 0 & \dfrac{1}{a_2^2}\end{bmatrix},\end{align*} $$
the equation of
$\mathcal {E}$
can be abbreviated as
$\mathcal {E}=\{Bx\cdot x=1\}$
. In particular, for any
$x \in \mathcal {E}$
, the vector
$Bx$
is collinear with the normal to
$\mathcal {E} $
at x; in fact,
$Bx$
points outside the convex domain bounded by
$\mathcal {E}$
. For
$\unicode{x3bb} \in (0,1)$
, let
$f_{\unicode{x3bb} } \colon \mathbb {A}\to \mathbb {A}$
be the associated dissipative billiard map where, in such a case, it is convenient to describe the phase-space with the
$(x,v)$
coordinates:
$$ \begin{align*}\{(x,v)\in \mathcal{E} \times T^1 \mathcal{E}: Bx\cdot v\leq 0\}.\end{align*} $$
With an abuse of notation, we will refer to this set of coordinates
$\{(x,v)\in \mathcal {E} \times T^1 \mathcal {E}: Bx\cdot v\leq 0\}$
also as
${\mathbb A}$
. To lighten notation,
$f_{\unicode{x3bb} }$
will also denote the dissipative billiard map in
$(x,v)$
-coordinates.
Since a point
$(s,r)\in \mathbb {A}$
is
$2$
-periodic for
$f_{\unicode{x3bb} }$
if and only if it is
$2$
-periodic for the standard billiard map
$f_1$
, the set of
$2$
-periodic points is reduced to
$$ \begin{align*} \mathrm{II}:=\{E_1,E_2,H_1,H_2\}, \end{align*} $$
where we denote by
$$ \begin{align*} \{E_1,E_2=f_{\unicode{x3bb}}(E_1)\} \quad \text{and} \quad \{H_1,H_2=f_{\unicode{x3bb}}(H_1)\} \end{align*} $$
the
$2$
-periodic orbits of
$f_{\unicode{x3bb} }$
corresponding to the minor and the major axis, respectively.
The next result is a direct outcome of Lemma 3.7.
Lemma 4.4. Let
$f_{\unicode{x3bb} } \colon \mathbb {A} \to \mathbb {A}$
be a dissipative billiard map within an ellipse
$\mathcal {E}$
of non-zero eccentricity. The
$2$
-periodic orbit
$\{E_1,E_2=f_{\unicode{x3bb} }(E_1)\}$
, corresponding to the minor axis, is a sink. The
$2$
-periodic orbit
$\{H_1, H_2 =f_{\unicode{x3bb} }(H_1)\}$
, corresponding to the major axis, is a saddle.
Proof. The statement immediately follows from Lemma 3.7. Indeed, as in §3.3, for a 2-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}=\{p,f_1(p)\}$
, denote by
$\tau $
and
$\mathcal {K}$
respectively the distance between the two bounces and the common curvature at these points, and as in equation (3.6), let
$$ \begin{align*} k_{1,2}(p):=(\tau \mathcal{K} + 1)^2\geq 0. \end{align*} $$
On the one hand, when
$p\in \{E_1,E_2\}$
,
$\tau \mathcal {K} = 2 a_2 (-{a_2}/{a_1^2}) = -2 ({a_2}/{a_1})^2 \in (-2,0)$
. Thus,
$k_{1,2}(p)\in [0,1)$
and then, the
$2$
-periodic orbit
$\{E_1,E_2=f_{\unicode{x3bb} }(E_1)\}$
is a sink. On the other hand, when
$p\in \{H_1,H_2\}$
,
$\tau \mathcal {K} = 2 a_1( -{a_1}/{a_2^2}) = -2 ({a_1}/{a_2})^2 <-2$
. Thus,
$k_{1,2}(p)>1$
and then, the
$2$
-periodic orbit
$\{H_1,H_2=f_{\unicode{x3bb} }(H_1)\}$
is a saddle.
Notation 4.5. For
$i = 1,2$
, we denote by
$\mathcal {W}^s(H_i;f_{\unicode{x3bb} }^2)$
(respectively
$\mathcal {W}^u(H_i;f_{\unicode{x3bb} }^2)$
) the (one-dimensional) stable (respectively unstable) manifold of
$H_i$
for
$f_{\unicode{x3bb} }^2$
. To lighten the notation, for
$*=s,u$
,
$i=1,2$
, we also denote by
$\mathcal {W}^*(\mathcal {O}_{\unicode{x3bb} }(H_i))$
the union
$\mathcal {W}^*(H_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}^*(H_2;f_{\unicode{x3bb} }^2)$
. Similarly, let
$\mathcal {W}^s(E_i;f_{\unicode{x3bb} }^2)$
be the (two-dimensional) stable manifold of
$E_i$
for
$f_{\unicode{x3bb} }^2$
. Again, to lighten the notation, for
$i=1,2$
, we denote by
$\mathcal {W}^s(\mathcal {O}_{\unicode{x3bb} }(E_i)) $
the union
$\mathcal {W}^s(E_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}^s(E_2;f_{\unicode{x3bb} }^2)$
.
The main result of the present section is the following characterization of the Birkhoff attractor for dissipative billiard maps within an ellipse.
Theorem 4.6. Let
$f_{\unicode{x3bb} } \colon \mathbb {A} \to \mathbb {A}$
be a dissipative billiard map within an ellipse
$\mathcal {E}$
of non-zero eccentricity. The corresponding Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
is equal to the attractor
$\Lambda _{\unicode{x3bb} }^0$
, and we have
$$ \begin{align} \Lambda_{\unicode{x3bb}}^0=\Lambda_{\unicode{x3bb}}=\mathcal{W}^u(\mathcal{O}_{\unicode{x3bb}}(H_1))\cup \{E_1,E_2\} = \overline{\mathcal{W}^u(\mathcal{O}_{\unicode{x3bb}}(H_1))}. \end{align} $$
Moreover, for
$i=1,2$
,
$\mathcal {W}^u(H_i;f_{\unicode{x3bb} }^2)\setminus \{H_i\}$
is the disjoint union of two branches
$\mathscr {C}_i^1,\mathscr {C}_i^2$
, with
$\mathscr {C}_i^j\subset \mathcal {W}^s(E_j;f_{\unicode{x3bb} }^2)$
,
$j=1,2$
.
The next two propositions will be used in the proof of Theorem 4.6.
Proposition 4.7. Let
$f_{\unicode{x3bb} } \colon \mathbb {A} \to \mathbb {A}$
be a dissipative billiard map within an ellipse
$\mathcal {E}$
of non-zero eccentricity. The function
$$ \begin{align*} L\colon \mathbb{A} \to \mathbb R, \quad (x,v) \mapsto Bx \cdot v \end{align*} $$
is a Lyapunov function for
$f_{\unicode{x3bb} }$
. Moreover, its neutral set
$\mathcal {N}(L)$
is equal to
$ f_{\unicode{x3bb} }^{-1}({\mathbb A}_{\perp })$
, where
${\mathbb A}_{\perp } $
is the set of points
$\{(x,v) \in {\mathbb A} : v \text { is collinear to } x\}$
. More precisely, there exists a continuous function
$\delta \colon \mathbb {R}_+\to \mathbb {R}_+$
with
$\lim _{\varepsilon \to 0} \delta (\varepsilon )=0$
such that for any
$(x,v)\in \mathbb {A}$
,
$$ \begin{align} |L(f_{\unicode{x3bb}}(x,v))-L(x,v)|<\varepsilon\implies d((x,v),f_{\unicode{x3bb}}^{-1}({\mathbb A}_{\perp}))<\delta(\varepsilon), \end{align} $$
where d is the usual distance on
${\mathbb A}$
.
Proof. For
$(x,v)\in \mathbb {A}$
, let
$(x',v'):=f_{\unicode{x3bb} } (x,v)$
. We first observe that
$$ \begin{align*} B(x'-x)\cdot (x+x') = Bx' \cdot x + Bx' \cdot x' - Bx \cdot x -Bx \cdot x' = 0,\end{align*} $$
since
$x,x' \in \mathcal {E}$
and the matrix B is symmetric. As
$x'-x$
is collinear to the vector v, the previous relation yields
$Bv \cdot (x+x')=0$
; by the symmetry of B, we thus obtain
$$ \begin{align} -Bx' \cdot v=Bx \cdot v. \end{align} $$
Moreover, due to the reflection law,
$Bx' \cdot (v+v') < 0$
, except when
$v'$
is collinear with the normal at
$x'$
, in which case,
$Bx' \cdot (v+v')=0$
. By equation (4.3), we conclude that
$$ \begin{align*} L(x',v')=Bx' \cdot v' \le Bx\cdot v=L(x,v), \end{align*} $$
with equality exactly when the bounce at
$x'$
is perpendicular to
$\mathcal {E}$
. This means that
$|L(x',v')-L(x,v)|\ll 1$
if and only if
$d((x,v),f_{\unicode{x3bb} }^{-1}({\mathbb A}_{\perp }))\ll 1$
.
Remark 4.8. It is worth noting that
$f^{-1}_{\unicode{x3bb} }({\mathbb A}_{\perp })$
can be alternatively detected as a neutral set in the following way. For
$\zeta \in [0,a_2)\cup (a_2,a_1)$
, let us consider the family of quadrics:
$$ \begin{align*} \mathcal{E}_{\zeta}:=\bigg\{x=(x_1,x_2)\in \mathbb R^2: \frac{x_1^2}{a_1^2-\zeta^2}+\frac{x_2^2}{a_2^2-\zeta^2}=1\bigg\}. \end{align*} $$
For
$\zeta \in [0,a_2)$
,
$\mathcal {E}_{\zeta }$
is an ellipse confocal to
$\mathcal {E}$
and, for
$\zeta \in (a_2,a_1)$
,
$\mathcal {E}_{\zeta }$
is a hyperbola confocal to
$\mathcal {E}$
. Let
$F_1=(-c,0)$
and
$F_2=(c,0)$
be the two foci of
$\mathcal {E}$
, where
${c:=\sqrt {a_1^2-a_2^2}}$
. We extend the previous definition by letting
$\mathcal {E}_{a_2}:=(-\infty ,-c]\cup [c,+\infty )\times \{0\}$
and
$\mathcal {E}_{a_1}:=\{0\}\times \mathbb R$
. By the theory of usual elliptic billiards, for
$(x,v)\in \mathbb {A}\setminus \mathrm {II}$
, there exists a unique
$\zeta =\zeta (x,v)>0$
such that any orbit segment of the
$f_1$
-trajectory starting at
$(x,v)$
is tangent to
$\mathcal {E}_{\zeta }$
; moreover,
$\mathcal {E}_{\zeta }$
is an ellipse when the segment
$[x,x']$
does not intersect
$[F_1,F_2]$
, and it is a (possibly degenerate) hyperbola when
$[x,x']$
intersects
$[F_1,F_2]$
. Finally, we set
$\zeta (H_1)=\zeta (H_2):=a_2$
and
$\zeta (E_1)=\zeta (E_2):=a_1$
. Then, comparing the standard reflection law to the dissipative one, it can be proved that the function
$-\zeta $
is a Lyapunov function for
$f_{\unicode{x3bb} }$
, with neutral set
$\mathcal {N}(-\zeta )=f_{\unicode{x3bb} }^{-1}({\mathbb A}_{\perp })$
(Figure 5).
The Lyapunov function
$\zeta $
.
Remark 4.9
-
(a) Let us recall that
$\mathrm {II}:= \{E_1,E_2,H_1,H_2\}$
is the set of
$2$
-periodic points. By Proposition 4.7, the function
$\mathcal {L}_{\unicode{x3bb} }:=L + L \circ f_{\unicode{x3bb} }$
is also a Lyapunov function for
$f_{\unicode{x3bb} }$
, with neutral set Indeed, an orbit with two consecutive perpendicular bounces is necessarily
$$ \begin{align*} \mathcal{N}(\mathcal{L}_{\unicode{x3bb}}) = f_{\unicode{x3bb}}^{-1}({\mathbb A}_{\perp}) \cap f_{\unicode{x3bb}}^{-2}({\mathbb A}_{\perp}) =f_{\unicode{x3bb}}^{-1}(f_{\unicode{x3bb}}^{-1}({\mathbb A}_{\perp}) \cap {\mathbb A}_{\perp})=f_{\unicode{x3bb}}^{-1}(\mathrm{II})=\mathrm{II}. \end{align*} $$
$2$
-periodic. Moreover, by equation (4.2), there exists a continuous function
$\hat \delta \colon \mathbb {R}_+\to \mathbb {R}_+$
with
$\lim _{\varepsilon \to 0} \hat \delta (\varepsilon )=0$
such that for any
$(x,v)\in \mathbb {A}$
, (4.4)where d is the usual distance on
$$ \begin{align} 0\leq \mathcal{L}_{\unicode{x3bb}}(x,v)-\mathcal{L}_{\unicode{x3bb}}(f_{\unicode{x3bb}}(x,v))<\varepsilon\implies d((x,v),\mathrm{II})<\hat \delta(\varepsilon), \end{align} $$
${\mathbb A}$
.
-
(b) For any
$(x,v)\in {\mathbb A}_{\perp }$
,
$Bx$
and v are collinear, with opposite orientations, and hence Therefore,
$$ \begin{align*} L(x,v) =-\Vert Bx\Vert= -\sqrt{\frac{x_1^2}{a_1^4}+\frac{x_2^2}{a_2^4}}=-\sqrt{\frac{x_2^2}{a_2^2}\bigg(\frac{1}{a_2^2}-\frac{1}{a_1^2}\bigg)+\frac{1}{a_1^2}}. \end{align*} $$
$L|_{{\mathbb A}_{\perp }}$
is maximal when
$x_2=0$
(and takes the value
$-{1}/{a_1}$
), that is, for
$(x,v)\in \{H_1,H_2\}$
, and
$L|_{{\mathbb A}_{\perp }}$
is minimal when
$x_2^2=a_2^2$
(and takes the value
$-{1}/{a_2}$
), that is, for
$(x,v)\in \{E_1,E_2\}$
.
-
(c) Let
$X\subset \mathrm {int}({\mathbb A})$
be a
$f_{\unicode{x3bb} }$
-invariant set. Then, both
$-L\vert _{X}$
and
$-\mathcal {L}_{\unicode{x3bb} }\vert _{X}$
are Lyapunov functions for
$f_{\unicode{x3bb} }^{-1}$
.
Proposition 4.10. Let
$f_{\unicode{x3bb} } \colon \mathbb {A} \to \mathbb {A}$
be a dissipative billiard map within an ellipse
$\mathcal {E}$
of non-zero eccentricity. All the orbits are attracted by a
$2$
-periodic orbit, that is, for any
$(s,r)\in \mathbb {A}$
, there exists
$p_{+}=p_+(s,r)\in \mathrm {II} = \{E_1,E_2,H_1,H_2\}$
such that
$$ \begin{align*} \lim_{n \to +\infty}f_{\unicode{x3bb}}^{2n}(s,r)=p_{+},\quad \lim_{n \to +\infty}f_{\unicode{x3bb}}^{2n+1}(s,r)=f_{\unicode{x3bb}}(p_{+}). \end{align*} $$
In particular, the set of periodic points for
$f_{\unicode{x3bb} }$
is reduced to the set
$\mathrm {II}$
of
$2$
-periodic points. Moreover,
$\mathrm {II}\subset \Lambda _{\unicode{x3bb} }\subset \Lambda _{\unicode{x3bb} }^0$
, and for any
$(s,r)\in \Lambda _{\unicode{x3bb} }^0\setminus \mathrm {II}$
, there exist
$i_-,i_+\in \{1,2\}$
such that
$$ \begin{align*} \lim_{n \to -\infty}f_{\unicode{x3bb}}^{2n}(x,v)=H_{i_-},\quad \lim_{n \to -\infty}f_{\unicode{x3bb}}^{2n-1}(x,v)=f_{\unicode{x3bb}}(H_{i_-}),\\ \lim_{n \to +\infty}f_{\unicode{x3bb}}^{2n}(x,v)=E_{i_+},\quad \lim_{n \to +\infty}f_{\unicode{x3bb}}^{2n-1}(x,v)=f_{\unicode{x3bb}}(E_{i_+}). \end{align*} $$
Proof. Fix
$(s,r)\in \mathbb {A}$
. By Remark 4.9(a), the function
$\mathcal {L}_{\unicode{x3bb} }$
is a Lyapunov function for
$f_{\unicode{x3bb} }$
whose neutral set is
$\mathrm {II}$
; consequently, the omega-limit set
$\omega _{f_{\unicode{x3bb} }}(s,r)$
satisfies
$\omega _{f_{\unicode{x3bb} }}(s,r)\subset \mathcal {N}(\mathcal {L}_{\unicode{x3bb} })=\mathrm {II}$
. For each
$n\geq 0$
, we set
$u_{n}:=\mathcal {L}_{\unicode{x3bb} }(f_{\unicode{x3bb} }^{n}(s,r)). $
The sequence
$(u_{n})_{\geq 0}$
is decreasing and bounded from below by
$\min _{\mathbb {A}} \mathcal {L}_{\unicode{x3bb} }>-\infty $
, and hence is convergent. In particular,
$\lim _{n \to +\infty } (u_n-u_{n+1})= 0$
; by equation (4.4), we deduce that
$$ \begin{align*}\lim_{n \to +\infty}\,d(f_{\unicode{x3bb}}^{n}(s,r),\mathrm{II})=0,\end{align*} $$
where d is the usual distance inherited from
${\mathbb A}$
. Recall that
$\mathrm {II}$
is formed of only four different points and let
$\epsilon :=\tfrac 13\min _{p\neq q \in \mathrm {II}}d(p,q)>0$
. From the previous limit, there exists
$n_0 \in \mathbb {N}$
such that for any
$n \geq n_0$
, we have
$d(f_{\unicode{x3bb} }^{n}(s,r),\mathrm {II})<\epsilon $
. Actually, for
$n \geq n_0$
, there exists a unique point
$p(s,r,n)\in \mathrm {II}$
such that
$d(f_{\unicode{x3bb} }^{n}(s,r),p(s,r,n))<\epsilon $
. In particular, since the bounce at
$f_{\unicode{x3bb} }^{n+1}(s,r)$
gets closer and closer to being perpendicular as
$n\to +\infty $
, we have that
$$ \begin{align*}\lim_{n\to+\infty}\,d(f_{\unicode{x3bb}}^{n+2}(s,r),f_{\unicode{x3bb}}^{n}(s,r))=0.\end{align*} $$
Let us then fix
$n_1\geq n_0$
such that for any
$n\geq n_1$
, it holds that
$$ \begin{align*} d(f_{\unicode{x3bb}}^{n+2}(s,r),f_{\unicode{x3bb}}^{n}(s,r))<\epsilon. \end{align*} $$
Then, for any
$n\geq n_1$
, we have
$$ \begin{align*} &d(p(s,r,n),p(s,r,n+2))\\ &\leq d(p(s,r,2n),f_{\unicode{x3bb}}^{n}(s,r))+d(f_{\unicode{x3bb}}^{n}(s,r),f_{\unicode{x3bb}}^{n+2}(s,r))+d(f_{\unicode{x3bb}}^{n+2}(s,r),\\&\quad\ p(s,r,n+2))<3 \epsilon. \end{align*} $$
By the choice of
$\epsilon> 0$
, it follows that
$p(s,r,n+2)=p(s,r,n)$
for any
$n \geq n_1$
. Let us then set
$p_+=p_+(s,r):=p(s,r,2n)$
for any
$2n\geq n_1$
(it is well defined by the previous discussion). Then we conclude that
$$ \begin{align*} \lim_{n \to +\infty}f_{\unicode{x3bb}}^{2n}(s,r)=p_{+},\quad \lim_{n \to +\infty}f_{\unicode{x3bb}}^{2n+1}(s,r)=f_{\unicode{x3bb}}(p_{+}). \end{align*} $$
In particular, we deduce also that
$p_+(f_{\unicode{x3bb} }(s,r))=f_{\unicode{x3bb} }(p_+(s,r))$
and that
$\omega _{f_{\unicode{x3bb} }}(s,r)=\{p_+,f_{\unicode{x3bb} }(p_+)\}$
.
The sets
$\Lambda _{\unicode{x3bb} },\Lambda _{\unicode{x3bb} }^0$
are
$f_{\unicode{x3bb} }$
-invariant; moreover,
$f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }}$
, respectively
$f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }^0}$
, is invertible, and
$\tilde {\mathcal {L}}_{\unicode{x3bb} }:=-\mathcal {L}_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }}$
, respectively
$\tilde {\mathcal {L}}_{\unicode{x3bb} }^0:=-\mathcal {L}_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }^0}$
, is a Lyapunov function for
$(f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }})^{-1}$
, respectively
$(f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }^0})^{-1}$
, see Remark 4.9(c). Since
$\Lambda _{\unicode{x3bb} },\Lambda _{\unicode{x3bb} }^0$
are compact, for any
${(s,r)\in \Lambda _{\unicode{x3bb} }}$
, respectively
$(s,r)\in \Lambda _{\unicode{x3bb} }^0$
, we have that the alpha-limit set
$\alpha _{f_{\unicode{x3bb} }}(s,r)$
satisfies
$\emptyset \neq \alpha _{f_{\unicode{x3bb} }}(s,r)\subset \mathcal {N}(\tilde {\mathcal {L}}_{\unicode{x3bb} })\cap \Lambda _{\unicode{x3bb} }\subset \mathrm {II}$
, respectively
$\emptyset \neq \alpha _{f_{\unicode{x3bb} }}(s,r)\subset \mathcal {N}(\tilde {\mathcal {L}}_{\unicode{x3bb} }^0)\cap \Lambda _{\unicode{x3bb} }^0\subset \mathrm {II}$
; in particular, we deduce that
$\emptyset \neq \Lambda _{\unicode{x3bb} } \cap \mathrm {II} \subset \Lambda _{\unicode{x3bb} }^0 \cap \mathrm {II}$
. Fix
$(s,r)\in \Lambda _{\unicode{x3bb} }$
, respectively
${(s,r)\in \Lambda _{\unicode{x3bb} }^0}$
. Arguing as above, we see that there exists
$p_-=p_-(s,r)\in \mathrm {II}\cap \Lambda _{\unicode{x3bb} }$
, respectively
$p_-=p_-(s,r)\in \mathrm {II}\cap \Lambda _{\unicode{x3bb} }^0$
, such that
$$ \begin{align*} \lim_{n \to -\infty}f_{\unicode{x3bb}}^{2n}(s,r)=p_{-}, \quad \lim_{n \to -\infty}f_{\unicode{x3bb}}^{2n-1}(s,r)=f_{\unicode{x3bb}}(p_{-}).\end{align*} $$
Then, it holds that
$\alpha _{f_{\unicode{x3bb} }}(s,r)=\{p_-,f_{\unicode{x3bb} }(p_-)\}$
. There are two cases for the point
${(s,r)\in \Lambda _{\unicode{x3bb} }}$
, respectively
$\Lambda ^0_{\unicode{x3bb} }$
:
-
• either
$\alpha _{f_{\unicode{x3bb} }}(s,r) \cap \omega _{f_{\unicode{x3bb} }}(s,r)\neq \emptyset $
, and then the whole orbit
$(f_{\unicode{x3bb} }^k(s,r))_{k \in \mathbb {Z}}$
is in the neutral set
$\mathcal {N}(\mathcal {L}_{\unicode{x3bb} })$
, that is,
$(s,r)=p_-(s,r)=p_+(s,r)$
is
$2$
-periodic; -
• otherwise,
$(s,r)\notin \mathrm {II}$
(this case clearly occurs, as
$\Lambda _{\unicode{x3bb} }$
separates
$\mathbb {A}$
, while
$\mathrm {II}$
is a finite set) and the
$2$
-periodic orbits
$\alpha _{f_{\unicode{x3bb} }}(s,r)=\{p_-,f_{\unicode{x3bb} }(p_-)\}$
and
$\omega _{f_{\unicode{x3bb} }}(s,r)=\{p_+,f_{\unicode{x3bb} }(p_+)\}$
are distinct. By Remark 4.9(b), and since the orbits of
$p_+$
and
$p_-$
are different, we have that
$\mathcal {L}_{\unicode{x3bb} }(p_-)>\mathcal {L}_{\unicode{x3bb} }(p_+)$
and actually,
$\alpha _{f_{\unicode{x3bb} }}(s,r)=\{H_1,H_2\}$
and
$\omega _{f_{\unicode{x3bb} }}(s,r)=\{E_1,E_2\}$
. As
$\Lambda _{\unicode{x3bb} },\Lambda _{\unicode{x3bb} }^0$
are closed, we also deduce that
$\mathrm {II}\subset \Lambda _{\unicode{x3bb} }\subset \Lambda _{\unicode{x3bb} }^0$
.
Remark 4.11. When the dissipation
$\unicode{x3bb} $
is constant, the fact that
$\mathrm {II}\subset \Lambda _{\unicode{x3bb} }\subset \Lambda _{\unicode{x3bb} }^0$
proven in Proposition 4.10 also follows from Lemma 3.3, due to the symmetries of the ellipse
$\mathcal {E}$
.
Let us also note that, by Proposition 4.10, for any
$(s,r) \in \mathbb {A}\setminus \mathcal {W}^s(H_1;f_{\unicode{x3bb} })$
, the forward orbit of
$(s,r)$
converges to the
$2$
-periodic orbit
$\{E_1,f_{\unicode{x3bb} }(E_1) = E_2\}$
.
We are now ready to give the proof of Theorem 4.6.
Notation 4.12. Given some small
$\delta>0$
, we denote by
$\mathcal {W}_{\delta }^{u}(H_i;f_{\unicode{x3bb} }^2)$
the
$\delta $
-local unstable manifold of
$H_i$
with respect to
$f_{\unicode{x3bb} }^2$
, that is, the set
$$ \begin{align*}\mathcal{W}_{\delta}^{u}(H_i;f_{\unicode{x3bb}}^2) := \{(s,r)\in{\mathbb A} :d(f_{\unicode{x3bb}}^{2n}(s,r),H_i)\leq \delta \text{ for all } n\leq 0\}.\end{align*} $$
Similarly, the
$\delta $
-local stable manifold of
$H_i$
with respect to
$f_{\unicode{x3bb} }^2$
is
$$ \begin{align*}\mathcal{W}_{\delta}^{s}(H_i;f_{\unicode{x3bb}}^2):=\{(s,r)\in{\mathbb A}:d(f_{\unicode{x3bb}}^{2n}(s;r),H_i)\leq \delta \text{ for all } n\geq 0\}.\end{align*} $$
For
$*=s,u$
,
$i=1,2$
, the notation
$\mathcal {W}_{\delta }^{*}(\mathcal {O}_{\unicode{x3bb} }(H_i))$
refers to
$\mathcal {W}_{\delta }^{*}(H_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}_{\delta }^{*}(H_2;f_{\unicode{x3bb} }^2)$
. We denote by
$$ \begin{align*}\mathcal{W}_{\delta}^{s}(E_i;f_{\unicode{x3bb}}^2):=\{(s,r)\in{\mathbb A}:d(f_{\unicode{x3bb}}^{2n}(s,r),E_i)\leq\delta\text{ for all } n\geq 0\},\end{align*} $$
the
$\delta $
-local stable manifold of
$E_i$
with respect to
$f_{\unicode{x3bb} }^2$
. Similarly, for
$i=1,2$
, the notation
$\mathcal {W}_{\delta }^{s}(\mathcal {O}_{\unicode{x3bb} }(E_i))$
refers to
$\mathcal {W}_{\delta }^{s}(E_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}_{\delta }^{s}(E_2;f_{\unicode{x3bb} }^2)$
.
Proof of Theorem 4.6
By Proposition 4.10, for any
$(s,r) \in \Lambda _{\unicode{x3bb} }\setminus \{E_1,E_2\}$
, respectively
$(s,r) \in \Lambda _{\unicode{x3bb} }^0\setminus \{E_1,E_2\}$
, there exists
$i_-\in \{1,2\}$
such that
$\lim _{n \to -\infty }f_{\unicode{x3bb} }^{2n}(s,r)=H_{i_-}$
, and hence
$(s,r)\in \mathcal {W}^{u}(\mathcal {O}_{\unicode{x3bb} }(H_{i_-}))$
. We deduce that
$$ \begin{align} \Lambda_{\unicode{x3bb}}\subset \Lambda_{\unicode{x3bb}}^0\subset \mathcal{W}^{u}(\mathcal{O}_{\unicode{x3bb}}(H_1))\cup \{E_1,E_2\}=\overline{ \mathcal{W}^{u}(\mathcal{O}_{\unicode{x3bb}}(H_1))}, \end{align} $$
where the last equality follows again from Proposition 4.10; indeed, the points
$E_1,E_2$
are accumulated by the forward orbit of any point
$(s,r) \in \Lambda _{\unicode{x3bb} }\setminus \{E_1,E_2\}\subset \mathcal {W}^{u}(\mathcal {O}_{\unicode{x3bb} }(H_1))$
. From equation (4.5) and applying Lemma 2.7 to
$\overline { \mathcal {W}^{u}(\mathcal {O}_{\unicode{x3bb} }(H_1))}$
, we deduce that
$\overline { \mathcal {W}^{u}(\mathcal {O}_{\unicode{x3bb} }(H_1))}$
separates
${\mathbb A}$
. Since it is also compact, connected, and
$f_{\unicode{x3bb} }$
-invariant, it holds that
$\overline { \mathcal {W}^{u}(\mathcal {O}_{\unicode{x3bb} }(H_1)) } \in \mathcal {X}(f_{\unicode{x3bb} })$
.
Claim 4.13. There is
$\delta>0$
such that, for any
$x\in \mathcal {W}_{\delta }^{u}(\mathcal {O}_{\unicode{x3bb} }(H_1))$
,
$\overline { \mathcal {W}^{u}(\mathcal {O}_{\unicode{x3bb} }(H_1)) } \setminus \{x\}$
does not separate the annulus.
Proof of the claim
Let
$\eta>0$
be small enough such that the balls of radius
$\eta $
centered at points in
$\{H_1,H_2,E_1,E_2\}$
are pairwise disjoint. From Proposition 4.10, for
$i=1,2$
, we have that
$\mathcal {W}^u(H_i;f_{\unicode{x3bb} }^2)\setminus \{H_i\}$
is contained in the stable manifold of
$\{E_1,E_2\}$
; in particular, there exists
$N\in \mathbb N$
such that for every
$n>N$
, it holds that
$$ \begin{align} f_{\unicode{x3bb}}^n(\mathcal{W}^u_{\eta}(\mathcal{O}_{\unicode{x3bb}}(H_1))\setminus f_{\unicode{x3bb}}^N(\mathcal{W}^u_{\eta}(\mathcal{O}_{\unicode{x3bb}}(H_1))\subset B(E_1,\eta)\cup B(E_2,\eta). \end{align} $$
We can then choose
$\delta>0$
small enough such that, for
$i=1,2$
, the
$\delta $
-local unstable manifold of
$H_i$
is a
$C^1$
graph over the first coordinate projection of
$B(H_i,\delta )$
and such that
$B(H_i,\delta )\cap \mathcal {W}^u(H_i;f_{\unicode{x3bb} }^2)= \mathcal {W}^u_{\delta }(H_i;f_{\unicode{x3bb} }^2)$
, that is, the unstable manifold meets the ball only at the local unstable manifold. This last property is possible thanks to equation (4.6). The
$\delta $
-local unstable manifold
$\mathcal {W}^u_{\delta }(H_i;f_{\unicode{x3bb} }^2)$
separates the ball
$B(H_i,\delta )$
, that is, we have
$B(H_i,\delta )\setminus \mathcal {W}^u_{\delta }(H_i;f_{\unicode{x3bb} }^2)= \mathcal {U}\cup \mathcal {V}$
for two disjoint connected open sets
$\mathcal {U}$
and
$\mathcal {V}$
. For any
$x\in \mathcal {W}^u_{\delta }(H_i;f_{\unicode{x3bb} }^2)$
, the set
$B(H_i,\delta )\setminus (\mathcal {W}^u_{\delta }(H_i;f_{\unicode{x3bb} }^2)\setminus \{x\})$
is path-connected. We conclude that
$\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1))}\setminus \{x\}$
does not separate the annulus if
$$ \begin{align*} \mathcal{U}\subset U_{\overline{\mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(H_1))}}\quad \text{and}\quad \mathcal{V}\subset V_{\overline{\mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(H_1))}}, \end{align*} $$
where
${\mathbb A}\setminus \overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1))}=U_{\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1))}}\cup V_{\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1))}}$
. This follows from the fact that
$H_i\in \Lambda _{\unicode{x3bb} }$
and that
$\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1))}\in \mathcal {X}(f_{\unicode{x3bb} })$
: indeed, by Proposition 2.5(2),
$H_i\in \mathrm {Fr}(U_{\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1))}}\cap \mathrm {Fr}(V_{\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1))}})$
and, in particular, the two connected open sets
$\mathcal {U}$
and
$\mathcal {V}$
cannot be contained in the same connected component of
${\mathbb A}\setminus \overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1))}$
.
Thus, by Lemma 2.6, we have that
$$ \begin{align} \mathcal{W}_{\delta}^u(\mathcal{O}_{\unicode{x3bb}}(H_1))\subset \Lambda_{\unicode{x3bb}}. \end{align} $$
Let us also recall that for
$i=1,2$
, we have
$\mathcal {W}^u(\mathcal {O}_{\unicode{x3bb} }(H_1))=\bigcup _{j\geq 0}f_{\unicode{x3bb} }^{j}(\mathcal {W}_{\delta }^u(\mathcal {O}_{\unicode{x3bb} }(H_1)))$
. By equation (4.7), and as
$\Lambda _{\unicode{x3bb} }$
is
$f_{\unicode{x3bb} }$
-invariant and closed, we obtain
$$ \begin{align} \overline{\mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(H_1))}\subset \Lambda_{\unicode{x3bb}}. \end{align} $$
Comparing equations (4.5) and (4.8), we deduce that all the inclusions are actually equalities, which concludes the proof of equation (4.1).
By the previous discussion,
$\Lambda _{\unicode{x3bb} } \setminus \mathrm {II}$
is the disjoint union of four connected components
$\mathscr {C}_1,\mathscr {C}_1^{\prime },\mathscr {C}_2=f_{\unicode{x3bb} }(\mathscr {C}_1),\mathscr {C}_2'=f_{\unicode{x3bb} }(\mathscr {C}_1^{\prime })$
, where
$\mathscr {C}_i$
and
$\mathscr {C}_i'$
correspond to the two branches of
$ \mathcal {W}^u(H_i;f_{\unicode{x3bb} }^2)\setminus \{H_i\}$
for
$i=1,2$
. Moreover, by Corollary 4.10,
$\mathscr {C}_1\subset \mathcal {W}^s(E_j;f_{\unicode{x3bb} }^2)$
and
$\mathscr {C}_1^{\prime }\subset \mathcal {W}^s(E_k;f_{\unicode{x3bb} }^2)$
for some
$j,k\in \{1,2\}$
. We claim that
$j\neq k$
. Assume by contradiction that
$j=k$
and set
$\widehat {\mathscr {C}}_1:=\mathscr {C}_1\cup \mathscr {C}_1^{\prime }\cup \{H_1,E_j\}=\mathcal {W}^u(H_1;f_{\unicode{x3bb} }^2)\cup \{E_j\}$
. Since
$\mathcal {W}^u(H_1;f_{\unicode{x3bb} }^2)$
has no self-intersection, we have that
$\widehat {\mathscr {C}}_1$
is an
$f_{\unicode{x3bb} }^2$
-invariant simple closed curve. We distinguish between two cases:
-
(1) either
$\widehat {\mathscr {C}}_1$
separates the annulus
${\mathbb A}$
; then, we would have
$\Lambda _{\unicode{x3bb} }=\widehat {\mathscr {C}}_1\sqcup f_{\unicode{x3bb} }(\widehat {\mathscr {C}}_1)$
, where
$\widehat {\mathscr {C}}_1$
and
$f_{\unicode{x3bb} }(\widehat {\mathscr {C}}_1)$
are compact, connected, and both separate
${\mathbb A}$
. This would imply that
${{\mathbb A}\setminus \Lambda _{\unicode{x3bb} }}$
is the disjoint union of three connected open sets, one of which is an
$f_{\unicode{x3bb} }$
-invariant bounded open set. This would contradict the dissipative character of the map; -
(2) or the curve
$\widehat {\mathscr {C}}_1$
is homotopic to a point; in particular, it bounds an
$f_{\unicode{x3bb} }^2$
-invariant open set. Again, this would contradict the dissipative character of
$f_{\unicode{x3bb} }^2$
.
Thus,
$j\neq k$
. Setting
$\mathscr {C}_1^j:=\mathscr {C}_1$
,
$\mathscr {C}_1^k:=\mathscr {C}_1^{\prime }$
,
$\mathscr {C}_2^k:=\mathscr {C}_2=f_{\unicode{x3bb} }(\mathscr {C}_1^j)$
,
$\mathscr {C}_2^j:=\mathscr {C}_2'=f_{\unicode{x3bb} }(\mathscr {C}_1^k)$
, this concludes the proof.
In the next corollary, we prove that the conclusion of Theorem 4.6 remains true for strictly convex domains whose boundary is sufficiently
$C^2$
-close to an ellipse. For simplicity, in the rest of this section, we will assume that the dissipation
$\unicode{x3bb} $
is constant.
Corollary 4.14. Let
$\mathcal {E}$
be an ellipse and fix
$\unicode{x3bb} {\kern-1pt}\in{\kern-1pt} (0,1)$
. Then, there exists
${\epsilon =\epsilon (\mathcal {E},\unicode{x3bb} ){\kern-1pt}>{\kern-1pt}0}$
such that for any domain
$\Omega \subset \mathbb R^2$
, whose boundary
$\partial \Omega $
is
$C^k$
,
$k \geq 2$
, and satisfies
$d_{C^2}(\partial \Omega ,\mathcal {E})<\epsilon $
, the following holds. Let
$f_{\unicode{x3bb} }\colon \mathbb {A} \to \mathbb {A}$
be the dissipative billiard map within
$\Omega $
. There exist
$2$
-periodic orbits
$\{H_1(\Omega ),H_2(\Omega )\}$
and
$\{E_1(\Omega ),E_2(\Omega )\}$
of saddle and sink type, respectively, and the Birkhoff attractor is equal to
$$ \begin{align*} \Lambda_{\unicode{x3bb}}= \mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(H_1(\Omega)) \cup \{E_1(\Omega),E_2(\Omega)\} = \overline{\mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(H_1(\Omega)) }, \end{align*} $$
where
$ \mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1(\Omega )) :=\mathcal {W}^u(H_1(\Omega );f_{\unicode{x3bb} }^2) \cup \mathcal {W}^u(H_2(\Omega );f_{\unicode{x3bb} }^2)$
. Moreover, the function
$(\mathcal {E},\unicode{x3bb} )\mapsto \epsilon (\mathcal {E},\unicode{x3bb} )$
can be chosen to be continuous.
Proof. Denote by
$g_{\unicode{x3bb} }$
the dissipative billiard map of
$\mathcal {E}$
with dissipation parameter
$\unicode{x3bb} \in (0,1)$
. Let
$\{H_1,H_2\}$
and
$\{E_1,E_2\}$
be the
$2$
-periodic orbits for
$g_{\unicode{x3bb} }$
of saddle and sink type, respectively, see Lemma 4.4. For
$\partial \Omega $
sufficiently
$C^2$
-close to
$\mathcal {E}$
, the associated domain
$\Omega $
is still strongly convex, as the curvature function depends continuously on the domain in the
$C^2$
-topology, and
$\mathcal {E}$
is strongly convex. Without loss of generality, we can assume that the perimeter of
$\partial \Omega $
is still one (as the dynamics is invariant under rescaling), so that the dissipative billiard map
$f_{\unicode{x3bb} }$
is defined on the same phase space
${\mathbb A}$
as
$g_{\unicode{x3bb} }$
. Moreover, for any
$\eta>0$
, there exists
$\epsilon _0(\eta )>0$
such that if
$d_{C^2}(\partial \Omega ,\mathcal {E})<\epsilon _0(\eta )$
, then
$d_{C^1}(f_{\unicode{x3bb} },g_{\unicode{x3bb} })<\eta $
. Indeed, by equation (3.4), the differentials of
$f_{\unicode{x3bb} },g_{\unicode{x3bb} }$
depend continuously on the curvature function. Fix
$\eta>0$
small enough such that for any
$\Omega $
with
$d_{C^2}(\partial \Omega ,\mathcal {E})<\epsilon _0:=\epsilon _0(\eta )$
, the
$2$
-periodic orbits
$\{H_1,H_2\}$
and
$\{E_1,E_2\}$
have continuations
$\{H_1(\Omega ),H_2(\Omega )\}$
(of saddle type) and
$\{E_1(\Omega ),E_2(\Omega )\}$
(of sink type) for
$f_{\unicode{x3bb} }$
.
Let
$\delta _0>0$
be such that for every
$0<\delta <\delta _0$
, the balls
$B(E_1,\delta ), B(E_2,\delta )$
are both contractible, that is, any closed path contained in
$B(E_1,\delta )$
, respectively
$B(E_2,\delta )$
, is homotopic to a point. As already noticed with inclusion in equation (4.6) in Proposition 4.10, we can fix
$0<\delta <\delta _0$
small enough such that there exists
$N\in \mathbb N$
with the property that for any
$n>N$
, the set
$$ \begin{align*} g_{\unicode{x3bb}}^n(\mathcal{W}^u_{\delta/2}(\mathcal{O}_{g_{\unicode{x3bb}}}(H_1)))\setminus g_{\unicode{x3bb}}^N(\mathcal{W}^u_{\delta/2}(\mathcal{O}_{g_{\unicode{x3bb}}}(H_1))) \subset B(E_1,{\delta/2})\cup B(E_2,{\delta/2}), \end{align*} $$
where
$\mathcal {W}^u_{\delta /2}(\mathcal {O}_{g_{\unicode{x3bb} }}(H_1)):=\mathcal {W}^u_{\delta /2}(H_1;g_{\unicode{x3bb} }^2)\cup \mathcal {W}^u_{\delta /2}(H_2;g_{\unicode{x3bb} }^2)$
. By the Hadamard–Perron the- orem—see e.g. [Reference Brin and StuckBS15, Proposition 5.6.1]—local invariant manifolds of hyperbolic fixed points depend continuously on the dynamics in the
$C^1$
-topology (and hence depend continuously on
$\partial \Omega $
in the
$C^2$
-topology). Consequently, there exists
$0<\epsilon _1<\epsilon _0$
such that for every domain
$\Omega $
with
$d_{C^2}(\partial \Omega ,\mathcal {E})<\epsilon _1$
, for any
$n>N$
, the set
$$ \begin{align*} f_{\unicode{x3bb}}^n(\mathcal{W}^u_{\delta}(\mathcal{O}_{f_{\unicode{x3bb}}}(H_1(\Omega))))\setminus f_{\unicode{x3bb}}^N(\mathcal{W}^u_{\delta}(\mathcal{O}_{f_{\unicode{x3bb}}}(H_1(\Omega))))\end{align*} $$
is contained in
$B(E_1,{\delta })\cup B(E_2,{\delta })$
, where
$\mathcal {W}^u_{\delta }(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1(\Omega ))):=\mathcal {W}^u_{\delta }(H_1(\Omega );f_{\unicode{x3bb} }^2)\cup \mathcal {W}^u_{\delta }(H_2(\Omega );f_{\unicode{x3bb} }^2)$
. Observe that
$E_i(\Omega )\in B(E_i,\delta )$
for
$i=1,2$
. In particular, the
$2$
-periodic orbit
$\{E_1(\Omega ),E_2(\Omega )\}$
belongs to
$$ \begin{align*}\overline{\mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(H_1(\Omega))}:=\overline{\mathcal{W}^u(H_1(\Omega);f_{\unicode{x3bb}}^2)\cup\mathcal{W}^u(H_2(\Omega);f_{\unicode{x3bb}}^2)}\end{align*} $$
and the latter is an
$f_{\unicode{x3bb} }$
-invariant, compact, and connected set.
Claim 4.15. The set
$\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1(\Omega ))}$
separates
${\mathbb A}$
.
Proof of the claim
As for
$\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1(\Omega ))}$
, we use the notation
$$ \begin{align*}\overline{\mathcal{W}^u(\mathcal{O}_{g_{\unicode{x3bb}}}(H_1))} := \overline{\mathcal{W}^u(H_1(\Omega);g_{\unicode{x3bb}}^2) \cup \mathcal{W}^u(H_2(\Omega);g_{\unicode{x3bb}}^2)}. \end{align*} $$
Observe that the second one separates the annulus because, by Theorem 4.6, it is the Birkhoff attractor of
$g_{\unicode{x3bb} }$
. The claim then follows if we show that they are homotopic. From the previous choice of
$\epsilon _1>0$
, we can decompose
$$ \begin{align*} \overline{\mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(H_1(\Omega)))}=\gamma_1\cup\gamma_2\cup\hat\gamma_1\cup \hat \gamma_2, \end{align*} $$
where for
$i=1,2$
,
$\gamma _i\subset B(E_i,\delta )$
is a continuous path containing
$E_i(\Omega )$
whose endpoints
$q_i^1,q_i^2$
lie on the circle
$\mathcal {C}(E_i,\delta )$
(centered at
$E_i$
of radius
$\delta $
), while
$\hat \gamma _i\subset f_{\unicode{x3bb} }^N(\mathcal {W}^u_{\delta }(H_i(\Omega );f_{\unicode{x3bb} }^2))$
is an unstable arc with endpoints
$q_1^i$
and
$q_2^i$
. Similarly,
$$ \begin{align*} \overline{\mathcal{W}^u(\mathcal{O}_{g_{\unicode{x3bb}}}(H_1))}=\Gamma_1\cup\Gamma_2\cup\hat\Gamma_1\cup \hat \Gamma_2, \end{align*} $$
where for
$i=1,2$
,
$\Gamma _i\subset B(E_i,\delta )$
is a continuous path containing
$E_i$
whose endpoints
$Q_i^1,Q_i^2$
lie on the circle
$\mathcal {C}(E_i,\delta )$
, while
$\hat \Gamma _i\subset g_{\unicode{x3bb} }^N(\mathcal {W}^u_{\delta }(H_i;g_{\unicode{x3bb} }^2))$
is an unstable arc with endpoints
$Q_1^i$
and
$Q_2^i$
.
Fix
$0<\delta '\ll \delta $
. We can retract a
$\delta '$
-neighborhood of the circles
$\mathcal {C}(E_1,\delta )$
and
$\mathcal {C}(E_2,\delta )$
in such a way that the respective images
$\{\gamma _i',\hat \gamma _i',\Gamma _i',\hat \Gamma _i'\}_{i=1,2}$
of
$\{\gamma _i,\hat \gamma _i,\Gamma _i,\hat \Gamma _i\}_{i=1,2}$
after retraction satisfy that for
$i=1,2$
,
$\gamma _i',\Gamma _i'$
have the same endpoints and
$\hat \gamma _i',\hat \Gamma _i'$
have the same endpoints. In particular, it holds that
$\gamma _1\cup \gamma _2\cup \hat \gamma _1\cup \hat \gamma _2$
is homotopic to
$\Gamma _1\cup \Gamma _2\cup \hat \Gamma _1\cup \hat \Gamma _2$
if and only if
$\gamma _1^{\prime }\cup \gamma _2'\cup \hat \gamma _1^{\prime }\cup \hat \gamma _2'$
is homotopic to
$\Gamma _1^{\prime }\cup \Gamma _2'\cup \hat \Gamma _1^{\prime }\cup \hat \Gamma _2'$
. We denote by
$B_1^{\prime },B_2'$
the respective images of
$B(E_1,\delta )$
and
$B(E_2,\delta )$
after retraction.
Since for
$i=1,2$
, the set
$B_i'$
is contractible and
$\gamma _i'\cup \Gamma _i'$
is a closed loop, we deduce that
$\gamma _i'$
and
$\Gamma _i'$
are homotopic. In addition, by the continuous dependence of the local unstable manifolds on the dynamics, the unstable arcs
$\hat \gamma _i$
and
$\hat \Gamma _i$
are
$C^0$
-close to each other, and then, the paths
$\hat \gamma _i'$
and
$\hat \Gamma _i'$
are homotopic. We conclude that
$\gamma _1^{\prime }\cup \gamma _2'\cup \hat \gamma _1^{\prime }\cup \hat \gamma _2'$
is homotopic to
$\Gamma _1^{\prime }\cup \Gamma _2'\cup \hat \Gamma _1^{\prime }\cup \hat \Gamma _2'$
; by construction, it follows that
$\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1(\Omega )))}=\gamma _1\cup \gamma _2\cup \hat \gamma _1\cup \hat \gamma _2$
is also homotopic to
$ \overline {\mathcal {W}^u(\mathcal {O}_{g_{\unicode{x3bb} }}(H_1))}=\Gamma _1\cup \Gamma _2\cup \hat \Gamma _1\cup \hat \Gamma _2$
.
As a consequence of the previous claim, we have
$\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1(\Omega ))} \in \mathcal {X}(f_{\unicode{x3bb} })$
, and hence, by Proposition 2.5, the Birkhoff attractor of
$f_{\unicode{x3bb} }$
is contained in it. Since the Birkhoff attractor cannot be reduced to
$\{E_1(\Omega ),E_2(\Omega )\}$
, it must contain points in the unstable manifold of the saddle periodic orbit. Since the Birkhoff attractor is invariant and closed, it holds that
$H_1(\Omega ), H_2(\Omega )\in \Lambda _{\unicode{x3bb} }$
. Repeating the proof of Claim 4.13, we can show that for any point x in the local unstable manifold of
$\{H_1(\Omega ),H_2(\Omega )\}$
, the set
$\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1(\Omega )))} \setminus \{x\}$
does not separate the annulus. By Lemma 2.6, we deduce that the local unstable manifold is contained in the Birkhoff attractor. Again, since
$\Lambda _{\unicode{x3bb} }$
is invariant and closed, we deduce that
$\Lambda _{\unicode{x3bb} }=\overline {\mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(H_1(\Omega ))) }$
.
Finally, the fact that the function
$(\mathcal {E},\unicode{x3bb} )\mapsto \epsilon (\mathcal {E},\unicode{x3bb} )$
can be chosen to be continuous follows from the fact that the objects for which certain conditions have to be satisfied while choosing
$\epsilon _0,\epsilon _1$
above depend continuously on the dynamics
$f_{\unicode{x3bb} }$
(in the
$C^1$
-topology), which itself depends continuously on the eccentricity of the ellipse
$\mathcal {E}$
and on the dissipation parameter
$\unicode{x3bb} \in (0,1)$
.
5. Birkhoff attractors for strong dissipation
In the previous section, as a first example, we have seen that, in the case of a circular table, the Birkhoff attractor is the simplest possible, that is, the graph of the zero function. We are thus naturally led to investigate when
$\Lambda _{\unicode{x3bb} }$
is topologically simple, that is, it is a graph. The main result of the present section is proving that the geometric condition contained in Definition D together with the hypothesis that the dissipation is strong, that is, with
$\unicode{x3bb} $
close to
$0$
, are sufficient for the Birkhoff attractor to be a graph. The corresponding result (Theorem 5.7) contains also details on dynamics’ and graphs’ regularity and uses the notions of dominated splitting for an invariant set and of a normally contracted (called also hyperbolic) manifold. These definitions are recalled at the beginning of the next section.
Throughout this section, for simplicity, we will assume that the dissipative billiard maps
$f_{\unicode{x3bb} }$
have constant dissipation
$\unicode{x3bb} \in (0,1)$
. Yet, we will argue in Remark 5.17 that the same results can be obtained for a general dissipative billiard map as in Definition A.
5.1. Normally contracted Birkhoff attractors for strong dissipation
Let
$f_{\unicode{x3bb} }\colon {\mathbb A} \to {\mathbb A}, (s,r) \mapsto (s',r')$
be the dissipative billiard map within a convex domain
$\Omega \subset \mathbb R^2$
. We use the notation of §3.1.
Proposition 5.1. Assume that
$\Omega \in \mathcal {D}^k$
, where
$\mathcal {D}^k$
has been introduced in Definition D. Then, there exists
$\unicode{x3bb} (\Omega )\in (0,1)$
and a cone-field
$\mathcal {C}=(\mathcal {C}(s,r))_{(s,r)\in {\mathbb A}}$
containing the horizontal direction:
$$ \begin{align*} \mathcal{C}(s,r):=\{v\in T_{(s,r)}{\mathbb A} :v=(v_s,v_r), \vert v_r\vert \leq\eta(r)\vert v_s\vert\}, \end{align*} $$
where
$\eta :[-1,1]\to \mathbb R_+$
is a continuous function, such that for each
$ \unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
and for each
$(s,r)\in \mathbb {T}\times [-\unicode{x3bb} (\Omega ),\unicode{x3bb} (\Omega )]$
,
$$ \begin{align*} Df_{\unicode{x3bb}}(s,r)\mathcal{C}(s,r)\subset \mathrm{int}\, \mathcal{C} (f_{\unicode{x3bb}}(s,r)) \cup\{0\}. \end{align*} $$
Proof. We recall that for any
$(s,r)\in {\mathbb A}$
and
$(s',r'):=f_{\unicode{x3bb} }(s,r)$
,
$\tau (s,r):=\ell (s,s')=\|\Upsilon (s)-\Upsilon (s')\|$
is the Euclidean distance between the consecutive bounces
$\Upsilon (s)$
,
$\Upsilon (s')$
, so that the quantity
$\tau (s)$
in Definition D is merely
$\tau (s) = \tau (s,0)$
. Moreover, since
$\Omega \in \mathcal {D}^k$
, there exists a constant
$c_0>0$
such that
$$ \begin{align*} \max_{s \in \mathbb{T}} \tau(s,0)\mathcal{K}(s)<-1-c_0. \end{align*} $$
By compactness and continuity of the involved functions, we can fix
$\delta _0> 0$
and
$K_0>0$
such that
$$ \begin{align*} \max_{(s,r)\in{\mathbb A}}\tau(s,r)\leq \mathrm{diam}\: \Omega<\delta_0,\quad \max_{s \in \mathbb{T}}|\mathcal{K}(s)|<\mathcal{K}_0. \end{align*} $$
We can find
$\unicode{x3bb} _1\in (0,1)$
small enough such that
where
$\nu (r):= \sqrt {1 - r^2}$
. By equation (3.4), for each
$(s,r)\in \mathrm {int}(\mathbb {A})$
, we have
$$ \begin{align} Df_{\unicode{x3bb}}(s,r) e_1= \begin{bmatrix} - \dfrac{\tau \mathcal{K} + \nu}{\nu'} \\ \unicode{x3bb}(\tau \mathcal{K} \mathcal{K}' + \mathcal{K} \nu'+ \mathcal{K}' \nu ) \end{bmatrix},\quad Df_{\unicode{x3bb}}(s,r) e_2= \begin{bmatrix} \dfrac{\tau}{\nu \nu'} \\ \\ -\unicode{x3bb} \dfrac{\tau \mathcal{K}'+ \nu'}{\nu} \end{bmatrix}, \end{align} $$
where
$e_{1}=(1,0)^T$
,
$e_2(0,1)^T$
are the vectors of the canonical basis. Moreover,
$\mathcal {K}$
,
$\mathcal {K}'$
denote the curvatures at the points corresponding to
$s,s'$
, while
$\nu := \nu (r) = \sqrt {1 - r^2}$
,
$\nu ' := \nu ' (r') = \sqrt {1 - ({r'}/{\unicode{x3bb} })^2}$
.
Let
$\alpha _0:={c_0}/{2\delta _0}>0$
. At each
$(s,r)\in {\mathbb A}$
, we identify
$T_{(s,r)}{\mathbb A}$
with
$\mathbb R^2$
, and define the cone
$$ \begin{align*} \mathcal{C}^{\alpha_0}(s,r):=\{u=a e_1+ b e_2\in \mathbb{R}^2: |b|\leq \alpha_0\nu(r) |a|\}. \end{align*} $$
We note that this cone always contains the horizontal direction
$\mathbb R\times \{0\}$
. With the now fixed
$\unicode{x3bb} \in (0,\unicode{x3bb} _1)$
, let
$(s,r)\in f_{\unicode{x3bb} }(\mathbb {A})=\mathbb {T} \times [-\unicode{x3bb} ,\unicode{x3bb} ]$
,
$(s',r'):=f_{\unicode{x3bb} }(s,r)$
. By equations (5.1) and (5.2), for any
$u=ae_1+be_2\in \mathcal {C}^{\alpha _0}(s,r)$
, its image
$u'$
by
$Df_{\unicode{x3bb} }(s,r)$
is equal to
$$ \begin{align*}u' = \bigg[-a \frac{\tau \mathcal{K} + \nu}{\nu'} + b \frac{\tau}{\nu \nu'} , a \unicode{x3bb}(\tau \mathcal{K} \mathcal{K}' + \mathcal{K} \nu'+ \mathcal{K}' \nu ) - b\unicode{x3bb} \frac{\tau \mathcal{K}'+ \nu'}{\nu} \bigg]^T=: a'e_1+b'e_2,\end{align*} $$
where
$\nu =\nu (r),\nu '=\nu '(r')$
. Thus, we have
$$ \begin{align*} \nu(r') |a'|& \geq \nu'(r') |a'| \geq |a|c_0-|b|\delta_0\geq \frac{c_0}{2}|a|,\\ |b'|&\leq |a| \unicode{x3bb} ((\delta_0\mathcal{K}_0^2+2\mathcal{K}_0)+\alpha_0(\delta_0 \mathcal{K}_0+1)). \end{align*} $$
Now given
$\mu _0\in (0,1)$
, it holds that
$$ \begin{align*} Df_{\unicode{x3bb}}(s,r) \mathcal{C}^{\alpha_0}(s,r)\subset \mathcal{C}^{\mu_0 \alpha_0}(s',r'), \end{align*} $$
provided that
$ \unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
, with
Setting
$\mathcal {C}(s,r)=\mathcal {C}^{\alpha _0}(s,r)$
for each
$(s,r)\in {\mathbb A}$
, we conclude the proof.
To continue on this section, we need to recall some notions and results: the definition of dominated splitting for an invariant set, the definition of normally contracted (hyperbolic) manifold (see e.g. [Reference SambarinoSam16, Definition 2.2] and [Reference Berger and BounemouraBB13], respectively) and a theorem by Hirsch, Pugh, and Shub on the regularity of such normally contracted manifolds (see [Reference Hirsch, Pugh and ShubHPS77]).
Definition 5.2. (Dominated splitting)
Let M be a compact Riemannian manifold without boundary. Let
$f\colon M \to M$
be a
$C^\ell $
diffeomorphism onto its image,
$\ell \geq 1$
. Let K be an invariant set for f. Then, K has a dominated splitting if the tangent bundle over K splits into two subbundles
$T_KM=E\oplus F$
such that:
-
(1) E and F are invariant by
$Df$
; -
(2) the subbundles E and F vary continuously with respect to the point
$x\in K$
; -
(3) there exist
$C>0$
and
$0<\nu <1$
such that for any
$x\in K$
,
$$ \begin{align*} \Vert Df^n(x)\vert_{E}\Vert \,\cdot \Vert Df^{-n}(f^n(x))\vert_{F}\Vert \leq C\, \nu^n\quad \text{for all } n \geq 0. \end{align*} $$
Roughly speaking, the previous definition says that any direction not contained in the subbundle E converges exponentially fast to the direction F under iteration of
$Df$
. For the following definitions of (
$\ell $
-)normal contraction, we refer to [Reference Crovisier and PotrieCP15, Reference Hirsch, Pugh and ShubHPS77].
Definition 5.3. (Normally contracted manifold)
Let M be a compact Riemannian manifold without boundary. Let
$f\colon M \to M$
be a
$C^\ell $
diffeomorphism onto its image,
$\ell \geq 1$
. Let N be a closed
$C^1$
manifold, invariant under f. Then, we say that N is normally contracted if N has a dominated splitting
$T_NM=E^s \oplus TN$
such that
$E^s$
is uniformly contracted, that is, there exists
$n_0\in \mathbb N$
and
$\mu \in (0,1)$
such that for any
$n\geq n_0$
, it holds that
$$ \begin{align*} \|Df^n(x)|_{E^s}\|\leq\mu^n\quad \text{for all } x \in N. \end{align*} $$
Moreover, we say that N is
$\ell $
-normally contracted if the above splitting satisfies the following stronger condition: there exist
$C>0$
and
$0<\nu <1$
such that for any
$x\in N$
and for any
$1 \leq j \leq \ell $
,
$$ \begin{align*} \Vert Df^n(x)\vert_{E^s}\Vert \,\cdot \Vert Df^{-n}(f^n(x))\vert_{TN}\Vert^j \leq C\, \nu^n\quad \text{for all } n \geq 0. \end{align*} $$
Once we have an
$\ell $
-normally contracted manifold, then the following theorem by Hirsch, Pugh, and Shub assures that the manifold is as regular as the dynamics.
Theorem 5.4. [Reference Hirsch, Pugh and ShubHPS77]
Let M be a compact Riemannian manifold without boundary. Let
$f\colon M \to M$
be a
$C^\ell $
diffeomorphism onto its image,
$\ell \geq 1$
. Let N be a closed
$C^1$
manifold, invariant under f. If N is
$\ell $
-normally contracted, then N is actually a
$C^\ell $
manifold.
We can now state an interesting outcome of Proposition 5.1.
Proposition 5.5. Let
$\Omega \in \mathcal {D}^k$
,
$k \geq 2$
. Let
$\unicode{x3bb} (\Omega )\in (0,1)$
be given by Proposition 5.1. Then, for any
$\unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
, the attractor
$\Lambda _{\unicode{x3bb} }^0$
has a dominated splitting
$ E^s\oplus E^c, $
where the bundle
$E^s$
is uniformly contracted, and each point
$(s,r)\in \Lambda _{\unicode{x3bb} }^0$
has a stable manifold
$\mathcal {W}^s(s,r)$
, which is transverse to the horizontal. Moreover, there exists
$0<\unicode{x3bb} '(\Omega )<\unicode{x3bb} (\Omega )$
such that for some
$C>0$
and
$0<\nu <1$
, we have that for any
$\unicode{x3bb} \in (0,\unicode{x3bb} '(\Omega ))$
, for any
$x\in \Lambda _{\unicode{x3bb} }^0$
, and for any
$1 \leq j \leq k-1$
,
$$ \begin{align} \Vert Df_{\unicode{x3bb}}^n(x)\vert_{E^s}\Vert \,\cdot \Vert Df_{\unicode{x3bb}}^{-n}(f_{\unicode{x3bb}}^n(x))\vert_{E^c}\Vert^j \leq C\, \nu^n\quad \text{for all } n \geq 0. \end{align} $$
Proof. Let
$\unicode{x3bb} (\Omega )\in (0,1)$
be as in Proposition 5.1. Take
$\unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
. By the cone-field criterion (see e.g. [Reference Crovisier and PotrieCP15, Theorem 2.6] and [Reference SambarinoSam16, Proposition 2.2]) for the cone-field
$\mathcal {C}=(\mathcal {C}(s,r))_{(s,r)\in {\mathbb A}}$
constructed in Proposition 5.1, we deduce that the attractor
${\Lambda _{\unicode{x3bb} }^0\subset f_{\unicode{x3bb} }(\mathbb {A})}$
has a dominated splitting
$E_{\unicode{x3bb} }^s\oplus E_{\unicode{x3bb} }^c=E^s\oplus E^c$
, where
$E^c(s,r)$
is contained in the horizontal cone
$\mathcal {C}(s,r)$
for each
$(s,r)\in \Lambda _{\unicode{x3bb} }^0$
. Moreover, the fiber bundle
$E^s$
is uniformly contracted. Indeed, by the domination, there exist
$\mu \in (0,1)$
and
$n_0 \in \mathbb {N}$
such that for each
$(s,r)\in \Lambda _{\unicode{x3bb} }^0$
and for each
$n \geq n_0$
,
$$ \begin{align*} \|Df_{\unicode{x3bb}}^n(s,r)|_{E^s}\|\leq \mu^n \|Df_{\unicode{x3bb}}^n(s,r)|_{E^c}\|. \end{align*} $$
Observe that, by equation (3.2), we have
$\det Df_{\unicode{x3bb} }^n(s,r)=\unicode{x3bb} ^n$
. As the angle between
$E^s$
and
$E^c$
is bounded away from zero at each point of
$\Lambda _{\unicode{x3bb} }^0$
, thus uniformly as
$\Lambda _{\unicode{x3bb} }^0$
is compact, and by a change of basis, we conclude that there exists
$n_1 \in \mathbb {N}$
such that for each
$(s,r)\in \Lambda _{\unicode{x3bb} }^0$
and for each
$n \geq n_1$
,
$$ \begin{align*} \|Df_{\unicode{x3bb}}^n(s,r)|_{E^s}\|\leq \unicode{x3bb}^{n/2}, \end{align*} $$
that is, the bundle
$E^s$
is uniformly contracted. In particular, by the stable manifold theorem (see e.g. [Reference Hirsch, Pugh and ShubHPS77] or [Reference Crovisier and PotrieCP15]), each point
$(s,r)\in \Lambda _{\unicode{x3bb} }^0$
has a stable manifold
$\mathcal {W}^s(s,r)$
, which is uniformly transverse to the cone-field
$\mathcal {C}$
which contains the horizontal direction.
The center bundle
$E^c$
is contained in a cone around the horizontal direction and independent of
$\unicode{x3bb} $
. By equation (5.2), the modulus of the projection over the first coordinate of
$Df_{\unicode{x3bb} }(s,r)(1,0)^T$
does not depend on
$\unicode{x3bb} $
. Since the central direction
$E^c$
is contained in a cone around the horizontal direction, we deduce that there exist constants
$0<C_1<C_2$
such that for any
$\unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
, it holds that
$$ \begin{align} C_1\leq \|Df_{\unicode{x3bb}}(s,r)|_{E^c}\|\leq C_2\quad \text{for all } (s,r)\in \Lambda_{\unicode{x3bb}}^0. \end{align} $$
Since
$\det Df_{\unicode{x3bb} }(s,r)=\unicode{x3bb} $
, reasoning as above, we deduce that there exists a constant
$C_3>0$
such that for any
$\unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
, it holds that
$$ \begin{align} \|Df_{\unicode{x3bb}}(s,r)|_{E^s}\|\leq C_3 \unicode{x3bb}\quad \text{for all } (s,r)\in \Lambda_{\unicode{x3bb}}^0. \end{align} $$
By equations (5.4) and (5.5), we conclude that for
$\unicode{x3bb} '(\Omega )\in (0,\unicode{x3bb} (\Omega ))$
sufficiently small, equation (5.3) holds for any
$\unicode{x3bb} \in (0,\unicode{x3bb} '(\Omega ))$
, for any
$x\in \Lambda _{\unicode{x3bb} }^0$
, and for any
$1 \leq j \leq k-1$
.
Remark 5.6. Observe that if we could say a priori that
$\Lambda ^0_{\unicode{x3bb} }$
is a
$C^1$
manifold, then Proposition 5.5 would be saying that
$\Lambda ^0_{\unicode{x3bb} }$
is
$\ell $
-normally contracted.
The following proposition guarantees that the center space
$E^c$
of the dominated splitting of
$\Lambda _{\unicode{x3bb} }^0$
integrates uniquely to the Birkhoff attractor (see [Reference Bonatti and CrovisierBC16] for related results in this direction).
Theorem 5.7. Let
$\Omega \in \mathcal {D}^k$
,
$k \geq 2$
, and let
$\unicode{x3bb} (\Omega )\in (0,1)$
be given by Proposition 5.1. Then, for
$\unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
of
$f_{\unicode{x3bb} }$
coincides with the attractor
$\Lambda _{\unicode{x3bb} }^0$
and is a normally contracted
$C^{1}$
graph over
$\mathbb {T}\times \{0\}$
. Let
$\unicode{x3bb} '(\Omega )<\unicode{x3bb} (\Omega )$
be given by Proposition 5.5. Then, for
$\unicode{x3bb} \in (0,\unicode{x3bb} '(\Omega ))$
,
$\Lambda _{\unicode{x3bb} }=\Lambda _{\unicode{x3bb} }^0$
is actually a
$C^{k-1}$
graph and
$\Lambda _{\unicode{x3bb} }$
converges in the
$C^1$
topology to the zero section
${\mathbb T} \times \{0\}$
as
$\unicode{x3bb} \to 0$
.
Proof. Fix
$\unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
and let
$\mathcal {C}=(\mathcal {C}(s,r))_{(s,r)\in {\mathbb A}}$
be the cone-field in
$\mathbb {T}\times [-\unicode{x3bb} ,\unicode{x3bb} ]$
constructed in Proposition 5.1; let us recall that it contains the horizontal direction, as
$\mathcal {C}(s,r)=\{v\in T_{(s,r)}{\mathbb A} : v=(v_s,v_r), \vert v_r\vert \leq \alpha _0\,\nu (r)\,\vert v_s\vert \}$
. Let
$$ \begin{align*} \mathscr{F}:=\{\gamma\colon\mathbb{T}\to [-\unicode{x3bb},\unicode{x3bb}]\text{ such that }\gamma\in C^1({\mathbb T})\text{ and }(1,\gamma'(s))\in \mathcal{C}(s,\gamma(s))\text{ for all } s \in {\mathbb T} \}. \end{align*} $$
The map
$f_{\unicode{x3bb} }$
acts on
$\mathscr {F}$
by the graph transform
$$ \begin{align*} \mathscr{G}_{f_{\unicode{x3bb}}}\colon \mathscr{F} \to \mathscr{F},\quad \gamma \mapsto (s \mapsto \pi_2 \circ f_{\unicode{x3bb}} (g_{\unicode{x3bb}}^{-1}(s),\gamma(g_{\unicode{x3bb}}^{-1}(s)) )), \end{align*} $$
where
$\pi _1,\pi _2$
denote the projection on the first and second coordinate, respectively, and
$g_{\unicode{x3bb} }\colon {\mathbb T} \to {\mathbb T}$
is the map
$s\mapsto \pi _1 \circ f_{\unicode{x3bb} }(s,\gamma (s))$
. Indeed, the cone-field
$\mathcal {C}$
around the horizontal direction is contracted by the dynamics, that is,
$Df_{\unicode{x3bb} }(s,r)\mathcal {C}(s,r)\subset \mathrm {int}(\mathcal {C}(f_{\unicode{x3bb} }(s,r)))\cup \{0\}$
, and hence for any
$\gamma \in \mathscr {G}_{f_{\unicode{x3bb} }}$
, the image by
$f_{\unicode{x3bb} }$
of the graph of
$\gamma $
is still the graph of a
$C^1$
function, such that the vector tangent to
$f_{\unicode{x3bb} }(\mathrm {graph}(\gamma ))$
is in
$\mathcal {C}$
, and
$\pi _1 \circ f_{\unicode{x3bb} }|_{\mathrm {graph}(\gamma )}$
is a homeomorphism between
$\mathrm {graph}(\gamma )$
and
$\mathbb {T}$
. In particular,
$f_{\unicode{x3bb} }(\mathrm {graph}(\gamma ))=\mathrm {graph}(\mathscr {G}_{f_{\unicode{x3bb} }}(\gamma ))$
for a well-defined function
$\mathscr {G}_{f_{\unicode{x3bb} }}(\gamma )\in \mathscr {F}$
.
For any
$k\in \mathbb {N}$
, let us denote
$\mathbb {A}_{k}:=f_{\unicode{x3bb} }^{k}(\mathbb {A})$
and let
$\mathscr {F}_k$
be the subset of functions
$\gamma \in \mathscr {F}$
whose graph is contained in
${\mathbb A}_k$
. Note that
$\mathscr {F}_{k+1}\subset \mathscr {F}_k$
and, by construction, it holds that
$\mathscr {F}= \mathscr {F}_1$
. Moreover, if
$\gamma \in \mathscr {F}_k$
,
$k \geq 1$
, then it holds that
$\mathscr {G}_{f_{\unicode{x3bb} }}(\gamma )\in \mathscr {F}_{k+1}$
.
In the following, let
$\|\cdot \|_\infty $
be the sup-norm on the space
$C^0({\mathbb T},[-1,1])$
. That is, for
$\gamma _1,\gamma _2\in C^0({\mathbb T},[-1,1])$
, we let
$\|\gamma _2-\gamma _1\|_\infty :=\max _{s \in \mathbb {T}}|\gamma _2(s)-\gamma _1(s)|$
. The graph transform acts as a contraction on
$\mathscr {F}$
for
$\|\cdot \|_\infty $
.
Claim 5.8. There exists a constant
$c>0$
such that for any
$\gamma _1,\gamma _2\in \mathscr {F}$
, it holds that
$$ \begin{align*} \|\mathscr{G}_{f_{\unicode{x3bb}}}^n(\gamma_2)-\mathscr{G}_{f_{\unicode{x3bb}}}^n(\gamma_1)\|_\infty\leq c\unicode{x3bb}^{n/2} \|\gamma_2-\gamma_1\|_\infty\quad \text{for all } n \geq 0. \end{align*} $$
Proof of the claim
For
$k_0 \geq 1$
sufficiently large,
$\mathbb {A}_{k_0}$
is foliated by stable leaves
$\{\mathcal {W}^s(x)\cap \mathbb {A}_{k_0}:x\in \Lambda _{\unicode{x3bb} }^0\}$
, and by the transversality between
$E^s$
and
$\mathcal {C}$
on
$\Lambda _{\unicode{x3bb} }^0$
, there exists
$\theta _0>0$
such that
$$ \begin{align} \angle(T_{x}\mathcal{W}^s(x),T_x\Gamma_\gamma)\geq \theta_0\quad \text{for all } x\in \mathrm{graph}(\gamma),\, \gamma \in \mathscr{F}_{k_0}, \end{align} $$
where
$\angle $
denotes the (non-oriented) angle between the considered vector subspaces.
Let
$\gamma _1,\gamma _2\in \mathscr {F}_{k_0}$
. For each
$s \in \mathbb {T}$
, we denote by
$H_{\gamma _1,\gamma _2}^s(s)\in \mathrm {graph}(\gamma _2)$
the image of
$(s,\gamma _1(s))\in \mathrm {graph}(\gamma _1)$
by the holonomy map from
$\mathrm {graph}(\gamma _1)$
to
$\mathrm {graph}(\gamma _2)$
along the leaves of
$\mathcal {W}^s$
. That is, follow the stable leaf passing through
$(s,\gamma _1(s))$
until it intersects the graph of
$\gamma _2$
: such intersection point is
$H^s_{\gamma _1,\gamma _2}(s)$
.
For
$n \geq 1$
, denote by
$\gamma _i^n$
the image
$\mathscr {G}_{f_{\unicode{x3bb} }}^n(\gamma _i)$
,
$i=1,2$
. Then, by equation (5.6), there exists a constant
$c>1$
such that for each
$\gamma _1,\gamma _2\in \mathscr {F}_{k_0}$
,
$$ \begin{align} c^{-1} |\gamma_2(s)-\gamma_1(s)|\leq d_{\mathcal{W}^s}((s,\gamma_1(s)),H_{\gamma_1,\gamma_2}^s(s))\leq c |\gamma_2(s)-\gamma_1(s)|\quad \text{for all } s \in \mathbb{T}, \end{align} $$
where
$d_{\mathcal {W}^s}$
denotes the distance along a stable leaf, induced by the restriction to this leaf of the Riemannian metric. Moreover, since
$(s,\gamma _1(s))$
and
$H_{\gamma _1,\gamma _2}^s(s))$
belong to the same stable leaf, there exists
$n_0\in \mathbb {N}$
such that for
$n \geq n_0$
,
$$ \begin{align} d_{\mathcal{W}^s}(f_{\unicode{x3bb}}^n (s,\gamma_1(s)),f_{\unicode{x3bb}}^n \circ H_{\gamma_1,\gamma_2}^s(s))\leq \unicode{x3bb}^{n/2} d_{\mathcal{W}^s}((s,\gamma_1(s)),H_{\gamma_1,\gamma_2}^s(s))\quad \text{for all } s \in \mathbb{T}. \end{align} $$
For each
$s\in \mathbb {T}$
and
$n \geq 0$
, let us set
$s_{-n}:=g_{\unicode{x3bb} }^{-n}(s)$
, with
$g_{\unicode{x3bb} } \colon s\mapsto \pi _1 \circ f_{\unicode{x3bb} }(s,\gamma _1(s))$
as above, so that
$f_{\unicode{x3bb} }^n (s_{-n},\gamma _1(s_{-n}))=(s,\gamma ^n_1(s))$
and
$f_{\unicode{x3bb} }^n \circ H_{\gamma _1,\gamma _2}^s(s_{-n})=H_{\gamma _1^n,\gamma _2^n}^s(s)$
. Indeed, for
$i=1,2$
,
$f_{\unicode{x3bb} }^n$
sends the graph of
$\gamma _i$
to the graph of
$\gamma _i^n$
, and it sends stable leaves to stable leaves. Then, by equations (5.7)–(5.8), for
$n \geq n_0$
, it holds that
$$ \begin{align*} |\gamma_2^n(s)-\gamma_1^n(s)|&\leq c d_{\mathcal{W}^s}((s,\gamma_1^n(s)),H_{\gamma_1^n,\gamma_2^n}^s(s))\\ &=c d_{\mathcal{W}^s}(f_{\unicode{x3bb}}^n (s_{-n},\gamma_1(s_{-n})),f_{\unicode{x3bb}}^n \circ H_{\gamma_1,\gamma_2}^s(s))\\ &\leq c\unicode{x3bb}^{n/2} d_{\mathcal{W}^s}((s_{-n},\gamma_1(s_{-n})),H_{\gamma_1,\gamma_2}^s(s_{-n}))\\ &\leq c^2 \unicode{x3bb}^{n/2} |\gamma_2(s_{-n})-\gamma_1(s_{-n})|\leq c^2 \unicode{x3bb}^{n/2} \|\gamma_2-\gamma_1\|_\infty. \end{align*} $$
Now, for any
$\bar \gamma _1,\bar \gamma _2\in \mathscr {F}=\mathscr {F}_1$
, their images under
$\mathscr {G}_{f_{\unicode{x3bb} }}^{k_0-1}$
are in
$\mathscr {F}_{k_0}$
, and hence, up to enlarging the constant c, we conclude that for any
$n \geq 0$
,
$$ \begin{align*} \|\mathscr{G}_{f_{\unicode{x3bb}}}^n(\bar\gamma_2)-\mathscr{G}_{f_{\unicode{x3bb}}}^n(\bar\gamma_1)\|_\infty\leq c^2\unicode{x3bb}^{n/2} \|\bar\gamma_2-\bar\gamma_1\|_\infty.\\[-37pt] \end{align*} $$
Let
$\gamma _0^\pm \colon s\in \mathbb {T} \mapsto \pm \unicode{x3bb} \in (-1,1)$
; clearly,
$\gamma _0^\pm \in \mathscr {F}$
. For
$n \geq 0$
, let
$\gamma _{n}^\pm :=\mathscr {G}_{f_{\unicode{x3bb} }}^{n}(\gamma _0^\pm )$
. Observe that for any
$n\geq 0$
,
$\gamma ^\pm _{n}$
is in
$\mathscr {F}_{n+1}$
. By Claim 5.8, the sequences
$(\gamma _n^\pm )_{n \geq 0}$
are Cauchy sequences, and hence converge to a continuous function
$\gamma _\infty ^\pm $
.
Claim 5.9. The equality
$\gamma _\infty ^+=\gamma _\infty ^-=:\gamma _\infty $
holds and
$\Lambda _{\unicode{x3bb} }^0=\Lambda _{\unicode{x3bb} }=\mathrm {graph}(\gamma _\infty )$
.
Proof. Let us denote by
$\Gamma _\infty ^\pm $
the graph of
$\gamma _\infty ^\pm $
. By construction,
$\Gamma _\infty ^\pm $
is invariant under
$f_{\unicode{x3bb} }$
, it is compact, connected, and separates the annulus, that is,
$\Gamma _\infty ^\pm \in \mathcal {X}(f_{\unicode{x3bb} })$
. On one hand, by Proposition 2.5, we have
$\Lambda _{\unicode{x3bb} }\subset \Gamma _\infty ^\pm $
. On the other hand, by the graph property, for any
$x \in \Gamma _\infty ^\pm $
, the set
$\Gamma _\infty ^\pm \setminus \{x\}$
does not separate the annulus. By Lemma 2.6, we conclude that
$\Gamma _\infty ^+=\Gamma _\infty ^-=\Lambda _{\unicode{x3bb} }$
.
We can now show that the Birkhoff attractor coincides with the attractor. By definition of the attractor
$\Lambda _{\unicode{x3bb} }^0$
, we have
$\Lambda _{\unicode{x3bb} }^0 \subset \bigcap _{n \geq 0}{\mathbb A}_n$
. Observe that for each
$n \geq 0$
,
${\mathbb A}_n$
is a cylinder bounded by the graphs of
$\gamma _n^\pm $
. Since the sequences
$(\mathrm {graph}(\gamma _n^\pm ))_{n \geq 0}$
converge to the same limit graph
$\Gamma _\infty ^+=\Gamma _\infty ^-=\Lambda _{\unicode{x3bb} }$
, it follows that
$\Lambda _{\unicode{x3bb} }^0\subseteq \Gamma _\infty ^+=\Gamma _\infty ^-=\Lambda _{\unicode{x3bb} }\subseteq \Lambda ^0_{\unicode{x3bb} }$
.
Claim 5.10. The function
$\gamma _\infty $
is
$C^1$
, and
$(1,\gamma ^{\prime }_\infty (s))\in E^c(s,\gamma _\infty (s))$
for every
$s\in {\mathbb T}$
.
Proof. The function
$\gamma _\infty =\gamma _\infty ^+$
is the
$C^0$
limit of the sequence of
$C^1$
functions
$(\gamma _n^+)_{n \geq 0}$
. To show that
$\gamma _\infty $
is
$C^1$
, it suffices to show that the derivatives
$((\gamma _n^+)')_{n \geq 0}$
also converge uniformly. For each
$n \geq 0$
, let us denote by
$\Gamma _n^+$
the graph of
$\gamma _n^+$
. With the same notation as in Claim 5.8, for each
$s \in \mathbb {T}$
, it holds that
$$ \begin{align*} T_{(s,\gamma_n^+(s))}\Gamma_{n}^+=Df_{\unicode{x3bb}}^n(s_{-n},\gamma_0^+(s_{-n}))T_{(s_{-n},\gamma_0^+(s_{-n}))}\Gamma_{0}^+\subset Df_{\unicode{x3bb}}^n(s_{-n},\gamma_0^+(s_{-n})) \mathcal{C}(s_{-n},\gamma_0^+(s_{-n})), \end{align*} $$
and by the cone-field criterion, the cone
$Df_{\unicode{x3bb} }^n(s_{-n},\gamma _0^+(s_{-n})) \mathcal {C}(s_{-n},\gamma _0^+(s_{-n}))$
is exponentially small with respect to n, uniformly in
$s \in {\mathbb T}$
, that is, the amplitude of each cone
$Df_{\unicode{x3bb} }^n \mathcal {C}$
is, up to a uniform constant, equal to
$\mu ^n$
times the amplitude of the cone
$\mathcal {C}$
for some uniform constant
$\mu \in (0,1)$
. We conclude that the sequence
$((\gamma _n^+)')_{n \geq 0}$
converges in the
$C^1$
-topology. Moreover, it also implies that at any point
$(s,\gamma _\infty (s))=\lim _{n \to +\infty } (s,\gamma _n^+(s))$
, the tangent space of the limit graph
$\Gamma _{\infty }$
is equal to
$E^c(s,\gamma _\infty (s))$
.
So far, we have shown that the (Birkhoff) attractor
$\Lambda _{\unicode{x3bb} }^0=\Lambda _{\unicode{x3bb} }$
is the graph of a
$C^1$
function
$\gamma _\infty $
. Moreover, the graph of
$\gamma _\infty $
is tangent to
$E^c$
, and so
$\Lambda _{\unicode{x3bb} }$
is a normally contracted
$C^1$
graph, since
$$ \begin{align*} T{\mathbb A}|_{\Lambda_{\unicode{x3bb}}}=E^s\oplus E^c=E^s \oplus T\Lambda_{\unicode{x3bb}}. \end{align*} $$
Now, let
$\unicode{x3bb} '(\Omega )<\unicode{x3bb} (\Omega )$
be given by Proposition 5.5. The domain
$\partial \Omega $
is of class
$C^k$
, and thus, the dissipative billiard map
$f_{\unicode{x3bb} }$
is of class
$C^{k-1}$
for any
$\unicode{x3bb} \in (0,1)$
. Fix
$\unicode{x3bb} \in (0,\unicode{x3bb} '(\Omega ))$
. Since
$T\Lambda _{\unicode{x3bb} }=E^c$
and by equation (5.3), we deduce that
$\Lambda _{\unicode{x3bb} }$
is
$(k-1)$
-normally contracted in the sense of Definition 5.3, and hence by Theorem 5.4, the function
$\gamma _\infty $
is actually
$C^{k-1}$
.
By construction,
$\Lambda _{\unicode{x3bb} }=\Lambda _{\unicode{x3bb} }^0\subset f_{\unicode{x3bb} }({\mathbb A})={\mathbb T} \times [-\unicode{x3bb} ,\unicode{x3bb} ]$
, and hence,
$\Lambda _{\unicode{x3bb} }$
converges to the zero section
${\mathbb T} \times \{0\}$
in the
$C^0$
-topology. To show the convergence in the
$C^1$
-topology, it suffices to show that
$T \Lambda _{\unicode{x3bb} }=E^c$
converges uniformly to the horizontal space. By construction, at any
$s \in {\mathbb T}$
,
$E^c(s,\gamma ^\infty (s))\subset Df_{\unicode{x3bb} }(s_{-1},\gamma _0^+(s_{-1})) \mathcal {C}(s_{-1},\gamma _0^+(s_{-1}))$
, and by equation (5.2), the vertical component of vectors in the latter cone is less than
$\tilde c \unicode{x3bb} $
for some constant
$\tilde c>0$
.
5.2. Examples and further consequences
As a first consequence of Theorem 5.7, we prove that, when the dissipation is strong, that is,
$\unicode{x3bb} $
is close to zero, the Birkhoff attractor of ellipses is a normally contracted
$C^1$
graph. Given an ellipse
$\mathcal {E}$
of non-zero eccentricity, we let
$\{E_1,E_2\}$
be the
$2$
-periodic orbit corresponding to the minor axis; it is a sink, by Lemma 4.4. Then, we define
that is,
$\unicode{x3bb} _-(\mathcal {E})=\unicode{x3bb} _-(p)$
for
$p = E_1$
or
$E_2$
as in equation (3.7).
By Theorem 4.6, we have
$\Lambda _{\unicode{x3bb} }= \overline {\mathcal {W}^u(H_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}^u(H_2;f_{\unicode{x3bb} }^2)}$
, and for
$i=1,2$
,
$\mathcal {W}^u(H_i;f_{\unicode{x3bb} }^2)\setminus \{H_i\}$
is the disjoint union of two branches
$\mathscr {C}_i^1,\mathscr {C}_i^2$
, with
$\overline {\mathscr {C}_i^j}=\mathscr {C}_i^j\cup \{H_i,E_j\}$
,
$j=1,2$
. Thus,
$\Lambda _{\unicode{x3bb} }$
is a manifold if and only if for
$i,j \in \{1,2\}$
, the tangent space
$T_x \mathscr {C}_i^j=T_x \Lambda _{\unicode{x3bb} }$
has a limit
$V_i^j$
as
$\mathscr {C}_i^j\ni x \to E_j$
. Indeed, by Corollary 3.4, if so, we necessarily have
$V_1^j=V_2^j$
. Clearly, a necessary condition for this to hold is that the eigenvalues of
$Df_{\unicode{x3bb} }^2(E_i)$
are real for
$i=1,2$
, that is,
$\unicode{x3bb} \in (0,\unicode{x3bb} _-(\mathcal {E}))$
. Actually, the following holds.
Corollary 5.11. Let
$f_{\unicode{x3bb} } \colon \mathbb {A} \to \mathbb {A}$
be the dissipative billiard map inside an ellipse
$\mathcal {E}$
with eccentricity
$e\in (0,{\sqrt 2}/{2})$
. Then, there exists
$\unicode{x3bb} (\mathcal {E})<\unicode{x3bb} _-(\mathcal {E})$
such that, for
$\unicode{x3bb} \in (0,\unicode{x3bb} (\mathcal {E}))$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }= \overline {\mathcal {W}^u(H_1;f_{\unicode{x3bb} }^2)\cup \mathcal {W}^u(H_2;f_{\unicode{x3bb} }^2)}$
is a normally contracted
$C^1$
graph, which is actually
$C^\infty $
except possibly at
$E_i$
,
$i=1,2$
, where
$\Lambda _{\unicode{x3bb} }$
is tangent to the weak stable space of
$Df_{\unicode{x3bb} }^2(E_i)$
.
Proof. Let
$\Upsilon \colon \mathbb {T} \to \mathbb R^2$
be a parameterization of the boundary by arclength such that
$\Upsilon (0),\Upsilon (\tfrac 12)$
correspond to the trace on
$\mathcal {E}$
of the
$2$
-periodic orbit of maximal length. For each
$s \in \mathbb {T}$
, let
$\tau (s)=\tau (s,0):=\|\Upsilon (s)-\Upsilon (s')\|$
, where
$\Upsilon (s),\Upsilon (s')$
are the two points of intersection of
$\mathcal {E}$
and the normal to
$\mathcal {E}$
at
$\Upsilon (s)$
, and let
$\mathcal {K}(s)\leq 0$
be the curvature at
$\Upsilon (s)$
. The function
$[0,\tfrac 14]\ni s \mapsto \tau (s) \mathcal {K}(s)$
is increasing, with
$\tau (\tfrac 14)\mathcal {K}(\tfrac 14)=-2({a_2}/{a_1})^2=2(e^2-1)$
, where
$e>0$
is the eccentricity of
$\mathcal {E}$
. Thus, if
$e\in (0,{\sqrt {2}}/{2})$
, the domain bounded by
$\mathcal {E}$
is in
$\mathcal {D}^\infty $
(recall Definition D), and hence by Proposition 5.5 and Theorem 5.7, there exists
$\unicode{x3bb} (\mathcal {E})<\unicode{x3bb} _-(\mathcal {E}))$
such that for
$\unicode{x3bb} \in (0,\unicode{x3bb} (\mathcal {E}))$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
is a normally contracted
$C^1$
graph. In fact, it is
$C^\infty $
everywhere except possibly at
$E_1,E_2$
; indeed, near any other point, it coincides with some piece of the unstable manifold of
$H_1$
or
$H_2$
, which is
$C^\infty $
. By Lemma 3.5, the eigenvalues
$\mu _1,\mu _2$
of
$Df_{\unicode{x3bb} }^2(E_i)$
satisfy
$\unicode{x3bb} ^2<\mu _1<\mu _2<1$
. As
$\Lambda _{\unicode{x3bb} }$
is
$C^1$
and
$f_{\unicode{x3bb} }$
-invariant, for
$i=1,2$
, any tangent vector
$v\in T_{E_i}\Lambda _{\unicode{x3bb} }$
is an eigenvector of
$Df_{\unicode{x3bb} }^2(E_i)$
; since
$\Lambda _{\unicode{x3bb} }$
is normally contracted, any such v has to be in the eigenspace associated to the weak eigenvalue
$\mu _2$
.
Remark 5.12. If the eccentricity is larger than
${\sqrt 2}/{2}$
, then we loose the graph property, even for small dissipation parameters
$\unicode{x3bb} \in (0,1)$
, see Proposition 5.16 below.
Remark 5.13. It was asked to us by Viktor Ginzburg whether the phase space
${\mathbb A}$
of dissipative billiards admits an invariant foliation by curves homotopic to the zero-section
${\mathbb T} \times \{0\}$
. Indeed, in the case of a dissipative billiard within a circle considered at the beginning of §4, it is clear that the horizontal foliation
$\{{\mathbb T} \times \{r\}\}_{r \in [-1,1]}$
is preserved by any dissipative map
$f_{\unicode{x3bb} }$
,
$\unicode{x3bb} \in (0,1)$
. More generally, the existence of such foliations seems much less rigid than in the conservative case, where it is related to the famous Birkhoff conjecture (see e.g. [Reference Avila, De Simoi and KaloshinADSK16, Reference Bialy and MironovBM22, Reference Kaloshin and SorrentinoKS18] for recent progress in this direction).
Indeed, fix a domain
$\Omega \in \mathcal {D}^k$
,
$k \geq 2$
, and a dissipation parameter
$\unicode{x3bb} \in (0,\unicode{x3bb} (\Omega ))$
. With the notation of Theorem 5.7, for any
$k\geq 0$
, let
${\mathbb A}_k:=f_{\unicode{x3bb} }^k({\mathbb A})$
and let
$\mathcal {F}_1$
be a foliation of
${\mathbb A}_1\setminus {\mathbb A}_2$
defined as follows. Note that
${\mathbb A}_1\setminus {\mathbb A}_2$
has two connected components
${\mathbb A}_1^+$
and
${\mathbb A}_1^-$
, where
${\mathbb A}_1^\pm $
is bounded by the leaves
${\mathbb T} \times \{\pm \unicode{x3bb} \}$
(in
${\mathbb A}_1^\pm $
) and
$f_{\unicode{x3bb} }({\mathbb T} \times \{\pm \unicode{x3bb} \})$
(in the complement of
${\mathbb A}_1^\pm $
). Let then
$\mathcal {F}_1$
be the disjoint union of two foliations
$\mathcal {F}_1^+$
and
$\mathcal {F}_1^-$
, where
$\mathcal {F}_1^\pm $
is a foliation of
${\mathbb A}_1^\pm $
by
$C^1$
graphs over
${\mathbb T}\times \{0\}$
whose tangent space remains in the cone-field
$\mathcal {C}$
constructed in Proposition 5.1, and whose boundary leaves are
${\mathbb T} \times \{\pm \unicode{x3bb} \}$
and
$f_{\unicode{x3bb} }({\mathbb T} \times \{\pm \unicode{x3bb} \})$
. For
$k \geq 0$
, let
$\mathcal {F}_k$
be the foliation of
${\mathbb A}_{k}\setminus {\mathbb A}_{k+1}$
whose leaves are images by
$f_{\unicode{x3bb} }^{k-1}$
of the leaves of
$\mathcal {F}_1$
. Since the cone-field
$\mathcal {C}|_{{\mathbb T}\times [-\unicode{x3bb} ,\unicode{x3bb} ]}$
is contracted under forward iteration, for each
$k \geq 1$
, the leaves of
$\mathcal {F}_k$
are
$C^1$
graphs over
${\mathbb T} \times \{0\}$
whose tangent space is contained in the cone-field
$\mathcal {C}$
. Moreover, the same argument as in the proof of Theorem 5.7 says that the collection of leaves of
$\mathcal {F}_k$
converges uniformly to the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
in the
$C^1$
-topology as
$k \to +\infty $
. Let
$\mathcal {F}_0:=f_{\unicode{x3bb} }^{-1}(\mathcal {F}_1)$
, and let
$\mathcal {F}$
be the foliation
$\mathcal {F}:=\sqcup _{k \geq 0} \mathcal {F}_k\sqcup \Lambda _{\unicode{x3bb} }$
. By construction, it is a foliation of
${\mathbb A}$
by
$C^1$
curves and it is (forward-)invariant under
$f_{\unicode{x3bb} }$
. Moreover, the leaves of
$\mathcal {F}\cap {\mathbb A}_1$
are
$C^1$
graphs over
${\mathbb T}\times \{0\}$
.
As
$\unicode{x3bb} $
gets increasingly smaller, the Birkhoff attractor of
$f_{\unicode{x3bb} }$
is contained in an inccreasingly smaller strip around the zero section; actually, we can use the
$C^1$
convergence of the Birkhoff attractor to
${\mathbb T}\times \{0\}$
to deduce interesting information on the dynamics of
$f_{\unicode{x3bb} }\vert _{\Lambda _{\unicode{x3bb} }}$
, when
$\unicode{x3bb} \in (0,1)$
is small, from the degenerate one-dimensional dynamics of
$f_0$
, namely when
$\unicode{x3bb} =0$
.
Theorem 5.14. Let
$k \geq 2$
. For a
$C^k$
-generic billiard
$\Omega \in \mathcal {D}^k$
, there exists
$\unicode{x3bb} "(\Omega )\in (0,1)$
such that for any
$\unicode{x3bb} \in (0,\unicode{x3bb} "(\Omega ))$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
is a
$C^{k-1}$
normally contracted graph of rotation number
$\tfrac 12$
, and, moreover,
$$ \begin{align*} \Lambda_{\unicode{x3bb}}=\bigcup_{i=1}^\ell \overline{\mathcal{W}^u(H_i;f^2_{\unicode{x3bb}}) \cup \mathcal{W}^u(f_{\unicode{x3bb}}(H_i);f^2_{\unicode{x3bb}})}, \end{align*} $$
for some finite collection
$\{H_i,f_{\unicode{x3bb} }(H_i)\}_{i=1,\ldots ,\ell }$
of
$2$
-periodic orbits of saddle type.
Proof. Let
$\Omega \in \mathcal {D}^k$
be a
$C^k$
-generic billiard as in Corollary 3.9 and let
$\unicode{x3bb} '(\Omega ) \in (0,1)$
be given by Theorem 5.7. For any
$\unicode{x3bb} \in (0,\unicode{x3bb} '(\Omega ))$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
of
$f_{\unicode{x3bb} }$
is normally contracted, and is equal to the graph
$\Gamma _{\gamma _{\unicode{x3bb} }}$
of some
$C^{k-1}$
function
$\gamma _{\unicode{x3bb} } \colon {\mathbb T} \to [-1,1]$
. We let
$g_{\unicode{x3bb} } \colon {{\mathbb T}}\to {{\mathbb T}}$
be the circle map
${{\mathbb T}}\ni s \mapsto \pi _1 \circ f_{\unicode{x3bb} }(s,\gamma _{\unicode{x3bb} }(s))$
induced by
$f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }}$
, where
$\pi _1\colon {\mathbb A}={{\mathbb T}} \times [-1,1] \to {{\mathbb T}}$
is the projection over the first coordinate. For any
$s \in {\mathbb T}$
, let
$(s_1,\gamma _{\unicode{x3bb} }(s_1)):=f_{\unicode{x3bb} }(s,\gamma _{\unicode{x3bb} }(s))$
. By equation (3.4), it holds that
$$ \begin{align} g_{\unicode{x3bb}}'(s)= -\frac{\tau(s,\gamma_{\unicode{x3bb}}(s)) \mathcal{K}(s) + \nu(s)}{\nu'(s)} +\frac{\tau(s,\gamma_{\unicode{x3bb}}(s))}{\nu(s) \nu'(s)}\gamma_{\unicode{x3bb}}'(s), \end{align} $$
where
$\tau (s,\gamma _{\unicode{x3bb} }(s))$
is the length of the orbit segment for
$f_{\unicode{x3bb} }$
(also of
$f_1$
) connecting the points
$\Upsilon (s)$
and
$\Upsilon (s_1)$
,
$\mathcal {K}(s)$
is the curvature at
$\Upsilon (s)$
, and
$\nu (s) = \sqrt {1 - \gamma _{\unicode{x3bb} }^2(s)}$
,
$\nu ' (s):= \sqrt {1 - ({\gamma _{\unicode{x3bb} }(s_1)}/{\unicode{x3bb} })^2}$
.
Let us note that the function
${\mathbb T}\ni s\mapsto \pi _1\circ f_{\unicode{x3bb} }(s,0)$
is independent of the value of
$\unicode{x3bb} \in [0,1]$
. In particular, for any
$\unicode{x3bb} \in (0,1)$
, it holds that
$\pi _1\circ f_{\unicode{x3bb} }|_{{\mathbb T} \times \{0\}}=\pi _1\circ f_1|_{{\mathbb T} \times \{0\}}=\pi _1\circ f_0|_{{\mathbb T} \times \{0\}}$
. We denote such a function by
$g_0$
. Note that the function
$g_{\unicode{x3bb} }=\pi _1\circ f_{\unicode{x3bb} } |_{\Lambda _{\unicode{x3bb} }}$
is
$C^0$
-converging to
$g_0$
as
$\unicode{x3bb} \to 0$
, since
$\Lambda _{\unicode{x3bb} }$
converges to the zero section by Theorem 5.7. The extended family
$(g_{\unicode{x3bb} })_{\unicode{x3bb} \in [0,\unicode{x3bb} '(\Omega ))}$
satisfies the following claim.
Claim 5.15. The family of maps
$(g_{\unicode{x3bb} })_{\unicode{x3bb} \in [0,\unicode{x3bb} '(\Omega ))}$
depends continuously on
$\unicode{x3bb} $
in the
$C^1$
-topology.
Proof. By the theory of normally contracted invariant manifolds and their persistence (see e.g. [Reference Berger and BounemouraBB13, Theorem 2.1 and Corollary 2.2]),
$\gamma _{\unicode{x3bb} }$
depends continuously on
$\unicode{x3bb} \in (0,\unicode{x3bb} '(\Omega ))$
in
$C^{k-1}$
-topology, and hence
$g_{\unicode{x3bb} }$
also depends continuously on
$\unicode{x3bb} $
in
$C^{k-1}$
-topology. Moreover, by Theorem 5.7,
$\gamma _{\unicode{x3bb} }$
converges to the zero function
$0$
in the
$C^1$
-topology as
$\unicode{x3bb} \to 0$
, and hence
$g_{\unicode{x3bb} }$
converges to the map
$g_0$
in the
$C^1$
-topology.
In particular,
$g_{\unicode{x3bb} }$
converges to the map
$g_0$
in the
$C^1$
-topology, with
$g_0^{\prime }\colon s \mapsto -\tau (s,0) \mathcal {K}(s)-1$
, where
$\tau (s,0)$
is the length of the first orbit segment for
$f_0$
(also for
$f_1$
) starting at
$(s,0)$
and
$\mathcal {K}(s)$
is the curvature at
$\Upsilon (s)$
. Note that
$-\tau (s,0) \mathcal {K}(s)-1\neq 0$
, since
$\Omega $
is in
$\mathcal {D}^k$
. In particular,
$g_0$
is a circle diffeomorphism. Let us denote by
$\mathrm {II}$
the set of
$2$
-periodic points of the family
$\{f_{\unicode{x3bb} }\}_{\unicode{x3bb} \in [0,1]}$
. As already observed, the set
$\Pi $
is common to every
$f_{\unicode{x3bb} }$
. Since the set
$\mathrm {II}$
is contained in the zero section
${\mathbb T}\times \{0\}$
, the circle diffeomorphism
$g_0$
has rotation number
$\tfrac 12$
. Moreover, by Corollary 3.9, for a
$C^k$
-generic domain
$\Omega $
, for any
$\unicode{x3bb} \in [0,1)$
, all the
$2$
-periodic points of the billiard map
$f_{\unicode{x3bb} }$
are either saddles or sinks. In particular, the latter persist under perturbation, even when we consider the one-dimensional dynamics on the corresponding Birkhoff attractor, as we are going to show. In fact, for any
$2$
-periodic point
$p=(s,0)\in \mathrm {II}$
, denoting by
$\mathcal {K}_1,\mathcal {K}_2$
the curvatures at the two bounces and by
$\tau $
the Euclidean distance between the two bounces, according to equation (5.9), the multiplier
$(g_0^2)'(s)=({d}/{ds})g_0(g_0(s))$
is equal to
$k_{1,2}:=(\tau \mathcal {K}_1 + 1)(\tau \mathcal {K}_2 + 1)$
. In particular, for the circle diffeomorphism
$g_0$
, the
$2$
-periodic point s is repelling when
$|k_{1,2}|>1$
and attracting when
$|k_{1,2}|<1$
. By Claim 5.15, for
$\unicode{x3bb}>0$
small, the circle diffeomorphism
$g_{\unicode{x3bb} }$
is
$C^1$
-close to
$g_0$
. Thus, for any
$p=(s,0)\in \mathrm {II}$
, the
$2$
-periodic point s for
$g_0$
admits a continuation for
$g_{\unicode{x3bb} }$
. Since the set
$\Pi $
is common to all functions
$f_{\unicode{x3bb} }$
and since generically
$2$
-periodic points are isolated, we deduce that s is also
$2$
-periodic for
$g_{\unicode{x3bb} }$
. Therefore, there exists
$\unicode{x3bb} "(\Omega )\in (0,\unicode{x3bb} '(\Omega ))$
such that for any
$\unicode{x3bb} \in (0,\unicode{x3bb} "(\Omega ))$
, the restriction
$f_{\unicode{x3bb} }|_{\Lambda _{\unicode{x3bb} }}$
still has rotation number
$\tfrac 12$
. Observe that, on the one hand, if
$p=(s,0)$
is a sink for
$f_{\unicode{x3bb} }$
, then s is an attracting
$2$
-periodic point for
$g_{\unicode{x3bb} }$
; on the other hand, if
$p=(s,0)$
is of saddle type for
$f_{\unicode{x3bb} }$
, then s is a
$2$
-periodic repulsive point, because the Birkhoff attractor is normally contracted. By standard facts of the theory of circle homeomorphisms with rational rotation number, the
$\alpha $
-limit set
$\alpha _{f_{\unicode{x3bb} }^2}(s,r)$
of any point
$(s,r)\in \Lambda _{\unicode{x3bb} } \setminus \mathrm {II}$
is a
$2$
-periodic point
$H=H(s,r)$
, which has to be of saddle type (as sinks are repulsive for the past dynamics). Arguing as in the proof of Proposition 4.10, we deduce that
$(s,r) \in \mathcal {W}^u(H;f_{\unicode{x3bb} }^2)$
. Similarly,
$\omega _{f_{\unicode{x3bb} }^2}(s,r)=E\in \mathrm {II}$
, with E a sink periodic point in
$ \overline {\mathcal {W}^u(H;f_{\unicode{x3bb} }^2)}$
. We conclude that
$$ \begin{align*} \Lambda_{\unicode{x3bb}}=\bigcup_{i=1}^\ell \overline{\mathcal{W}^u(H_i;f^2_{\unicode{x3bb}}) \cup \mathcal{W}^u(f_{\unicode{x3bb}}(H_i);f^2_{\unicode{x3bb}})} \end{align*} $$
for some finite collection
$\{H_i,f_{\unicode{x3bb} }(H_i)\}_{i=1,\ldots ,\ell }$
of
$2$
-periodic orbits of saddle type, which concludes the proof.
We will now show that for any
$C^k$
convex domain in the interior of the complement of
$\mathcal {D}^k$
,
$k \geq 2$
, we loose the graph property of
$\Lambda _{\unicode{x3bb} }$
for small dissipation parameters
${\unicode{x3bb} \in (0,1)}$
. This is the case in particular for any ellipse
$\mathcal {E}$
of eccentricity e larger than
${\sqrt 2}/{2}$
. Indeed, if
$\{(s_0,0),f_1(s_0,0)\}$
is the
$2$
-periodic orbit along the minor axis of
$\mathcal {E}$
, then with the notation of Proposition 5.16, an easy computation shows that
$\tau (s_0,0)\mathcal {K}(s_0)=2(e^2-1)>-1$
, and hence the assumption of Proposition 5.16 is satisfied. As previously, given a strongly convex billiard
$\Omega $
, for
$(s,r)\in {\mathbb A}$
, we denote by
$\tau (s,r)$
the length of the first orbit segment for the (conservative) billiard map starting at
$(s,r)$
and by
$\mathcal {K}(s)<0$
the curvature at the point of
$\partial \Omega $
associated to s.
Proposition 5.16. Let
$k \geq 2$
and let
$\Omega $
be a strongly convex domain with
$C^k$
boundary in the complement of
$\mathcal {D}^k$
, such that there exists
$s_0 \in {\mathbb T}$
with
$\tau (s_0,0)\mathcal {K}(s_0)>-1$
. Then, for
$\unicode{x3bb} \in (0,1)$
sufficiently small, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
is not a graph over
${\mathbb T} \times \{0\}$
.
Proof. By contradiction, let us assume that there exists a sequence
$(\unicode{x3bb} _n)_{n \in \mathbb {N}}\in (0,1)^{\mathbb {N}}$
converging to
$0$
such that
$\Lambda _{\unicode{x3bb} _n}$
is the graph of some function
$\gamma _n\colon {\mathbb T} \to [-1,1]$
. As
$\Lambda _{\unicode{x3bb} _n}$
separates
${\mathbb A}$
, the function
$\gamma _n$
is necessarily continuous. We can then define the map
where
$\pi _1 \colon {\mathbb A} \to {\mathbb T}$
denotes the projection on the first coordinate. Note that by the graph hypothesis,
$g_{\unicode{x3bb} _n}$
is invertible. Let us also define
$g_0\colon s \mapsto \pi _1 \circ f_0(s,0)$
; note that
$g_0=\pi _1 \circ f_{\unicode{x3bb} }(s,0)$
for any
$\unicode{x3bb} \in [0,1]$
, and that
$g_0$
is
$C^1$
. By construction,
$\Lambda _{\unicode{x3bb} } \subset {\mathbb T} \times [-\unicode{x3bb} ,\unicode{x3bb} ]$
and
$f_{\unicode{x3bb} } \colon (s,r)\mapsto (s',\unicode{x3bb} r_1^{\prime })$
. Hence, for any
$\varepsilon>0$
, there exists
$n_\varepsilon \in \mathbb {N}$
such that for any
$n\geq n_\varepsilon $
,
$d_{C^0}(g_0,g_{\unicode{x3bb} _n})< \varepsilon $
. Now,
$g_0^{\prime }(s)=-(\tau (s)\mathcal {K}(s)+1)$
. On the one hand, if
$\{(s_1,0),(s_2,0)\}$
is a
$2$
-periodic orbit of maximal perimeter, then, as in the proof of Proposition 3.11,
$$ \begin{align*} (\tau(s_1)\mathcal{K}(s_1)+1)(\tau(s_2)\mathcal{K}(s_2)+1)\geq 1. \end{align*} $$
Since
$\mathcal {K}< 0$
, we conclude that there exists
$i\in \{1,2\}$
such that
$\tau (s_i)\mathcal {K}(s_i)+1\leq -1$
, that is,
$g_0^{\prime }(s_i)\geq 1$
. On the other hand, by assumption,
$g_0^{\prime }(s_0)<0$
. We conclude that there exist
$s_*\in {\mathbb T}$
and
$\eta _1,\eta _2>0$
such that
$g_0(s_*-\eta _1)=g_0(s_*+\eta _2)$
but
$g_0(s_*)\neq g_0(s_*-\eta _1)$
. Let
$\varepsilon :=\tfrac 13|g_0(s_*)-g_0(s_*-\eta _1)|$
. We deduce that for any
$n \geq n_\varepsilon $
,
By the continuity of
$g_{\unicode{x3bb} _n}$
, we deduce that
$g_{\unicode{x3bb} _n}$
is not injective, which is a contradiction.
We conclude this section by discussing the case where the dissipative billiard map
$f_{\unicode{x3bb} }$
has non-constant dissipation.
Remark 5.17. Let us consider a general dissipative billiard map
$f_{\unicode{x3bb} }$
as in Definition A for some
$C^{k-1}$
dissipation function
$\unicode{x3bb} \colon {\mathbb A} \to (0,1)$
. The results presented in this section can be obtained for such a map
$f_{\unicode{x3bb} }$
, as long as
$\|\unicode{x3bb} \|_{C^1}\ll 1$
.
6. Topologically complex Birkhoff attractors for mild dissipation
Birkhoff attractors for dissipative billiards described in §§4 and 5 do not make the idea of their possible topological complexity. In fact, following a celebrated result by Charpentier [Reference CharpentierCha34, §20], here Theorem 6.8, a Birkhoff attractor for a dissipative diffeomorphism may be an ‘indecomposable continuum’ (see Figure 6) and a sufficient condition for this occurrence is that two rotation numbers associated to the Birkhoff attractor itself are different. The aim of this section is proving that such a phenomenon occurs also for Birkhoff attractors of dissipative billiard maps. Moreover, we discuss some topological and dynamical consequences of this phenomenon.
An example of an indecomposable continuum (L Rempe-Gillen, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons).
The section is organized as follows. After recalling the main definition and properties of a twist map, we present the construction of the upper and the lower rotation numbers associated to the Birkhoff attractor, as well as the statement of Charpentier’s theorem. Finally, in the case of dissipative billiards, we give a sufficient condition assuring that the corresponding Birkhoff attractor has different upper and lower rotation numbers and we discuss the dynamical consequences of this fact.
6.1. Twist diffeomorphisms
Fix the standard metric and trivialization of the tangent space of
$\mathbb {A} := {\mathbb T}\times [-1,1]$
, as well as the counterclockwise orientation of the plane. Let
$\beta \in (0,{\pi }/{2})$
and denote by v the unitary vertical vector
$(0,1)$
. For any
$x\in {\mathbb A}$
, the cone
$C_+(x,\beta )$
is the set of vectors
$w\in T_x{\mathbb A}$
such that the angle
$\theta (v,w)$
(with respect to the fixed metric and trivialization) admits a lift in
$(-\pi +\beta ,-\beta )$
; similarly, the cone
$C_-(x,\beta )$
is the set of vectors
$w\in T_x{\mathbb A}$
such that the angle
$\theta (v,w)$
admits a lift in
$(\pi -\beta ,\beta )$
, see Figure 7. For the next definition, we refer to [Reference HermanHer83, §1.2].
The cones
$C_+(x,\beta )$
and
$C_-(x,\beta )$
.
Definition 6.1. Let U be an open subset of
${\mathbb A}$
. A
$C^1$
orientation-preserving diffeomorphism
$f\colon U\subset {\mathbb A} \to f(U)\subset {\mathbb A}$
is a positive, respectively negative, twist map on U if there exists
$\beta \in (0,{\pi }/{2})$
such that
$$ \begin{align*} Df(x)v\in C_+(f(x),\beta) \quad \text{respectively }Df(x)v\in C_-(f(x),\beta),\quad \text{for all } x \in U. \end{align*} $$
We are mostly interested in dissipative twist maps. Nevertheless, if we restrict to constant conformally symplectic twist maps, a variational setting can be described, following [Reference BangertBan88]. Let f be a constant conformally symplectic twist diffeomorphism of
$\text {int}(\mathbb {A})$
into its image of conformality ratio
$a> 0$
with respect to the area form
$\omega = dr \wedge ds = d\alpha $
, where
$\alpha = r\,ds$
is the Liouville 1-form. Denote by
$F\colon (S,r) \in \mathbb R\times [-1,1]\mapsto (S',r')\in \mathbb R\times [-1,1]$
a lift of f to the universal cover. The map
$f/a$
is an exact symplectic twist diffeomorphism of
$\text {int}(\mathbb {A})$
; this means that there exists a generating function
$H \in C^2(\mathbb R^2;\mathbb R)$
for
$F/a$
such that
$F^*\alpha - a \alpha = a\,dH$
, that is
$$ \begin{align} r'\,dS' - a r\,dS = a\,dH(S,S'). \end{align} $$
We define a (formal) action functional as
$$ \begin{align*} \mathcal{H}\colon \{S_i\}_{i\in\mathbb Z}\in\mathbb R^{\mathbb Z}\longmapsto \sum_{i\in\mathbb Z}\frac{H(S_i,S_{i+1})}{a^i}. \end{align*} $$
Definition 6.2. A bi-infinite sequence
$\{S_{i}\}_{i\in {\mathbb Z}}\in \mathbb R^{\mathbb Z}$
is stationary for
$\mathcal {H}$
if
$$ \begin{align*} \partial_1 H(S_i,S_{i+1})+a\partial_2H(S_{i-1},S_i)=0\quad\text{ for all } i\in\mathbb Z. \end{align*} $$
We can then characterize the orbits of F in terms of stationary sequences. Indeed, equality in equation (6.1) means that, for every
$S,S'\in \mathbb R$
,
$$ \begin{align*} \begin{cases} r = -\partial_1H(S,S'), \\ r' = a \partial_2H(S,S'). \end{cases} \end{align*} $$
As a consequence,
$\{ (S_i,r_i)\}_{i \in \mathbb {Z}}$
is an orbit of F if and only if for every
$i \in \mathbb {Z}$
, it holds that:
$$ \begin{align} -\partial_1H(S_i,S_{i+1}) = r_i = a \partial_2 H(S_{i-1},S_{i}). \end{align} $$
This implies the following.
Proposition 6.3. A bi-infinite sequence
$\{ (S_i,r_i)\}_{i \in \mathbb {Z}}$
is an orbit of F if and only if the bi-infinite sequence
$\{S_i\}_{i \in \mathbb {Z}}$
is stationary.
An important example of a twist map is given by the billiard map within a convex domain, as recalled in the following proposition.
Proposition 6.4. Let
$\Omega \subset \mathbb {R}^2$
be a convex domain, with
$C^k$
boundary,
$k \geq 2$
. Then, the associated billiard map
$f=f_1\colon {\mathbb A}\to {\mathbb A}$
given by equation (1.1) is a positive twist map when restricted to
$\mathrm {int}({\mathbb A})$
.
Proof. Let
$(s,r)\in \mathrm {int}({\mathbb A})$
. We consider the image of the vertical direction by the differential of f:
$$ \begin{align*} Df(s,r)\begin{bmatrix} 0\\1 \end{bmatrix}=\begin{bmatrix} \dfrac{\tau}{\nu\nu'} \\[8pt] -\dfrac{(\tau \mathcal{K}'+\nu')}{\nu} \end{bmatrix}\!. \end{align*} $$
To conclude that f is a positive twist map, it is sufficient to show that for some
$M>0$
, independent of the point
$(s,r)\in \mathrm {int}({\mathbb A})$
, it holds that
$$ \begin{align*} \dfrac{\vert \tau \mathcal{K}'+\nu'\vert/\nu}{\tau/\nu\nu'}=\vert \tau \mathcal{K}'+\nu'\vert \dfrac{\nu'}{\tau}\leq M. \end{align*} $$
Observe that for any point, we have
$\vert \tau \mathcal {K}'+\nu '\vert \leq \mathrm {diam}(\Omega )\mathcal {K}_0+1$
, where
$\mathcal {K}_0$
denotes the maximum in absolute value of the curvature of
$\partial \Omega $
. Thus, it suffices to get a uniform upper bound on
${\nu '}/{\tau }$
to conclude. Whenever
$\tau $
, which is the Euclidean distance between two consecutive points, is bounded away from zero, the quantity we are interested in is then clearly bounded. The points for which
$\tau $
is approaching zero are points increasingly closer to the boundary. Let then
$(s_n,r_n)_n\in (\mathrm {int}({\mathbb A}))^{\mathbb {N}}$
be a sequence of points converging to a point
$(s_\infty ,\pm 1)$
. Without loss of generality, assume that we converge to
$(s_\infty , 1)$
. Let
$\mathcal {K}_\infty \leq 0$
be the curvature at the point on
$\partial \Omega $
corresponding to
$s_\infty $
. We distinguish between two cases: either
$\mathcal {K}_\infty <0$
or
$\mathcal {K}_\infty =0$
.
In the first case, we let
$R_\infty :=|\mathcal {K}_\infty ^{-1}|>0$
be the radius of curvature at
$s_\infty $
. By approximating our convex domain with the osculating circle at the point
$s_\infty $
, we obtain
$$ \begin{align*} \tau_n \sim 2\nu_n'R_\infty, \end{align*} $$
denoting by
$\tau _n$
the Euclidean distance between the points corresponding to
$s_n$
and
$s_n'$
, where
$(s_n',r_n'):=f(s_n,r_n)$
, and with
$\nu _n':=\sqrt {1-(r_n')^2}$
(see e.g. [Reference DouadyDou82, Ch. 4, I.3.4.]). Thus,
$$ \begin{align*} \lim_{n \to+\infty}\dfrac{\nu^{\prime}_n}{\tau_n}=\frac{1}{2R_\infty}=-\frac{\mathcal{K}_\infty}{2}\leq \frac{\mathcal{K}_0}{2}, \end{align*} $$
which provides the required uniform bound.
In the second case, namely when
$\mathcal {K}_\infty =0$
, the boundary
$\partial \Omega $
is approximated up to order
$2$
by the tangent space at
$s_\infty $
. Let
$(\tilde s_n,\tilde r_n)$
and
$(\tilde s_n',\tilde r_n')$
be the respective approximations of
$(s_n,r_n)$
and
$(s_n',r_n'):=f(s_n,r_n)$
; then
$\tilde r_n=\tilde r_n'=1$
(and the corresponding
$\tilde \nu _n,\tilde \nu _n'$
satisfy
$\tilde \nu _n=\tilde \nu _n'=0$
). In addition, in our approximation,
$\tau _n$
is approximated by
$|\tilde s_n-\tilde s_n'|$
. This yields
$$ \begin{align*} \lim_{n \to+\infty}\dfrac{\nu^{\prime}_n}{\tau_n}=0. \end{align*} $$
In either case, we go get the required uniform bound.
6.2. Upper and lower rotation numbers and Charpentier’s result
We follow the presentation contained in [Reference Le CalvezLC88, §§4 and 5]. Recall from Definition 2.1 that
$$ \begin{align*}C=\{(s,r)\in{\mathbb A} :\phi^-(s)\leq r\leq \phi^+(s)\}\subset {\mathbb A},\end{align*} $$
where
$\phi ^-,\phi ^+\colon {\mathbb T}\to \mathbb R$
are continuous maps. For
$\unicode{x3bb} \in (0,1)$
, let
$f_{\unicode{x3bb} }$
be a dissipative (see Definition 2.1) positive twist map of C into its image and
$\Lambda _{\unicode{x3bb} }$
be its corresponding Birkhoff attractor (see Definition 2.5). Denote by
$C_{\unicode{x3bb} }^+$
(respectively
$C_{\unicode{x3bb} }^-$
) the connected component of
$C\setminus \Lambda _{\unicode{x3bb} }$
containing
$\{(s,\phi ^+(s))\in {\mathbb A} :s\in {\mathbb T}\}$
(respectively
$\{(s,\phi ^-(s))\in {\mathbb A} :s\in {\mathbb T}\}$
). For any
$(s,r)\in {\mathbb A}$
, the upper (respectively lower) vertical line is
$$ \begin{align*} V^+(s,r):=\{(s,y)\in{\mathbb A} :y\geq r\} \end{align*} $$
$\text {(respectively} \ V^-(s,r):=\{(s,y)\in {\mathbb A} :y\leq r\}$
). Let us now define (see Figure 8)
$$ \begin{align*} \Lambda_{\unicode{x3bb}}^+:=\{ x\in\Lambda_{\unicode{x3bb}} :V^+(x)\setminus\{x\}\subset C_{\unicode{x3bb}}^+\}\quad\text{and}\quad\Lambda_{\unicode{x3bb}}^-:=\{ x\in\Lambda_{\unicode{x3bb}} :V^-(x)\setminus\{x\}\subset C_{\unicode{x3bb}}^-\}. \end{align*} $$
The lower and upper verticals.
Therefore, we can define two functions
$\mu _{\unicode{x3bb} }^{\pm }\colon {\mathbb T}\to [-1,1]$
whose graphs
$\Gamma _{\mu _{\unicode{x3bb} }^{\pm }}$
satisfy
$\Gamma _{\mu _{\unicode{x3bb} }^\pm }=\Lambda _{\unicode{x3bb} }^\pm $
.
In the following, we fix a covering
$\pi \colon \mathbb R\times [-1,1]\to {\mathbb T}\times [-1,1]$
of
${\mathbb A}$
, and let
$\tilde \Lambda _{\unicode{x3bb} }:=\pi ^{-1}(\Lambda _{\unicode{x3bb} })$
,
$\tilde \Lambda _{\unicode{x3bb} }^\pm :=\pi ^{-1}(\Lambda _{\unicode{x3bb} }^\pm )$
; we also denote by
$\tilde \mu _{\unicode{x3bb} }^\pm \colon \mathbb R\to [-1,1]$
the lifts of
$\mu _{\unicode{x3bb} }^{\pm }\colon {\mathbb T}\to [-1,1]$
. Moreover, we let
$\pi _1\colon {\mathbb T}\times [-1,1]\to {\mathbb T}$
and
$\tilde \pi _1\colon \mathbb R\times [-1,1]\to \mathbb R$
be the first coordinate projections. The next propositions are [Reference Le CalvezLC88, Corollaries 4.8, 4.7, and 4.5], respectively.
Proposition 6.5. The map
$\tilde \mu ^+_{\unicode{x3bb} }\colon \mathbb R\to [-1,1]$
$(\text {respectively} \ \tilde \mu _{\unicode{x3bb} }^-\colon \mathbb R\to [-1,1])$
is upper (respectively lower) semi-continuous. Moreover,
$$ \begin{align*} \tilde\mu_{\unicode{x3bb}}^\pm(\tilde\theta')-\tilde\mu_{\unicode{x3bb}}^\pm(\tilde\theta)\leq (\tilde\theta'-\tilde\theta)\, \mathrm{cotan}\, \beta \quad \text{for all } \tilde\theta<\tilde\theta', \end{align*} $$
where
$\beta \in (0,{\pi }/{2})$
is the constant in Definition 6.1.
Proposition 6.6. The following properties hold:
-
(a)
$f_{\unicode{x3bb} }^{-1}(\Lambda _{\unicode{x3bb} }^\pm ) \subset \Lambda _{\unicode{x3bb} }^\pm $
and the order defined by the first coordinate projection is preserved by
$f_{\unicode{x3bb} }^{-1}$
; -
(b) let
$\hat {U}_{\unicode{x3bb} }^\pm := \{x \in C : \ V^\pm (x)\subset C_{\unicode{x3bb} }^\pm \}$
be the set of points radially accessible from below/above. If
$x\in f_{\unicode{x3bb} }(C) \cap \hat {U}^\pm _{\unicode{x3bb} }$
, then
$f_{\unicode{x3bb} }^{-1}(x)\in \hat {U}^\pm _{\unicode{x3bb} }$
and
$f_{\unicode{x3bb} }(V^\pm (f_{\unicode{x3bb} }^{-1}(x)))\subset \hat {U}^\pm _{\unicode{x3bb} }$
.
Let
$F_{\unicode{x3bb} }\colon \mathbb R\times [-1,1]\to F_{\unicode{x3bb} }(\mathbb R\times [-1,1])$
be a continuous lift of
$f_{\unicode{x3bb} }$
. The next result, due to Birkhoff, is [Reference Le CalvezLC88, Proposition 4.11].
Proposition 6.7. The sequence
$( ({\tilde \pi _1\circ F_{\unicode{x3bb} }^n-\tilde \pi _1})/{n} )_{n\in \mathbb N}$
converges uniformly on
$\tilde \Lambda _{\unicode{x3bb} }^+$
$(\text {respectively} \ \tilde \Lambda _{\unicode{x3bb} }^-)$
to a constant
$\rho _{\unicode{x3bb} }^+$
$(\text {respectively} \ \rho _{\unicode{x3bb} }^-)$
. The constants
$\rho _{\unicode{x3bb} }^+$
and
$\rho _{\unicode{x3bb} }^-$
—called upper and lower rotation numbers—do depend on the chosen lift, but not their difference.
From the previous result, we immediately conclude that if
$\Lambda _{\unicode{x3bb} }^+\cap \Lambda _{\unicode{x3bb} }^-\neq \emptyset $
(equivalently, if there is at least a point where
$\Lambda _{\unicode{x3bb} }$
is a graph), then
$\rho _{\unicode{x3bb} }^+=\rho _{\unicode{x3bb} }^-$
. We finally recall that when the upper and lower rotation numbers are different, then the corresponding Birkhoff attractor is topologically complicated, in the sense made precise by the following result of Charpentier (see [Reference CharpentierCha34, §20]).
Theorem 6.8. [Reference CharpentierCha34]
If
$\rho _{\unicode{x3bb} }^+-\rho _{\unicode{x3bb} }^->0$
, then
$\Lambda _{\unicode{x3bb} }$
is an indecomposable continuum, that is, it cannot be written as a union of two compact connected non-trivial sets.
6.3. The dissipative billiard case
In this section, for a dissipative billiard map, we give a sufficient condition assuring that the corresponding Birkhoff attractor has different upper and lower rotation numbers (see Proposition 6.10). Clearly, this is not the case of the billiard tables studied respectively in §§4 and 5. Indeed, for an ellipse, the corresponding Birkhoff attractor, independently from the dissipative parameter, is not an indecomposable continuum and, in particular, it holds that
$\rho _{\unicode{x3bb} }^+=\rho _{\unicode{x3bb} }^-=\tfrac 1 2\,\mod \mathbb Z$
. This can be proved even more directly. Indeed, since the rotation number is invariant under the dynamics and, for the dissipative billiard map on an ellipse, every point of the Birkhoff attractor is in the omega-limit set of a
$2$
-periodic point, we can deduce that every point of both
$\tilde \Lambda ^+_{\unicode{x3bb} }$
and
$\tilde \Lambda ^-_{\unicode{x3bb} }$
has rotation number equal to that of the
$2$
-periodic point, that is, equal to
$\tfrac 12$
. In §5, we study billiards whose Birkhoff attractor is a graph: in this case, we clearly have that
$\rho _{\unicode{x3bb} }^+=\rho _{\unicode{x3bb} }^-$
.
Let
$\Omega \subset \mathbb R^2$
be a strongly convex domain (that is, whose curvature never vanishes) with
$C^k$
,
$k \ge 2$
, boundary
$\partial \Omega $
. Then the associated (conservative) billiard map
$f = f_1$
is a
$C^{k-1}$
positive twist map (with respect to some
$\beta \in (0,{\pi }/{2})$
) of
${\mathbb A}:={\mathbb T} \times [-1,1]$
into itself. Consequently, for every
$\unicode{x3bb} \in (0,1)$
, the dissipative billiard map
$f_{\unicode{x3bb} }$
defined in §3.1 is a
$C^{k-1}$
positive twist map (with respect to some
$\beta ' \ge \beta \in (0,{\pi }/{2})$
) of
${\mathbb A}:={\mathbb T} \times [-1,1]$
into its image.
We recall that an essential curve in
${\mathbb A}$
is a topological embedding of
${\mathbb T}$
that is not homotopic to a point. The next proposition is an adaptation of [Reference Le CalvezLC88]: mainly, the only difference concerns the type of maps considered, but the proof follows the main lines of [Reference Le CalvezLC88, §8]. Some computations in the proof are simpler because of the kind of maps studied. More precisely, given a
$C^1$
function
$\unicode{x3bb} \colon {\mathbb A}\to (0,1)\subset \mathbb R$
, we consider compositions of twist maps with some homothety
$\mathcal {H}_{\unicode{x3bb} }$
of factor
$\unicode{x3bb} (s,r)$
in the second variable, but in the inverse order with respect to [Reference Le CalvezLC88]. This class of maps contains, in particular, the dissipative billiard maps considered in the present work. Let us recall the notion of instability region (see e.g. [Reference ArnaudArn16, Definition 2.18]).
Definition 6.9. Let
$C=\{(s,r)\in {\mathbb A}:\phi ^-(s)\leq r\leq \phi ^+(s)\}\subset {\mathbb A}$
, where
$\phi ^-,\phi ^+\colon {\mathbb T}\to \mathbb R$
are continuous maps. Let
$f\colon C \to f(C)$
be a twist map on
$\mathrm {int}(C)$
. Let
$\mathscr {V}(f)$
be the union of all f-invariant essential curves in C. An instability region
$\mathscr {I}$
is an open bounded connected component of
$C \setminus \mathscr {V}(f)$
that contains in its interior an essential curve.
Proposition 6.10. Let
$C{\kern-1.5pt}={\kern-1.5pt}\{(s,{\kern-1.5pt}r){\kern-1.5pt}\in{\kern-1.5pt} {\mathbb A}:\phi ^-(s){\kern-1.5pt}\leq{\kern-1.5pt} r{\kern-1.5pt}\leq{\kern-1.5pt} \phi ^+(s)\}{\kern-1.5pt}\subset{\kern-1.5pt} {\mathbb A}$
, where
$\phi ^-,\phi ^+{\kern-1.5pt}\colon{\kern-1.5pt} {\mathbb T}\to{\kern-1.5pt} \mathbb R$
are continuous maps. Let
$f\colon C\to C$
be an orientation-preserving homeomorphism, homotopic to the identity, such that:
-
(1) f preserves the standard
$2$
-form
$\omega =dr\wedge ds$
; -
(2)
$f\colon \mathrm {int}(C)\to \mathrm {int}(C)$
is a positive twist map on
$\mathrm {int}(C)$
with respect to
$\beta \in (0, \pi /2)$
; -
(3)
$\mathscr {I}:=\mathrm {int}(C)$
is an instability region for f that contains the zero section
${\mathbb T}\times \{0\}$
.
Then, there exists
$\bar \epsilon>0$
such that for any
$\epsilon \leq \bar \epsilon $
, for any
$C^1$
function
$\unicode{x3bb} \colon C\to (0,1)$
such that
${\epsilon }/{2}<d_{C^0}(\unicode{x3bb} ,1)<\epsilon $
, where the notation
$1$
stands for the constant function and
$\Vert D\unicode{x3bb} \Vert <\epsilon ^2$
, the Birkhoff attractor of
$f_{\unicode{x3bb} }:=\mathcal {H}_{\unicode{x3bb} }\circ f$
has
$\rho ^+_{\unicode{x3bb} }-\rho ^-_{\unicode{x3bb} }>0$
, where
$\mathcal {H}_{\unicode{x3bb} }\colon (s,r)\mapsto (s,\unicode{x3bb} (s,r) r)$
. Let us observe that by the assumptions on C, f, and by the definition of
$f_{\unicode{x3bb} }$
, the Birkhoff attractor of
$f_{\unicode{x3bb} }$
is contained in
$\mathrm {int}(C)$
.
Remark 6.11. Observe that, in Proposition 6.10, if we restrict to the class of constant functions
$\unicode{x3bb} $
, we are stating that there exists
$\unicode{x3bb} _0\in (0,1)$
such that, for any
$\unicode{x3bb} \in [\unicode{x3bb} _0,1)$
, the Birkhoff attractor of the dissipative map
$f_{\unicode{x3bb} }:=\mathcal {H}_{\unicode{x3bb} }\circ f$
has
$\rho ^+_{\unicode{x3bb} }-\rho ^-_{\unicode{x3bb} }>0$
, where
$\mathcal {H}_{\unicode{x3bb} }\colon (s,r)\mapsto (s,\unicode{x3bb} r)$
.
As a corollary of Proposition 6.10, we obtain a sufficient condition to assure that a dissipative billiard map has different rotation numbers.
Corollary 6.12. Let
$\Omega \subset \mathbb R^2$
be a strongly convex domain with
$C^k$
, boundary,
$k \ge 2$
. Let
$f=f_1$
be the associated (conservative) billiard map. If f admits an instability region
$\mathscr {I}$
that contains the zero section
$\mathbb {T}\times \{0\}$
, then there exists
$\unicode{x3bb} _0\in (0,1)$
such that, for any
$\unicode{x3bb} \in [\unicode{x3bb} _0,1)$
, the Birkhoff attractor of the corresponding dissipative billiard map
$f_{\unicode{x3bb} }$
has
$\rho _{\unicode{x3bb} }^+-\rho ^-_{\unicode{x3bb} }>0$
, with
$\tfrac 12\in (\rho _{\unicode{x3bb} }^-,\rho _{\unicode{x3bb} }^+)\,\mod \mathbb Z$
.
Remark 6.13. Both in Proposition 6.10 and in Corollary 6.12, the boundary of the instability region is made up of the graphs of two continuous functions
$\phi ^-<0<\phi ^+$
. These functions are actually Lipschitz by Birkhoff’s theorem, see [Reference BirkhoffBir22]. In Corollary 6.12, by the time-reversal symmetry of the conservative billiard map, it even holds
$\phi ^-=-\phi ^+$
.
Proof of Proposition 6.10
Since
$\unicode{x3bb} $
is, in particular, continuous on the compact set C, it takes values in
$(0,1)$
and since
$d_{C^0}(\unicode{x3bb} ,1)<\epsilon $
, there exist
$\unicode{x3bb} _{\min },\unicode{x3bb} _{\max }\in (1-\epsilon , 1)$
such that, for any
$(s,r)\in C$
, it holds that
Since
$\Vert D\unicode{x3bb} \Vert <\epsilon ^2$
, we also have that
$\unicode{x3bb} _{\max }-\unicode{x3bb} _{\min }<\epsilon ^2$
. Since also
$d_{C^0}(\unicode{x3bb} ,1)>{\epsilon }/{2}$
, we have that for every
$(s,r)\in C$
, it holds that
$\unicode{x3bb} (s,r)\leq \unicode{x3bb} _{\max }<1-{\epsilon }/{2}(1-2\epsilon )$
.
The map
$f_{\unicode{x3bb} }\colon C \to f_{\unicode{x3bb} }(C)\subset \mathrm {int}(C)$
, defined by
$f_{\unicode{x3bb} }:=\mathcal {H}_{\unicode{x3bb} }\circ f$
, is a dissipative map, according to Definition 2.1. Indeed, for every
$(s,r)\in \mathrm {int}(C)$
, it holds that
$$ \begin{align*} \mathrm{det}(Df_{\unicode{x3bb}}(s,r))= \mathrm{det}(D\mathcal{H}_{\unicode{x3bb}}(s',r'))=r'\partial_2\unicode{x3bb}(s',r')+\unicode{x3bb}(s',r')<1-\dfrac{\epsilon}{2}+2\epsilon^2, \end{align*} $$
where
$f(s,r)=(s',r')$
; there exists
$\epsilon _0$
small enough such that for every
$\epsilon <\epsilon _0$
, the value
$\mathrm {det}(Df_{\unicode{x3bb} }(s,r))$
is uniformly smaller than 1. Let F be a lift of f. We denote by
$\Lambda _{\unicode{x3bb} }$
the Birkhoff attractor of
$f_{\unicode{x3bb} }$
and by
$\rho _{\unicode{x3bb} }^\pm $
its lower and upper rotation numbers with respect to the lift
$\mathcal {H}_{\unicode{x3bb} }\circ F$
. Recall that f is a positive twist map with respect to
$\beta \in (0,{\pi }/{2})$
. Observe that, up to considering a smaller
$\epsilon _0$
, for any
$\epsilon \leq \epsilon _0$
, for any function
$\unicode{x3bb} $
that is
$\epsilon $
-
$C^1$
-close to
$1$
, the map
$f_{\unicode{x3bb} }$
is still a positive twist map on
$\mathrm {int}(C)$
with respect to
${\beta }/{2}\in (0,{\pi }/{4})$
.
Now consider the annulus
$\mathcal A$
bounded by
$\Gamma _{\phi ^-}:=\{(s,\phi ^-(s))\in {\mathbb A}:s\in {\mathbb T}\}$
and its image
$f_{\unicode{x3bb} }(\Gamma _{\phi ^-})$
. See Figure 9 for the following discussion. Since
$f(\Gamma _{\phi ^-})=\Gamma _{\phi ^-}$
, we have
$$ \begin{align*} f_{\unicode{x3bb}}(\Gamma_{\phi^-})= \mathcal{H}_{\unicode{x3bb}}\circ f(\Gamma_{\phi^-})=\mathcal{H}_{\unicode{x3bb}}(\Gamma_{\phi^-})=\Gamma_{\unicode{x3bb}\phi^-}, \end{align*} $$
and consequently
$m(\mathcal {A})=-\int _{\mathbb T}(1-\unicode{x3bb} (s,\phi ^-(s)))\phi ^-(s)\, ds$
; in particular, we have
We denote by
$C_{\unicode{x3bb} }^-$
the connected component of
$C\setminus \Lambda _{\unicode{x3bb} }$
containing
$\Gamma _{\phi ^-}=\{(s,\phi ^-(s))\in {\mathbb A}:s\in {\mathbb T}\}$
. Observe that, for every
$n\in \mathbb N^*$
, it holds that
Then, we have
where
$C^\pm :=\{(s,r)\in C,\, \pm r\geq 0\}$
. A consequence of equation (6.3) is that, up to choosing a smaller
$\epsilon _0$
, for every
$\epsilon \leq \epsilon _0$
,
$$ \begin{align} \text{for all } &\unicode{x3bb}\text{ such that } \frac{\epsilon}{2}<d_{C^0}(\unicode{x3bb},1)<\epsilon, \Vert D\unicode{x3bb}\Vert<\epsilon^2,\nonumber\\ &\quad\text{there exists at least one point } (s_{\unicode{x3bb}},r_{\unicode{x3bb}}) \in \Lambda_{\unicode{x3bb}}^-\text{ with } r \le \min_{s\in{\mathbb T}}\phi^+(s)/2. \end{align} $$
Here,
$\mathscr {I}^-:=\mathrm {int}(C^-)$
is the part of the instability region
$\mathscr {I}=\mathrm {int}(C)$
that lies below the zero section
$\mathbb {T} \times \{0\}$
, while
$\mathscr {I}_{\unicode{x3bb} }^-:=\mathrm {int}(C_{\unicode{x3bb} }^-)$
is the connected component of
$\mathscr {I}\setminus \Lambda _{\unicode{x3bb} }$
bounded by
$\Gamma _{\phi ^-}$
.
Indeed, if for some function
$\unicode{x3bb} $
every point of
$\Lambda _{\unicode{x3bb} }^-$
is contained in the interior of
$C^+=\{(s,r)\in C:r\geq \min _{s\in {\mathbb T}}\phi ^+(s)/2\}$
, then
$\int _{\mathbb T}\mu _{\unicode{x3bb} }^-(s)\, ds>\min _{s\in {\mathbb T}}\phi ^+(s)/2>0$
. We would then obtain
$m(C_{\unicode{x3bb} }^-)=m(C^-)+\int _{\mathbb T}\mu _{\unicode{x3bb} }^-(s)\, ds>m(C^-)+\min _{s\in {\mathbb T}}\phi ^+(s)/2$
. Nevertheless, by equation (6.3), it holds that
$m(C_{\unicode{x3bb} }^-)\leq ({1-\unicode{x3bb} _{\min }})/({1-\unicode{x3bb} _{\max }-\epsilon ^2})m(C^-)$
and we get
if
$\epsilon $
is small enough, this provides the required contradiction.
Denote by
$\mathscr {C}_\epsilon $
the set of
$C^1$
functions
$\unicode{x3bb} \colon C\to (0,1)$
such that
${\epsilon }/{2}<d_{C^0}(\unicode{x3bb} ,1)<\epsilon $
and
$\Vert D\unicode{x3bb} \Vert <\epsilon ^2$
. For every
$\epsilon \in (0,\epsilon _0)$
, let
$\unicode{x3bb} _\epsilon $
be a function in
$\mathscr {C}_\epsilon $
.
Proof of the claim
Let us show the claim when
$\pm =-$
. By contradiction, assume that this does not hold. In particular, there exists
$M>0$
and a sequence
$\epsilon _n\to 0$
as
$n\to +\infty $
such that, for every
$n\in \mathbb N$
, the function
$\unicode{x3bb} _{\epsilon _n}\in \mathscr {C}_{\epsilon _n}$
and it holds that
where
$H_M:=\{(s,r) : \phi ^-(s)\leq r\leq \phi ^-(s)+M\}\subset C$
(see Figure 10 for the following discussion).
Controlling the shape of
$\Lambda _{\unicode{x3bb} _{\epsilon _n}}^-$
.
By hypothesis, f preserves the standard
$2$
-form and it is a positive twist map on
$\mathrm {int}(C)$
. Thus, by [Reference BirkhoffBir32, §6] (see also [Reference HermanHer83, Proposition 5.9.2]), we have
$$ \begin{align} \Gamma_{\phi^+}=\{(s,\phi^+(s))\in {\mathbb A}:s\in{\mathbb T}\}\subset \overline{\bigcup_{k\in\mathbb Z}f^k(H_M)}=\overline{\bigcup_{n\in\mathbb N}f^{-n}(H_M)}. \end{align} $$
Denote by L the Lipschitz constant of
$\phi ^+$
and let
$E:=\min _{s\in {\mathbb T}}\phi ^+(s)>0$
. Moreover, recall that every
$f_{\unicode{x3bb} _{\epsilon _n}}$
is a positive twist map with respect to the constant
${\beta }/{2}\in (0,{\pi }/{4})$
, where
$\beta $
is the twist constant of f.
Choose
$j\in \mathbb N$
such that
$$ \begin{align} \dfrac{1}{j{(E/2)}-2-L}\leq \dfrac{\tan(\beta/2)}{2}. \end{align} $$
For
$i=0,\ldots ,j-1$
, denote
$s_i=i/j\,\mod 1$
and
$$ \begin{align*} B_i:=\bigg\{(s,r)\in C : |s-s_i|<\frac{1}{2j},\, |\phi^+(s)-r|<\frac{2}{j}\bigg\}. \end{align*} $$
By using the inclusion in equation (6.6), we deduce the existence of an index
$m_0\in \mathbb N$
such that
$$ \begin{align*} \bigcup_{0 \le m \le m_0}f^{-m}(H_M)\cap B_i\neq\emptyset \end{align*} $$
for every
$i=0,\ldots ,j-1$
. Therefore, since
$f_{\unicode{x3bb} _{\epsilon _n}}$
converges uniformly to f when
$\epsilon _n\to 0$
, there exists
$n_0 \in \mathbb N$
such that for all
$n\geq n_0$
and for all
$i=0,\ldots ,j-1$
,
Recall that
$V^-(s,r):=\{(s,y)\in {\mathbb A} :y\leq r\}$
and that, for any
$\unicode{x3bb} $
, the set
$C_{\unicode{x3bb} }^-$
is the connected component of
$C\setminus \Lambda _{\unicode{x3bb} }$
containing
$\{(s,\phi ^-(s))\in {\mathbb A}: s\in {\mathbb T}\}$
. Let us denote by
$\hat U^-_{\unicode{x3bb} }$
the set of points which are radially accessible from below, that is, the points
$(s,r)\in C$
such that
$V^-(s,r)\cap C\subset \overline {C_{\unicode{x3bb} }^-}$
. Clearly,
$H_M\subset \hat U_{\unicode{x3bb} }^-$
for any
$\unicode{x3bb} $
. For any
$n\geq n_0$
and any
$i\in \{0,\ldots ,j-1\}$
, by equation (6.8), there exist
$m =m(n,i)\in \{0,\ldots ,m_0\}$
and a point
$y=y(n,i,m)\in f^{-m}_{\unicode{x3bb} _{\epsilon _n}}(H_M)$
such that
This means that
and, by using Proposition 6.6(b), we get
$y=f^{-m}_{\unicode{x3bb} _{\epsilon _n}}(x) \in \hat U^-_{\unicode{x3bb} _{\epsilon _n}}$
. Since
$y \in B_i$
, it holds that
To conclude the proof, fix
$n\geq n_0$
. By equation (6.4), there exists a point
$(s_{\unicode{x3bb} _{\epsilon _n}},r_{\unicode{x3bb} _{\epsilon _n}})\in \Lambda _{\unicode{x3bb} _{\epsilon _n}}^-$
such that
$r_{\unicode{x3bb} _{\epsilon _n}}\leq \min _{s\in {\mathbb T}}\phi ^+(s)/2$
. Let us denote by
$(\tilde s_{\unicode{x3bb} _{\epsilon _n}},r_{\unicode{x3bb} _{\epsilon _n}})\in \tilde \Lambda _{\unicode{x3bb} _{\epsilon _n}}^-$
a lift of
$(s_{\unicode{x3bb} _{\epsilon _n}},r_{\unicode{x3bb} _{\epsilon _n}})$
. By equation (6.9), we can also find a point
$(\tilde s^{\prime }_{\unicode{x3bb} _{\epsilon _n}}, r^{\prime }_{\unicode{x3bb} _{\epsilon _n}})\in \tilde \Lambda _{\unicode{x3bb} _{\epsilon _n}}^-$
such that
Thus,
Since
$r_{\unicode{x3bb} _{\epsilon _n}}=\tilde \mu _{\unicode{x3bb} _{\epsilon _n}}^-(\tilde s_{\unicode{x3bb} _{\epsilon _n}})$
and
$r^{\prime }_{\unicode{x3bb} _{\epsilon _n}}=\tilde \mu _{\unicode{x3bb} _{\epsilon _n}}^-(\tilde s^{\prime }_{\unicode{x3bb} _{\epsilon _n}})$
, this contradicts Proposition 6.5 and completes the proof.
From the twist condition on the conservative map f, it holds that, once we fix a lift F of the map, the rotation numbers of the graphs of
$\phi ^+$
and
$\phi ^-$
are well defined. Denoting them by
$\rho ^+$
and
$\rho ^-$
, respectively, we also have that
$\rho ^-<\rho ^+$
. It is then sufficient to show that Claim 6.14 implies
We can follow verbatim the proof of [Reference Le CalvezLC88, Corollary 4.8] to deduce equation (6.10) and then conclude the proof.
Now, Corollary 6.12 is a straightforward consequence of Propositions 6.4, 6.10, Remarks 6.11, 6.13, and the following observation: with the notation of the above proof,
$\tfrac 12\in (\rho ^-,\rho ^+)\,\mod \mathbb Z$
, and hence for any
$C^1$
function
$\unicode{x3bb} $
whose
$C^0$
-distance from the constant function
$1$
is in
$(\epsilon /2, \epsilon )$
and such that
$\Vert D\unicode{x3bb} \Vert <\epsilon ^2$
, equation (6.10) implies that
$\tfrac 12\in (\rho _{\unicode{x3bb} }^-,\rho _{\unicode{x3bb} }^+)\,\mod \mathbb Z$
.
In the following, we show that the Birkhoff attractor of a dissipative billiard map may have different upper and lower rotation numbers, provided that the dissipation is mild enough. Moreover, we emphasize the main dynamical consequences of this fact. We start by recalling Corollary G, stated in §1.
Corollary G. Fix
$k\geq 3$
. There exists an open and dense subset
$\mathscr {U}$
of
$C^k$
strongly convex domains such that the following holds. For any
$\Omega \in \mathscr {U}$
, there exists
$\unicode{x3bb} _0(\Omega )\in (0,1)$
such that, for any
$\unicode{x3bb} \in [\unicode{x3bb} _0(\Omega ),1)$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
of the corresponding dissipative billiard map
$f_{\unicode{x3bb} }$
has
$\rho _{\unicode{x3bb} }^+-\rho ^-_{\unicode{x3bb} }>0$
, with
$\tfrac 12 \in (\rho _{\unicode{x3bb} }^-,\rho ^+_{\unicode{x3bb} })\,\mod \mathbb Z$
. Moreover, there exists
$\unicode{x3bb} _1(\Omega )\in [\unicode{x3bb} _0(\Omega ),1)$
such that for any
$\unicode{x3bb} \in [\unicode{x3bb} _1(\Omega ),1)$
and for any
$2$
-periodic point p of saddle type (e.g. when the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
corresponds to a diameter of the table), there exists a horseshoe
$K_{\unicode{x3bb} }(p)\subset \Lambda _{\unicode{x3bb} }$
in the homoclinic class
$H_{\unicode{x3bb} }(p):=\overline {\mathcal {W}^s(\mathcal {O}_{f_{\unicode{x3bb} }}(p))\pitchfork \mathcal {W}^u(\mathcal {O}_{f_{\unicode{x3bb} }}(p))}$
of p; more precisely, it holds that
$$ \begin{align*} K_{\unicode{x3bb}}(p)\subset H_{\unicode{x3bb}}(p):=\overline{\mathcal{W}^s(\mathcal{O}_{f_{\unicode{x3bb}}}(p))\pitchfork \mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(p))}\subset \overline{ \mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(p))}\subset \Lambda_{\unicode{x3bb}}. \end{align*} $$
The proof of Corollary G relies on the following proposition.
Proposition 6.15. For each
$k \geq 2$
, there exists an open and dense subset
$\mathscr {U}$
of
$C^k$
strongly convex domains such that for any
$\Omega \in \mathscr {U}$
, the corresponding billiard map has an instability region
$\mathscr {I}\subset \mathbb {A}:= \mathbb {T} \times [-1,1]$
that contains (a neighborhood of) the zero section
$\mathbb {T} \times \{0\}$
.
Proof. The argument follows the work [Reference Dias Carneiro, Oliffson Kamphorst and Pinto-de CarvalhoDCOKPdC07] of Dias Carneiro, Oliffson Kamphorst, and Pinto-de Carvalho. For a convex domain
$\Omega $
, let
$f=f_1\colon \mathbb {A} \to \mathbb {A}$
be the associated (conservative) billiard map. We denote by
$I \colon (s,r)\mapsto (s,-r)$
the time-reversal involution; recall that
$f \circ I =I \circ f^{-1}$
. Let
$\Gamma \subset {\mathbb A}$
be an f-invariant essential curve. In particular, by Birkhoff’s theorem (see [Reference BirkhoffBir22]), there exists a Lipschitz function
${\phi :{\mathbb T}\to \mathbb R}$
such that
$\Gamma =\{(s,\phi (s))\in {\mathbb A}:s\in {\mathbb T}\}$
. The symmetric graph
$I(\Gamma )=\{(s,-\phi (s))\in {\mathbb A}:s\in \mathbb {T}\}$
is also f-invariant, as
$f(I(\Gamma ))=I(f^{-1}(\Gamma ))=I(\Gamma )$
. Moreover, we observe that
$\Gamma \cap I(\Gamma )\subset \mathbb {T} \times \{0\}$
. In particular, any point in
$\Gamma \cap I(\Gamma )$
is a
$2$
-periodic point. Indeed, the intersection is also f-invariant: given any
$x_0=(s_0,0)\in \Gamma \cap I(\Gamma )$
, then also
$f(x_0)=(s_1,0)\in \Gamma \cap I(\Gamma )$
. Thus, the bounces at
$x_0$
and
$f(x_0)$
are perpendicular, and hence
$\{x_0,f(x_0)\}$
is a
$2$
-periodic orbit. We conclude that the rotation number of
$\Gamma $
(and so of
$I(\Gamma )$
) is equal to
$\tfrac 12$
.
Now, by [Reference Dias Carneiro, Oliffson Kamphorst and Pinto-de CarvalhoDCOKPdC07, §3], given a rational number
${p}/{q}\in \mathbb {Q}$
, there exists an open and dense subset
$\mathscr {U}_{p/q}$
of
$C^k$
strongly convex billiards which have no rotational invariant curve with rotation number
$p/q$
. Let us briefly recall the argument. If
$\gamma $
is such a curve, then the restriction
$f|_\gamma $
of the billiard map to
$\gamma $
is a homeomorphism of the circle, and since the rotation number is equal to
$p/q$
, there are periodic points on
$\gamma $
of type
$(p,q)$
. However, these cannot be linearly elliptic, since the curve is a Lipschitz graph over
${\mathbb T}$
, by Birkhoff’s theorem. Then, these periodic points are either degenerate, as in the circular billiard, or hyperbolic, in which case,
$\gamma $
will be a union of periodic points and saddle connections. By [Reference Dias Carneiro, Oliffson Kamphorst and Pinto-de CarvalhoDCOKPdC07, Theorems 1 and 2] (see also [Reference Dias Carneiro, Oliffson Kamphorst and Pinto-de CarvalhoDCOKPdC03, Reference Xia and ZhangXZ14]), both cases are not allowed for a strongly convex billiard
$\Omega $
in an open and dense subset of domains.
It follows from the previous discussion that there exists an open and dense subset
$\mathscr {U}=\mathscr {U}_{1/2}$
of
$C^k$
strongly convex domains such that any
$\Omega \in \mathscr {U}$
has no invariant essential curve crossing the zero section
$\mathbb {T} \times \{0\}$
, and thus, has an instability region containing a neighborhood of the zero section
$\mathbb {T} \times \{0\}$
.
Proof of Corollary G
Fix
$k \geq 3$
. As an immediate outcome of Corollary 6.12 and Proposition 6.15, there exists an open and dense subset
$\mathscr {U}$
of
$C^k$
strongly convex domains such that for any
$\Omega \in \mathscr {U}$
, there exists
$\unicode{x3bb} _0(\Omega )\in (0,1)$
such that, for any
$\unicode{x3bb} \in [\unicode{x3bb} _0(\Omega ),1)$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
of the corresponding dissipative billiard map
$f_{\unicode{x3bb} }$
has
$\rho _{\unicode{x3bb} }^+-\rho ^-_{\unicode{x3bb} }>0$
, with
$\tfrac 12 \in (\rho _{\unicode{x3bb} }^-,\rho ^+_{\unicode{x3bb} })\,\mod \mathbb Z$
. Let us denote by
$\mathrm {II}_{\max }$
the set of
$2$
-periodic points for
$f_1$
with locally maximal perimeter. As noted at the beginning of §3.3, for any
$p\in \mathrm {II}_{\max }$
,
$\{p,f_{\unicode{x3bb} }(p)\}$
is still a
$2$
-periodic orbit for each dissipative map
$f_{\unicode{x3bb} }$
,
$\unicode{x3bb} \in (0,1)$
, and by Proposition 3.11(a), it is actually of saddle type, for any
$\Omega $
in an open and dense subset of
$C^k$
domains. Moreover, for
$\unicode{x3bb} \in [\unicode{x3bb} _0(\Omega ),1)$
,
$\tfrac 12 \in (\rho _{\unicode{x3bb} }^-,\rho ^+_{\unicode{x3bb} })\,\mod \mathbb Z$
, and hence
$\mathrm {II}_{\max }\subset \Lambda _{\unicode{x3bb} }$
, by [Reference Le CalvezLC88, Proposition 14.2]. By Corollary 3.9(3), there exists
$\unicode{x3bb} _1(\Omega )\in [\unicode{x3bb} _0(\Omega ),1)$
such that, for any
$p \in \mathrm {II}_{\max }$
and for any
$\unicode{x3bb} \in [\unicode{x3bb} _1(\Omega ),1)$
, each branch of
$\mathcal {W}^s(p;f_{\unicode{x3bb} }^2)\setminus \{p\}$
and
$\mathcal {W}^u(p;f_{\unicode{x3bb} }^2)\setminus \{p\}$
contains a transverse homoclinic point. Now, Corollary 2.10 implies that for any
$\unicode{x3bb} \in [\unicode{x3bb} _1(\Omega ),1)$
, the Birkhoff attractor
$\Lambda _{\unicode{x3bb} }$
of
$f_{\unicode{x3bb} }$
contains a horseshoe
$K_{\unicode{x3bb} }(p)$
, with
$$ \begin{align*} K_{\unicode{x3bb}}(p)\subset H_{\unicode{x3bb}}(p):=\overline{\mathcal{W}^s(\mathcal{O}_{f_{\unicode{x3bb}}}(p))\pitchfork \mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(p))}\subset \overline{ \mathcal{W}^u(\mathcal{O}_{f_{\unicode{x3bb}}}(p))}\subset \Lambda_{\unicode{x3bb}}.\\[-36pt] \end{align*} $$
We can guarantee that the upper and lower rotation numbers of the Birkhoff attractor are different also in the case for every
$C^2$
-convex domain with a point with vanishing curvature, as explained in the following.
Corollary 6.16. Let
$\Omega $
be a convex domain with
$C^2$
boundary such that there is a point at which the curvature vanishes. Then for any
$\epsilon>0$
, there exists
$\unicode{x3bb} _0=\unicode{x3bb} _0(\Omega ,\epsilon )\in (0,1)$
such that for any
$\unicode{x3bb} \in [\unicode{x3bb} _0,1)$
, the Birkhoff attractor of
$f_{\unicode{x3bb} }$
has
$\rho ^+_{\unicode{x3bb} }-\rho ^-_{\unicode{x3bb} }\in (1-\epsilon ,1)$
.
Corollary 6.16 is a consequence of Propositions 6.4, 6.10, Remark 6.11, and the next well-known result by Mather (see [Reference MatherMat82] and also [Reference TabachnikovTab05, Corollary 5.29], [Reference Gutkin and KatokGK95, Theorem 1.1]).
Theorem 6.17. If the curvature of a
$C^{2}$
-convex billiard curve vanishes at some point, then the associated conservative billiard map has no invariant essential curves.
Proof of Corollary 6.16
Let
$\Omega $
be a convex domain with
$C^2$
boundary. Then, the associated billiard map
$f\colon {\mathbb A}\to {\mathbb A}$
is an orientation-preserving homeomorphism, homotopic to the identity, it preserves the standard
$2$
-form
$\omega =dr\wedge ds$
, and the restriction of f to
$\mathrm {int}({\mathbb A})$
is a
$C^1$
diffeomorphism, see [Reference Le CalvezLC90] and also [Reference DouadyDou82] for all details. By Proposition 6.4, it is a positive twist map on
$\mathrm {int}({\mathbb A})$
. Since there exists a point of zero curvature, by Theorem 6.17, the whole
$\mathrm {int}({\mathbb A})$
is an instability region. We conclude the proof by applying Proposition 6.10.
Let us conclude this section by the following remark, which provides a different proof of Proposition refprop different rho. We are grateful to Patrice Le Calvez for suggesting this argument. It is possible to show that, as
$\unicode{x3bb} $
tends to
$1$
in the
$C^0$
topology, the Birkhoff attractor comes iincreasingly closer to both
$\Gamma _{\phi ^+}$
and
$\Gamma _{\phi ^-}$
. From this, again following the argument of [Reference Le CalvezLC88, Corollary 4.8], it can be deduced that
$\lim _{\unicode{x3bb} \to 1}\rho _{\unicode{x3bb} }^\pm =\rho ^\pm $
, and thus, for
$\unicode{x3bb} $
close enough to
$1$
, it holds that
$\rho _{\unicode{x3bb} }^+-\rho _{\unicode{x3bb} }^->0$
, since
$\rho ^+-\rho ^->0$
.
Acknowledgements
We are grateful to Marie-Claude Arnaud for some useful discussions and suggestions, and to Patrice Le Calvez for suggesting an alternative proof of Proposition 6.10. O.B. was partially supported by the PRIN Project 2022FPZEES ‘Stability in Hamiltonian Dynamics and Beyond’. A.F. was partially supported by the ANR project ‘CoSyDy’ (ANR-CE40-0014) and the ANR project ‘GALS’ (ANR-23-CE40-0001). M.L. was partially supported by the ANR project ‘CoSyDy’ (ANR-CE40-0014) and the ANR project ‘PADAWAN’ (ANR-21-CE40-0012-01).
A Appendix. Proof of Lemmas 3.5 and 3.7: bifurcation of eigenvalues at
$2$
-periodic points for dissipative billiard maps
Let us recall that
$\mathrm {II}$
denotes the set of
$2$
-periodic points for the conservative billiard map
$f=f_1$
. For
$p=(s,0) \in \mathrm {II}$
, we denote by
$\tau =\ell (s,s'):=\|\Upsilon (s)-\Upsilon (s')\|$
the Euclidean distance between the points
$\Upsilon (s),\Upsilon (s')$
, where
$(s',0):=f(p)$
. We also denote by
$\mathcal {K}_1$
,
$\mathcal {K}_2$
the respective curvatures at
$\Upsilon (s),\Upsilon (s')$
.
Let us fix a
$C^{k-1}$
function
$\unicode{x3bb} \colon {\mathbb A}\to (0,1)$
such that
$f_{\unicode{x3bb} }:=\mathcal {H}_{\unicode{x3bb} } \circ f$
is a dissipative billiard map in the sense of Definition A, where
$\mathcal {H}_{\unicode{x3bb} }\colon (s,r)\mapsto (s,\unicode{x3bb} (s,r) r)$
. In particular,
$f_{\unicode{x3bb} }$
has the same set
$\mathrm {II}$
of
$2$
-periodic points as f, and for any
$2$
-periodic orbit
$\{p=(s,0),f_1(p)=f_{\unicode{x3bb} }(p)=(s',0)\}$
, we have
with
$\unicode{x3bb} _1:=\unicode{x3bb} (p)$
, and
$\unicode{x3bb} _2:=\unicode{x3bb} (f_{\unicode{x3bb} }(p))$
. By formula (3.4), we have
where we have set
$$ \begin{align*}k_{1,2}:=(\tau \mathcal{K}_1 + 1)(\tau \mathcal{K}_2 + 1).\end{align*} $$
Observe that
$\det Df_{\unicode{x3bb} }(p)=D\mathcal {H}_{\unicode{x3bb} }(f_{\unicode{x3bb} }(p))=\unicode{x3bb} _2$
, as f is conservative; similarly,
$\det Df_{\unicode{x3bb} }(f_{\unicode{x3bb} }(p))=\unicode{x3bb} _1$
. We thus obtain
In particular,
Proof of Lemma 3.5
We consider the case where the dissipation is constant, equal to some
$\unicode{x3bb} \in (0,1)$
. Let us denote by
$\mu _1=\mu _1(\unicode{x3bb} ),\mu _2=\mu _2(\unicode{x3bb} )$
the eigenvalues of
$Df_{\unicode{x3bb} }^2(p)$
, with
$|\mu _1|\leq |\mu _2|$
. In particular, with the above notations, we have
$\unicode{x3bb} _1=\unicode{x3bb} _2=\unicode{x3bb} $
, and
$$ \begin{align} \mu_1\mu_2=\det Df_{\unicode{x3bb}}^2(p)=\unicode{x3bb}^2,\quad \mathrm{tr}Df_{\unicode{x3bb}}^2(p)=(1+\unicode{x3bb})^2k_{1,2}-2\unicode{x3bb}. \end{align} $$
By (A.2), the characteristic polynomial
$\chi _{p,\unicode{x3bb} }(x)=\det (Df_{\unicode{x3bb} }^2(p)-x\, \mathrm {id})$
is equal to
$$ \begin{align*} \chi_{p,\unicode{x3bb}}(x)=x^2 -((1+\unicode{x3bb})^2k_{1,2}-2\unicode{x3bb}) x + \unicode{x3bb}^2, \end{align*} $$
with
$k_{1,2}=(1+\tau \mathcal {K}_1)(1+\tau \mathcal {K}_2)$
. Then,
$\chi _{p,\unicode{x3bb} }$
has discriminant
$\Delta =((1+\unicode{x3bb} )^2 k_{1,2}-2\unicode{x3bb} )^2-4 \unicode{x3bb} ^2=k_{1,2}(1+\unicode{x3bb} )^2((1+\unicode{x3bb} )^2k_{1,2}-4\unicode{x3bb} )$
, which has the same sign as
$$ \begin{align*} \tilde\Delta:=k_{1,2}((1+\unicode{x3bb})^2k_{1,2}-4\unicode{x3bb})=\unicode{x3bb}^2k_{1,2}^2 + 2\unicode{x3bb} k_{1,2}(k_{1,2}-2)+k_{1,2}^2. \end{align*} $$
The quantity
$\tilde \Delta $
is a quadratic polynomial in
$\unicode{x3bb} $
, whose discriminant is equal to
$$ \begin{align*} \delta=4k_{1,2}^2((k_{1,2}-2)^2-k_{1,2}^2) =-16k_{1,2}^2(k_{1,2}-1). \end{align*} $$
(a) If
$k_{1,2}>1$
, then
$\delta <0$
, hence
$\tilde \Delta> 0$
,
$\Delta> 0$
, and the eigenvalues
$\mu _1,\mu _2$
of
$Df_{\unicode{x3bb} }^2(p)$
are real, with
$|\mu _1|\leq |\mu _2|$
. Their product
$\mu _1 \mu _2=\mathrm {det}Df_{\unicode{x3bb} }^2(p)=\unicode{x3bb} ^2$
is positive; their sum
$\mu _1+\mu _2$
is also positive because
$$ \begin{align} \mu_1+\mu_2=\mathrm{tr}Df_{\unicode{x3bb}}^2(p)=(1+\unicode{x3bb})^2k_{1,2}-2\unicode{x3bb}> (1+\unicode{x3bb})^2-2\unicode{x3bb}=1+\unicode{x3bb}^2>0, \end{align} $$
where the first inequality comes from the hypothesis
$k_{1,2}>1$
. Therefore, both eigenvalues are positive, and
$0<\mu _1\leq \mu _2$
. In particular, by (A.2),
$0<\mu _1^2\leq \mu _1\mu _2=\unicode{x3bb} ^2$
and hence
$\mu _1\in (0,1)$
. Let us show that in fact,
$0<\mu _1\leq \unicode{x3bb} ^2<1<\mu _2$
. Indeed, for
$i=1,2$
, by (A.2)–(A.3), we have
$\mu _i+{\unicode{x3bb} ^2}/{\mu _i}>1+\unicode{x3bb} ^2$
, and hence
$P(\mu _i)>0$
, where
$P(X)=X^2-(1+\unicode{x3bb} ^2)X+\unicode{x3bb} ^2$
. Since the roots of P are
$\{\unicode{x3bb} ^2,1\}$
, and as
$\mu _1\in (0,1)$
, we deduce that
$\mu _1\in (0,\unicode{x3bb} ^2)$
, and then
$\mu _2={\unicode{x3bb} ^2}/{\mu _1}>1$
.
(b) If
$k_{1,2}=1$
, then
$\chi _{p,\unicode{x3bb} }(x)=x^2 -(1+\unicode{x3bb} ^2) x + \unicode{x3bb} ^2=(x-\unicode{x3bb} ^2)(x-1)$
, thus
$\mu _1=\unicode{x3bb} ^2$
,
$\mu _2=1$
.
If
$k_{1,2}<1$
,
$k_{1,2}\neq 0$
, then
$\delta> 0$
. Let
$\unicode{x3bb} _\pm :=-1+{2}/{k_{1,2}}\pm 2\sqrt {({1-k_{1,2}})/{k_{1,2}^2}}$
.
(c) Assume now that
$k_{1,2}\in (0,1)$
. Note that
$\unicode{x3bb} _+=-1+{2}/{k_{1,2}}(1+\sqrt {1-k_{1,2}})\geq -1+{2}/{k_{1,2}}> 1$
, and
$\unicode{x3bb} _-=-1+{2}/{k_{1,2}}(1-\sqrt {1-k_{1,2}})=({\xi ^2-2\xi +1})/ ({1-\xi ^2})= ({1-\xi })/ ({1+\xi })$
, with
$\xi :=\sqrt {1-k_{1,2}}\in (0,1)$
, so that
$\unicode{x3bb} _-\in (0,1)$
. Then:
-
(1) for
$\unicode{x3bb} \in (\unicode{x3bb} _-,1)$
,
$\tilde \Delta <0$
,
$\Delta <0$
, and hence the eigenvalues of
$Df_{\unicode{x3bb} }^2(p)$
are complex conjugate; -
(2) for
$\unicode{x3bb} \in (0,\unicode{x3bb} _-]$
,
$\tilde \Delta \geq 0$
,
$\Delta \geq 0$
, and hence the eigenvalues of
$Df_{\unicode{x3bb} }^2(p)$
are real. Moreover,
$\mu _1=\mu _2$
if and only if
$\unicode{x3bb} =\unicode{x3bb} _-$
.
In case (1), by (A.2), it holds that
$|\mu _1|=|\mu _2|=\sqrt {\mu _1\mu _2}=\sqrt {\unicode{x3bb} ^2}=\unicode{x3bb} \in (0,1)$
, and hence the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink.
Let us consider case (2). By (A.2),
$\mu _1,\mu _2$
have the same sign, and
$0<\mu _1^2\leq \mu _1\mu _2=\unicode{x3bb} ^2$
, and hence
$|\mu _1|\in (0,1)$
. Moreover, as
$k_{1,2}\in (0,1)$
, and by (A.2), for
$i=1,2$
, we have
$$ \begin{align} \mu_i+\frac{\unicode{x3bb}^2}{\mu_i}=\mathrm{tr}Df_{\unicode{x3bb}}^2(p)=(1+\unicode{x3bb})^2k_{1,2}-2\unicode{x3bb} \in (-2\unicode{x3bb},1+\unicode{x3bb}^2). \end{align} $$
Assume that
$\mu _1,\mu _2$
are negative. By (A.4), for
$i=1,2$
, it holds that
$\mu _i+{\unicode{x3bb} ^2}/{\mu _i}>-2\unicode{x3bb} $
, and hence
$0>\mu _i^2+2\unicode{x3bb} \mu _i+\unicode{x3bb} ^2=(\mu _i+\unicode{x3bb} )^2$
, a contradiction. Thus,
$\mu _1,\mu _2$
are positive, and then, by (A.4), for
$i=1,2$
, it holds that
$P(\mu _i)<0$
, where
$P(X)=X^2-(1+\unicode{x3bb} ^2)X+\unicode{x3bb} ^2$
. Since the roots of P are
$\{\unicode{x3bb} ^2,1\}$
, we deduce that
$\mu _1,\mu _2\in (\unicode{x3bb} ^2,1)$
, and then the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink as well.
(d) If
$k_{1,2}=0$
, then
$\chi _{p,\unicode{x3bb} }(x)=x^2 +2\unicode{x3bb} x + \unicode{x3bb} ^2=(x+\unicode{x3bb} )^2$
, and hence
$\mu _1=\mu _2=-\unicode{x3bb} $
.
(e),(f) Finally, assume that
$k_{1,2}<0$
. In that case, it is easy to check that
$\unicode{x3bb} _\pm =-1+{2}/{k_{1,2}}( 1\mp \sqrt {1-k_{1,2}} )<0$
, and then, for
$\unicode{x3bb} \in (0,1)$
,
$\Delta>0$
, and hence the eigenvalues of
$Df_{\unicode{x3bb} }^2(p)$
are real. By (A.2),
$\mu _1,\mu _2$
have the same sign, and
$0<\mu _1^2\leq \mu _1\mu _2=\unicode{x3bb} ^2$
, thus
$|\mu _1|\in (0,\unicode{x3bb} )$
. Moreover, as
$k_{1,2}<0$
, and by (A.2), for
$i=1,2$
, we have
$ \mu _i+ ({\unicode{x3bb} ^2}/{\mu _i}) = \mathrm {tr}Df_{\unicode{x3bb} }^2(p)<-2\unicode{x3bb} $
, and hence
$\mu _2<-\unicode{x3bb} <\mu _1<0$
. We have
$$ \begin{align*} \mu_2=\mu_2(\unicode{x3bb})=\tfrac{1}{2}\Big((1+\unicode{x3bb})^2 k_{1,2}-2\unicode{x3bb}-\sqrt{(1+\unicode{x3bb})^2k_{1,2}((1+\unicode{x3bb})^2 k_{1,2}-4 \unicode{x3bb})}\Big). \end{align*} $$
Observe that
$$ \begin{align*} \lim_{\unicode{x3bb} \to 0^+} \mu_2(\unicode{x3bb})=k_{1,2},\quad \lim_{\unicode{x3bb} \to 1^-} \mu_2(\unicode{x3bb})=-1+2k_{1,2}-2\sqrt{k_{1,2}(k_{1,2}- 1)}<-1. \end{align*} $$
By direct computation, we see that the equation
$\mu _2(\unicode{x3bb} )=-1$
admits a solution in
$(0,1)$
if and only if
$k_{1,2}\in (-1,0)$
; in that case, the only solution in
$(0,1)$
is
, with
We conclude that:
-
(e) if
$k_{1,2}\in (-1,0)$
, then
, and-
(i) for any
,
$-1<\mu _2<-\unicode{x3bb} <\mu _1<0$
, and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink; -
(ii) for
,
$\mu _1=-\unicode{x3bb} ^2$
,
$\mu _2=-1$
, and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is parabolic; -
(iii) for any
,
$\mu _2<-1<-\unicode{x3bb} ^2<\mu _1<0$
, and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle;
-
-
(f) if
$k_{1,2}\leq -1$
, then for any
$\unicode{x3bb} \in (0,1)$
,
$\mu _2<-1<-\unicode{x3bb} ^2<\mu _1<0$
, and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle.
Proof of Lemma 3.7
We now consider the case where
$\unicode{x3bb} \colon {\mathbb A}\to (0,1)$
is a general
$C^{k-1}$
function such that
$f_{\unicode{x3bb} }:=\mathcal {H}_{\unicode{x3bb} } \circ f$
is a dissipative billiard map in the sense of Definition A, where
$\mathcal {H}_{\unicode{x3bb} }\colon (s,r)\mapsto (s,\unicode{x3bb} (s,r) r)$
.
Fix a
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
for
$f_{\unicode{x3bb} }$
. Let us denote by
$\mu _1=\mu _1(\unicode{x3bb} ),\mu _2=\mu _2(\unicode{x3bb} )$
the eigenvalues of
$Df_{\unicode{x3bb} }^2(p)$
, with
$|\mu _1|\leq |\mu _2|$
. By (A.1), the characteristic polynomial
$\chi _{p,\unicode{x3bb} }(x)=\det (Df_{\unicode{x3bb} }^2(p)-x\, \mathrm {id})$
is equal to
with
$k_{1,2}=(1+\tau \mathcal {K}_1)(1+\tau \mathcal {K}_2)$
. Recall that we assume that
$k_{1,2}\geq 0$
.
On the one hand, if the eigenvalues
$\mu _1,\mu _2$
are not real, then they are complex conjugate, and as
$\mu _1\mu _2=\unicode{x3bb} _1\unicode{x3bb} _2\in (0,1)$
, their modulus is strictly less than
$1$
, and
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink.
On the other hand, if
$\mu _1,\mu _2\in \mathbb R$
, then as
$|\mu _1|\leq |\mu _2|$
, and
$\mu _1\mu _2=\unicode{x3bb} _1\unicode{x3bb} _2\in (0,1)$
, we deduce that
$|\mu _1|<1$
. Thus, the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle or a sink, unless
$\mu _2=1$
or
$\mu _2=-1$
. But
with
$(1+\unicode{x3bb} _1)(1+\unicode{x3bb} _2)>0$
, and hence
$\chi _{p,\unicode{x3bb} }(1)=0$
if and only if
$k_{1,2}=1$
. In that case, we have
and hence
$1$
is an eigenvalue no matter which
$\unicode{x3bb} $
we choose. In particular,
$\{p,f_{\unicode{x3bb} }(p)\}$
is parabolic for the conservative billiard map
$f_1$
.
Moreover, as
$k_{1,2}\geq 0$
, we have
thus
$-1$
is never an eigenvalue.
To conclude the proof, it remains to show that under the assumption that there is no parabolic
$2$
-periodic orbit, then for any
$2$
-periodic point p, for the dissipative billiard map
$f_{\unicode{x3bb} }$
, the point p is a saddle if and only if
$k_{1,2}>1$
, and a sink if and only if
$k_{1,2}<1$
.
Indeed, on the one hand, if
$k_{1,2}>1$
, then (A.5) above shows that
$\chi _{p,\unicode{x3bb} }(1)<0$
. Since
$\lim _{x \to +\infty } \chi _{p,\unicode{x3bb} }(x)=+\infty $
, we deduce that
$\chi _{p,\unicode{x3bb} }$
vanishes somewhere on
$(1,+\infty )$
, and hence
$\mu _2>1$
, and then the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a saddle.
On the other hand, if
$k_{1,2}<1$
, then (A.5) above shows that
$\chi _{p,\unicode{x3bb} }(1)>0$
. But
$1>x_{\min }$
, where
$x_{\min }\in \mathbb R$
is the point at which the quadratic polynomial
$\chi _{p,\unicode{x3bb} }$
attains its minimum; indeed, as
$k_{1,2}<1$
, we have
Thus,
$\chi _{p,\unicode{x3bb} }(1)>0$
implies that
$\chi _{p,\unicode{x3bb} }$
is positive on
$[1,+\infty )$
, and hence no eigenvalue has modulus
$>1$
(recall that if it were the case, then
$\mu _2>1$
would be a real zero of
$\chi _{p,\unicode{x3bb} }$
), and the
$2$
-periodic orbit
$\{p,f_{\unicode{x3bb} }(p)\}$
is a sink.
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