2.1 Geodesics and geodesic flow

Let $({\tilde {X}}, d_{{\tilde {X}}})$ be a CAT( $-1$ ) space and $\Gamma $ be a discrete group of isometries of ${\tilde {X}}$ acting freely, properly discontinuously, and cocompactly. (Recall that a group $\Gamma $ acting on X by isometries is discrete if for every ball $B = B(x, r) \subset X$ , $\{g \in G: gB \cap B \neq \varnothing \}$ is finite. See [Reference Bridson and HaefligerBH99] for background on CAT( $-1$ ) spaces.) The resulting quotient $X={\tilde {X}}/\Gamma $ , with the metric $d_X$ induced by $d_{{\tilde {X}}}$ , is a compact, locally CAT( $-1$ ) space.

To study the geodesic flow on X, we need an analog of the unit tangent bundle appropriate for this non-smooth setting which admits a unit-speed geodesic flow. This is provided by the following definition.

Definition 2.1. The space of (unit-speed) geodesics of ${\tilde {X}}$ is

$$ \begin{align*} G{\tilde{X}} := \{{\tilde{\gamma}}: {\mathbb{R}} \to {\tilde{X}}: d_{{\tilde{X}}}({\tilde{\gamma}}(s), {\tilde{\gamma}}(t)) = |s-t|\}. \end{align*} $$

The geodesic flow $g_t$ on $G{\tilde {X}}$ is given by $g_t {\tilde {\gamma }}(s) = {\tilde {\gamma }}(s+t)$ for any $t \in {\mathbb {R}}$ . Additionally, $GX = G{\tilde {X}} / \Gamma $ is the space of geodesics of X.

There are natural metrics on these spaces.

Definition 2.2. We equip $G{\tilde {X}}$ with the metric

$$ \begin{align*}d_{G{\tilde{X}}}({\tilde{\gamma}}_1, {\tilde{\gamma}}_2) := \int_{-\infty}^{\infty} d_{{\tilde{X}}}({\tilde{\gamma}}_1(t), {\tilde{\gamma}}_2(t)) e^{-2|t|} \; dt,\end{align*} $$

and $GX$ with the metric

$$ \begin{align*}d_{GX}(\gamma_1, \gamma_2) = \min\limits_{\widetilde{\gamma_1}, \widetilde{\gamma_2}}\int_{-\infty}^{\infty} d_{\widetilde{X}}(\widetilde{\gamma_1}(t) , \widetilde{\gamma_2}(t))e^{-2\lvert t \rvert}\,dt,\end{align*} $$

where the minimum is taken over all lifts $\widetilde {\gamma _1}$ and $\widetilde {\gamma _2}$ of $\gamma _1$ and $\gamma _2$ , respectively.

A straightforward computation shows that the geodesic flow is unit-speed with respect to $d_{GX}$ . (This is the reason for the normalizing factor 2 in the exponent.) We record here a few basic facts about $d_{GX}$ and its interaction with the geodesic flow.

A central tool in the geometry of CAT( $-1$ ) spaces is the boundary at infinity.

In CAT( $-1$ ) spaces, a pair $\{\xi , \eta \}$ of distinct points on $\partial ^\infty {\tilde {X}}$ uniquely determines an unparameterized, unoriented geodesic. Once we specify an orientation and the time- $0$ point on the geodesic, we have a geodesic in $G{\tilde {X}}$ . Therefore, $G{\tilde {X}}$ can be identified with $[(\partial ^\infty {\tilde {X}} \times \partial ^\infty {\tilde {X}}) \backslash \Delta ] \times {\mathbb {R}}$ , where $\Delta $ is the diagonal. For ease of notation, let $\partial ^{(2)}_\infty {\tilde {X}} = (\partial ^\infty {\tilde {X}} \times \partial ^\infty {\tilde {X}}) \backslash \Delta .$

2.2 Stable and unstable sets

Under the geodesic flow on a negatively curved manifold, the unit tangent bundle admits foliations by stable and unstable manifolds which are essential tools for studying the dynamics of these flows. In the CAT( $-1$ ) setting, we have a purely geometric description of analogous stable and unstable sets. Below, we will note that these are also analogous in their dynamical role.

Definition 2.6. Let $p \in {\tilde {X}}$ , $\xi \in \partial ^\infty {\tilde {X}}$ , and ${\tilde {\gamma }}$ be the geodesic ray from p to $\xi $ . The Busemann function centered at $\xi $ with basepoint p is defined as

$$ \begin{align*} B_p(-, \xi): {\tilde{X}} &\to {\mathbb{R}} \\ q & \mapsto \lim_{t\to \infty} (d_{{\tilde{X}}}(q, {\tilde{\gamma}}(t)) - t). \end{align*} $$

For convenience, given geodesic ray ${\tilde {\gamma }}(t)$ , the Busemann function determined by ${\tilde {\gamma }}$ is the Busemann function centered at ${\tilde {\gamma }}(+\infty )$ with basepoint ${\tilde {\gamma }}(0)$ :

$$ \begin{align*}B_{{\tilde{\gamma}}}(-):= B_{{\tilde{\gamma}}(0)}(-, {\tilde{\gamma}}(+\infty)).\end{align*} $$

The level sets for $B_p(-, \xi )$ are horospheres.

Definition 2.7. Let ${\tilde {\gamma }} \in G{\tilde {X}}$ .

• • The strong stable set through ${\tilde {\gamma }}$ is

  $$ \begin{align*}W^{ss}({\tilde{\gamma}}) = \{{\tilde{\gamma}}'\in G{\tilde{X}}: {\tilde{\gamma}}'(+\infty) = {\tilde{\gamma}}(+\infty) \text{ and } B_{{\tilde{\gamma}}}({\tilde{\gamma}}'(0)) = 0 \}.\end{align*} $$

• • For $\delta>0$ ,

  $$ \begin{align*} W^{ss}_\delta({\tilde{\gamma}}) = \{{\tilde{\gamma}}'\in G{\tilde{X}} : \ & {\tilde{\gamma}}'(+\infty) = {\tilde{\gamma}}(+\infty), \\ & B_{{\tilde{\gamma}}}({\tilde{\gamma}}'(0)) = 0, \text{and } d_{G{\tilde{X}}}({\tilde{\gamma}}, {\tilde{\gamma}}') < \delta\}. \end{align*} $$

• • The weak stable set through ${\tilde {\gamma }}$ is

  $$ \begin{align*}W^{cs}({\tilde{\gamma}}) = \bigcup_{t \in {\mathbb{R}}} g_t(W^{ss}({\tilde{\gamma}})).\end{align*} $$

• • Similarly, the strong and weak unstable sets through ${\tilde {\gamma }}$ are

  $$ \begin{align*}W^{uu}({\tilde{\gamma}}) = \{{\tilde{\gamma}}'\in G{\tilde{X}}: {\tilde{\gamma}}'(-\infty) = {\tilde{\gamma}}(-\infty) \text{ and } B_{-{\tilde{\gamma}}}({\tilde{\gamma}}'(0)) = 0 \},\end{align*} $$
  $$ \begin{align*} W^{uu}_\delta({\tilde{\gamma}}) = \{{\tilde{\gamma}}'\in G{\tilde{X}} : \ & {\tilde{\gamma}}'(+\infty) = {\tilde{\gamma}}(+\infty), \\ & B_{-{\tilde{\gamma}}}({\tilde{\gamma}}'(0)) = 0, \text{and } d_{G{\tilde{X}}}({\tilde{\gamma}}, {\tilde{\gamma}}') < \delta\}, \end{align*} $$
  and
  $$ \begin{align*}W^{cu}({\tilde{\gamma}}) = \bigcup_{t \in {\mathbb{R}}} g_t(W^{uu}({\tilde{\gamma}})).\end{align*} $$

(See Figure 1.)

Figure 1

Here, $\widetilde {\gamma '} \in W^{uu}(\widetilde {\gamma })$ and $\widetilde {\gamma "} \in W^{ss}(\widetilde {\gamma })$ . The 0-level sets of the Busemann functions are the dashed circles.

The next lemma outlines some useful, standard properties of Busemann functions.

Proof. Let q be any point in $\tilde X$ . Then,

$$ \begin{align*} B_{g_s{\tilde{\gamma}}}(q) & = \lim_{t\to \infty} (d_{{\tilde{X}}}(q, {\tilde{\gamma}}(s+t))-t) \\ & = \lim_{t'\to \infty} (d_{{\tilde{X}}}(q, {\tilde{\gamma}}(t'))-(t'-s)) \\ & = B_{{\tilde{\gamma}}}(q) + s, \end{align*} $$

which proves part (1).

Since $\tilde {\eta } \in W^{cs}({\tilde {\gamma }})$ , $g_r\tilde {\eta } \in W^{ss}({\tilde {\gamma }})$ for some $r \in {\mathbb {R}}$ and for $i\in \{1, 2\}$ , $g_{s_i}\tilde {\eta } \in W^{ss}(g_{s_i - r}{\tilde {\gamma }})$ . Therefore, $B_{g_{s_i-r}{\tilde {\gamma }}} (\tilde {\eta }(s_i)) = 0$ . By part (1), this implies $B_{{\tilde {\gamma }}}(\tilde {\eta }(s_i)) = r-s_i$ , from which part (2) follows immediately.

Part (3) holds as $B_{{\tilde {\gamma }}}$ is a limit of 1-Lipschitz functions. See also [Reference Bridson and HaefligerBH99, II.8] for how this fact follows naturally from a slightly different presentation of Busemann functions.

2.3 Metric Anosov flows

In [Reference PollicottPol87], Pollicott defines metric Anosov flows, generalizing the essential properties of Anosov flows to the non-smooth setting.

A continuous flow on a compact metric space is Anosov if, roughly speaking, it has a topological local product structure which coheres with the dynamics of the flow in a way which mimics the analogous structure for Anosov flows. Locally, Y looks like a product—two nearby points can always be connected by a small step along a ‘stable’ set, then a small step along an ‘unstable’ set, then a small move along the flow. This coheres with the dynamics of the flow in that the ‘stable’ set is truly stable in the dynamical sense: there are constants $C,\unicode{x3bb}>0$ such that if x and y are nearby points on the same stable set, then

$$ \begin{align*} d(\phi_t x, \phi_t y) \leq Ce^{-\unicode{x3bb} t}d(x,y) \quad \mbox{for } t\geq 0. \end{align*} $$

The corresponding expression for exponential contraction in backwards time holds for pairs of nearby points on a common unstable set. (See [Reference PollicottPol87] or [Reference Constantine, Lafont and ThompsonCLT20a] for more detail.)

Metric Anosov flows have many of the properties of Anosov flows. As a first example (which we will use below), we have the following proposition.

The flows we consider in this paper are metric Anosov.

Theorem 2.10. [Reference Constantine, Lafont and ThompsonCLT20a, Theorem 3.4]

For a compact, locally CAT( $-1$ ) space X, the geodesic flow on $GX$ is metric Anosov with $\unicode{x3bb} =1$ . Specifically, the strong stable and unstable sets of Definition 2.7 are the stable and unstable sets for the Anosov flow. That is, there exist small $\delta>0$ and $C>0$ such that

$$ \begin{align*}d_{G{\tilde{X}}}(g_t {\tilde{\gamma}}, g_t {\tilde{\gamma}}') \leq Ce^{-t} d_{G{\tilde{X}}}({\tilde{\gamma}}, {\tilde{\gamma}}') \quad\mbox{for } t\geq 0 \mbox{ and }{\tilde{\gamma}}' \in W^{ss}_\delta({\tilde{\gamma}});\end{align*} $$

$$ \begin{align*}d_{G{\tilde{X}}}(g_{-t} {\tilde{\gamma}}, g_{-t} {\tilde{\gamma}}') \leq Ce^{-t} d_{G{\tilde{X}}}({\tilde{\gamma}}, {\tilde{\gamma}}') \quad\mbox{for } t\geq 0 \mbox{ and }{\tilde{\gamma}}' \in W^{uu}_\delta({\tilde{\gamma}}).\end{align*} $$

2.4 Doubling and covering properties

In the proof of Theorem 1.1, we will need a technical fact about the geometry of $GX$ . We collect the necessary arguments leading to that fact here.

Let $(X, d)$ be a metric space. We introduce two useful properties of metric spaces.

One can prove that doubling implies packing (although the converse is not true).

Now, let X be locally CAT( $-1$ ). Provided the double boundary $\partial _\infty ^{(2)}{\tilde {X}}$ of $\widetilde {X}$ satisfies the doubling property, one can show an analogue of a technical lemma [Reference Lopes and ThieullenLT05, Lemma 22].

Proof. Let $\{x_l\}_{l \in L}$ be a maximal set in $\Sigma $ such that $d(x_{l_1}, x_{l_2}) \geq 2 \epsilon $ . By maximality, $\bigcup _{l \in L} B(x_l, 2\epsilon )$ covers $\Sigma $ . By the doubling condition, we can cover each $B(x_l, 2\epsilon )$ by at most $N_1$ $\epsilon $ -balls for some $N_1$ depending only on $\Sigma $ . For each $l \in L$ , we choose this collection minimally and use $\{B(y_{lm}, \epsilon )\}_{m \leq M(l)}$ to denote the collection covering $B(x_l, 2\epsilon )$ .

Note that if $B(y_{l_1m_1}, \epsilon ) \cap B(y_{l_2m_2}, \epsilon ) \neq \varnothing $ , then $d(x_{l_1}, x_{l_2}) < 6\epsilon $ . Indeed, if $x \in B(y_{l_1m_1}, \epsilon ) \cap B(y_{l_2m_2}, \epsilon )$ , then for $i = 1, 2$ , we have that $d(y_{l_im_i}, x_{l_im_i}) \leq 2\epsilon $ and $d(x, y_{l_im_i}) < \epsilon $ , so by the triangle inequality, $d(x_{l_1}, x_{l_2}) < 6\epsilon $ .

We claim that for each $l_1$ ,

(2.1) $$ \begin{align} \#\{y_{l_2m_2}: B(y_{l_1m_1}, \epsilon) \cap B(y_{l_2m_2}, \epsilon) \neq \varnothing\} &\leq N_1(\#\{x_{l_2}: d(x_{l_1}, x_{l_2}) < 6\epsilon\}) \nonumber \\ &\leq N_1^2N_2 = N_1^4. \end{align} $$

The first inequality follows from the bound obtained above on $d(x_{l_1},x_{l_2})$ and the fact that at most $N_1$ of the $y_{l_i m_i}$ terms are associated to each $x_{l_i}$ . For the second inequality, we can cover $B(x_{l_1},6\epsilon )$ by at most $N_1$ balls of radius $3\epsilon $ by doubling. Since the $\{x_{l_i}\}$ are $2\epsilon $ -separated, by packing, at most $N_2$ of the $x_l$ terms are in each of these balls.

We now use $\{B(y_{l_m})\}_{l \in L}$ to construct a finite-valence dual graph $\Gamma $ as follows:

• • $V(\Gamma ) = \{y_{l_m}\}_{l \in L, 1 \leq m \leq M(l)}$ (where $M(l) \leq N_1$ );

• • $E(\Gamma ) = \{[y_{l_1m_2}, y_{l_2m_2}] : B(y_{l_1m_1}, \epsilon ) \cap B(y_{l_2m_2}, \epsilon ) \neq \varnothing \}$ .

By equation (2.1), each vertex in $V(\Gamma )$ has valence at most $N_1^4$ . By Brooks’ theorem ([Reference BrooksBro41] or [Reference LovászLov75]), $\Gamma $ can be colored with at most $C:=N_1^4+1$ colors. Relabel the set $\{y_{lm}\}$ to $\{y_{jk}\}$ , where $k \in [1, C]$ indexes the colors in the coloring of $V(\Gamma )$ , and ${j \in [1, J(k)]}$ indexes the vertices colored with the kth color.

By definition of a coloring of a graph, for a fixed $k^{\ast }$ , the set $\{y_{jk^{\ast }}\}_{1 \leq j \leq J(k^{\ast })}$ is a totally disconnected subgraph of $\Gamma $ . Then, the balls in $\{B(y_{jk^{\ast }}, \epsilon )\}_{1 \leq j \leq J(k^{\ast })}$ are pairwise disjoint, as desired.

To use Lemma 2.14, we need a metric on $\partial _\infty ^{(2)}{\tilde {X}}$ which is doubling. Recall that the $\ell ^{\infty }$ product metric on the product $X \times Y$ of two metric spaces $(X, d_X)$ and $(Y, d_Y)$ is defined by

$$ \begin{align*}d_{X \times Y}((x_1, y_1), (x_2, y_2)) = \max\{d_X(x_1, x_2), d_Y(y_1, y_2)\}.\end{align*} $$

The proof of the following is relatively straightforward.

Recall that we can equip the boundary of a CAT( $-1$ ) space with a visual metric, which we now define. First, if X is CAT( $-1$ ), given a choice of basepoint $x_0 \in X$ , we define the Gromov product of $\xi , \eta \in \partial ^{\infty } X$ , which is denoted by $(\xi .\eta )_{x_0}$ :

$$ \begin{align*}(\xi.\eta)_{x_0} = \lim\limits_{\substack{x_n \rightarrow \xi \\ y_n \rightarrow \eta}} \tfrac{1}{2}(d(x_0, x_n) + d(x_0, y_n) - d(x_n, y_n)).\end{align*} $$

Definition 2.16. (Visual metric for CAT( $-1$ ) spaces, [Reference BourdonBou95])

If X is a proper CAT(- $1$ ) space, then we can define the visual metric to be

$$ \begin{align*} d_{x_0}(\eta, \xi) = \begin{cases}e^{-(\xi.\eta)_{x_0}} &\text{ if } \xi \neq \eta, \\ 0 &\text{ otherwise}.\end{cases} \end{align*} $$

Although we do not delve into the definition of visual metrics for Gromov hyperbolic spaces, in general, we remark that they exist (see [Reference Bridson and HaefligerBH99, Ch. III.H]).

Recall that a metric space is proper if every closed ball is compact. We now show that for a proper geodesic Gromov hyperbolic metric space X equipped with a discrete, cocompact action (e.g., the CAT( $-1$ ) space ${\tilde {X}}$ ), $\partial _\infty ^{(2)}{\tilde {X}}$ equipped with a ( $\ell ^{\infty }$ ) product of visual metrics is doubling. To do so, we must define some stronger properties of metric spaces.

Definition 2.17. A metric space X is Ahlfors Q-regular if there exists a positive Borel measure $\mu $ on X such that

$$ \begin{align*} C^{-1}R^Q \leq \mu(B(x, R)) \leq CR^Q, \end{align*} $$

where $R \leq \text {diam}(X)$ and $C \geq 1$ is a constant independent of x and R.

While we defined doubling for metric spaces, we can also say a measure admitted by a metric space is doubling.

Definition 2.18. A positive Borel measure $\mu $ on a metric space X is doubling if there exists a constant $C = C({\mu })$ such that

$$ \begin{align*}\mu(B(x, 2r)) \leq C\mu(B(x, r)).\end{align*} $$

We now briefly prove a lemma that is usually assumed in the literature due to the simplicity of the proof.

Proof. We first note that if a metric space is Ahlfors Q-regular, then it admits a doubling measure. Indeed, $\mu $ itself is a doubling measure; by the Ahlfors Q-regularity of X, we have

$$ \begin{align*} C^{-1}R^Q \leq \mu(B(x, R)) \leq CR^Q\quad \text{and}\quad C^{-1}2^QR^Q \leq \mu(B(x, 2R)) \leq C2^QR^Q. \end{align*} $$

Thus, $R^Q \leq C\mu (B(x, R))$ , so

$$ \begin{align*} \mu(B(x, 2R)) \leq C2^QR^Q \leq 2^QC^2\mu(B(x, R)). \end{align*} $$

Thus, $\mu $ is doubling with constant $2^QC^2$ . Finally, by [Reference Luukkainen and SaksmanLS98], we have that a metric space is doubling if and only if it admits a doubling measure.

This allows us to state the following theorem.

The original proof of Theorem 2.20 is actually due to Coornaert (see [Reference CoornaertCoo93, Proposition 7.4]), who shows that any $\Gamma $ -quasiconformal measure on $\partial ^{\infty } X$ with support contained in the limit set of $\Gamma $ is Ahlfors Q-regular, where $\Gamma $ is a discrete, cocompact isometric group action on X. In particular, if $\mu $ is any non-zero measure on $\partial ^{\infty } X$ , then $\mu $ is $\Gamma $ -quasiconformal (see [Reference CoornaertCoo93, Proposition 4.3]). As a consequence of Lemma 2.19, we thus have the following corollary.

Finally, we reach a key corollary.

Proof. Note that by properties of locally CAT( $-1$ ) spaces, $\widetilde {X}$ is a proper geodesic Gromov hyperbolic space that admits a discrete, cocompact isometric group action, so by Theorem 2.20, $\partial ^{\infty } \widetilde {X}$ equipped with the visual metric is doubling. By Lemma 2.15,

$$ \begin{align*} (\partial ^{\infty } \widetilde {X} \times \partial ^{\infty } \widetilde {X}) \setminus \Delta = \partial ^{(2)}_{\infty } \widetilde {X} \end{align*} $$

equipped with the $\ell ^{\infty }$ product of two visual metrics is doubling.

3.1 Sections and Markov proper families

Again, let $(Y, d_Y)$ be a compact metric space with a continuous metric Anosov flow $\{\phi _t\}$ . Assume that $\phi _t$ has no fixed points.

We will use the following notation related to sections.

Definition 3.2. Given a section D, we denote by $\text {Int}_{\phi }D$ the interior of D transverse to the flow; that is,

$$ \begin{align*}\text{Int}_\phi D = D \cap \bigcap_{\epsilon>0}(\phi_{(-\epsilon, \epsilon)} D )^\circ.\end{align*} $$

Let $\text {Proj}_D: \phi _{[-\tau , \tau ]}D \to D$ be the projection map defined by $\text {Proj}_D(\phi _t y) = y$ .

Given a collection $\mathcal {D} =\{D_1, \ldots , D_n \}$ of disjoint sections, if $\bigcup _{i=1}^n \phi _{(-\alpha ,0)}\text {Int}_\phi D_i =Y$ , then any orbit of $\phi _t$ crosses an infinite sequence of sections. We let $\psi $ be the first-return map for this collection, and let $\tau $ be the first-return time for the collection, defined by $\psi (x)=\phi _{\tau (x)}x$ for all $x\in \bigcup _{i=1}^n D_i$ .

Any orbit of $\phi _t$ can be encoded by the bi-infinite sequence of sections it crosses. To make this encoding useful, however, sections need to be chosen carefully. Bowen [Reference BowenBow73] and Pollicott [Reference PollicottPol87] provide a framework for doing this in the Anosov and metric Anosov flow settings, respectively. Reference [Reference Constantine, Lafont and ThompsonCLT20a] further describes this process. The end result is a Markov proper family.

References [Reference BowenBow73, Reference Constantine, Lafont and ThompsonCLT20a, Reference PollicottPol87] all discuss conditions under which Markov proper families exist. Condition (3) does not play a role in the present paper, and we will work with a weaker version of condition (4) (see Lemma 3.12 below). For our purposes, the key result is the following.

3.2 Markov codings and Markov proper families

In this section, let $(\mathcal {R}, \mathcal {S})$ be a Markov proper family at scale $\alpha>0$ for a metric Anosov flow $\{\phi _t\}$ on Y. We discuss how the sections index orbits as sequences in a subshift of finite type.

Definition 3.5. For a Markov proper family $(\mathcal {R}, \mathcal {S})$ for a metric Anosov flow, we define the canonical coding space to be

$$ \begin{align*}\Theta = \Theta(\mathcal{R}) = \bigg\{\theta \in \prod_{-\infty}^\infty\{1, \ldots, n\}: \text{ for all } l, k \geq 0, \bigcap_{j=-k}^{l} \psi^{-j}(\text{Int}_\phi R_{\theta_j}) \not= \emptyset\bigg\}.\end{align*} $$

In [Reference PollicottPol87, §2.3], the symbolic space $\Theta (\mathcal {R})$ is shown to be a shift of finite type. Let $\sigma : \Theta \to \Theta $ be the associated left shift map.

Definition 3.6. For any $\theta \in \Theta $ , we define the local stable and unstable sets of $\theta $ to be

$$ \begin{align*} W_{\mathrm{loc}}^s(\theta) & := \{y \in \text{Int}_\phi(R_{\theta_0}): \text{ for all }k\geq 0, \psi^k(y) \in R_{\theta_k}\}, \\ W_{\mathrm{loc}}^u(\theta) & := \{y \in \text{Int}_\phi(R_{\theta_0}): \text{ for all }k\geq 0, \psi^{-k}(y) \in R_{\theta_{-k}}\}. \end{align*} $$

There is a canonically defined map $\pi : \Theta (\mathcal {R}) \to \mathcal {R}$ given by

$$ \begin{align*}\pi(\theta) = W_{\mathrm{loc}}^s(\theta) \cap W_{\mathrm{loc}}^u(\theta) = \bigcap_{j=-\infty}^\infty \psi^{-j}(R_{\theta_j}).\end{align*} $$

Since the flow is expansive (Proposition 2.9), this intersection is a single point on $R_{\theta _0}$ and the map $\pi $ is well defined. However, for any point $y \in \mathcal {R}$ , we have a canonical sequence $\theta ^y \in \Theta (\mathcal {R})$ given by

$$ \begin{align*}\theta^y = (\ldots, \theta^y_{-1} \mid \theta^y_{0}, \theta^y_{1}, \ldots ) \quad \text{where } \psi^k(y) \in R_{\theta^y_k}.\end{align*} $$

Therefore, $\pi (\theta ^y) = y$ .

3.3 Sections for flow on $GX$

In this section, we utilize attributes of the metric Anosov flow to define the tools and verify the properties needed to construct a ‘discretized’ subaction. Let $(X, d_X)$ be a locally CAT( $-1$ ) space with geodesic flow $\{g_t\}$ which is metric Anosov (Theorem 2.10).

Let $\Sigma $ be the disjoint union of the rectangle sections $\{\Sigma _i\}_{i\in I}$ in the Markov proper family given by Proposition 3.4. Let $\psi : \Sigma \to \Sigma $ be the Poincaré return map and let ${\tau : \Sigma \to (0, \infty )}$ be the return time map with respect to the flow $\{g_t\}$ . While constructing collections of sections at arbitrarily small scales is straightforward (see Proposition 3.4), the key aspect of the next lemma is the ability to construct collections whose sections have arbitrarily small diameters and such that the associated return time is bounded below by a fixed constant, $\tau _*$ .

Proof. We first start with any collection of Poincaré sections $\widehat {\Sigma } = \{\widehat {\Sigma }_i\}_{i \in \hat I}$ , given by the collection $\mathcal {B}$ in Proposition 3.4, and suppose the associated return time is bounded below and above by $t_*$ and $t^{*}$ , respectively. Note that $t_*>0$ since the sections in $\widehat {\Sigma }$ are disjoint and closed. By definition, the $(0, t^{*})$ -flow boxes $g_{(0, t^{*})}\text {Int}_{g} \widehat {\Sigma }_i$ of $\widehat {\Sigma }_i$ cover $GX$ ; that is, $GX = \bigcup _{i\in I} g_{(0, t^{*})}\text {Int}_{g} \widehat {\Sigma }_i$ . Choose $\alpha ^{*}$ small enough that $2\alpha ^{*} < t_*$ and the $(0, t^{*} - 2\alpha ^{*})$ -flow boxes of $\widehat {\Sigma }^{\prime }_i$ still cover $GX$ , where

$$ \begin{align*}\widehat{\Sigma}^{\prime}_i = \{\gamma \in \widehat{\Sigma}_i: d(\gamma, GX \backslash \widehat{\Sigma}_i)> 2\alpha^{*}\}.\end{align*} $$

We want to apply Lemma 2.14 to $\widehat {\Sigma }_i'$ , but must take some care, as that lemma is proven on $\partial ^{(2)}_\infty \tilde X$ . Consider a lift $\Sigma ^{*}$ of $\widehat {\Sigma }^{\prime }_i$ to $G\tilde X$ . Since $G\tilde X \cong \partial ^{(2)}_\infty \tilde X \times \mathbb {R}$ , $\Sigma ^{*}$ projects to a subset $\Sigma ^{*}_\infty $ of $\partial ^{(2)}_\infty \tilde X$ by forgetting the flow coordinate. Apply Lemma 2.14 (with covering constant C) to $\Sigma ^{*}_\infty $ . Then, remembering the flow coordinate associated to $\Sigma ^{*}$ , we project the results of Lemma 2.14 back to $\Sigma ^{*}$ and then project down to $GX$ .

The result is as follows. Given any $\alpha < \alpha ^{*}$ , we can cover each $\widehat {\Sigma }^{\prime }_i$ with sets $\{B_{ijk}\}_{j \in J, \: 1\leq k \leq C}$ in such a way that for fixed $i, k,$ the sets $\{B_{ijk}\}_{j\in J}$ are pairwise disjoint, and (using Lemma 2.4) such that for any $t\in [0,\alpha ^{*}]$ , $\mathrm{diam}\: g_tB_{ijk}<\alpha $ . We note here that in the proof of Lemma 2.14, the sets $B_{ijk}$ are $\epsilon $ -balls with respect to the $\ell ^\infty $ product of visual metrics. It is not hard to see that by taking $\epsilon $ small enough, we can ensure that the resulting sets have $d_{GX}$ -diameter as small as we need.

We then ‘stack’ C copies of $\widehat {\Sigma }_i$ along the flow, letting

$$ \begin{align*}\tilde{\Sigma}_{ijk} = g_{({k}/{C})\alpha^{*}} \widehat{\Sigma}_{i} \quad \text{and} \quad \Sigma_{ijk} = g_{({k}/{C})\alpha^{*}} B_{ijk}.\end{align*} $$

Note that by the choice of $\alpha ^{*}$ , $\Sigma _{ijk} \subset \tilde {\Sigma }_{ijk}$ . Let $\tau $ denote the return time associated to the sections $\Sigma = \{ \Sigma _{ijk}\}_{i,j,k}$ .

By design, for fixed i, $\{\Sigma _{ijk}\}_{k, j}$ are pairwise disjoint, but additionally, since the maximum height of the stack is $\alpha ^{*} < ({t_*}/{2})$ , no two stacks intersect; therefore, $\{\Sigma _{ijk} \}_{i,j,k}$ are all pairwise disjoint. Also, by construction, $\tau $ is at least ${\alpha ^{*}}/{C} =: \tau _*$ . Finally, we note that the longest $\tau $ can be is the sum of the maximum return time for $\widehat {\Sigma }'$ and the maximum stack height; that is, $\tau < (t^{*} -2\alpha ^{*}) + \alpha ^{*} = t^{*} -\alpha ^{*} =: \tau ^{*}$ , and so the $(0, \tau ^{*})$ -flow boxes for $\Sigma $ cover $GX$ :

$$ \begin{align*}\bigcup_{i,j,k} g_{(0, \tau^{*})} \text{Int}_{g} \Sigma_{ijk} \supset \bigcup_{i} g_{(0, t^{*}-2\alpha^{*})} \text{Int}_{g} \widehat{\Sigma}^{\prime}_i = GX.\end{align*} $$

Importantly, we note that $\tau _*$ and $\tau ^{*}$ are independent of the size $\alpha $ of the sections $\Sigma $ (see Figure 2).

Figure 2

A visual aid for the proof of Lemma 3.7. Note that the stack associated to $\Sigma _{ijk}$ is covered by the flow boxes for $\Sigma _{i'jk}$ . Here, $C = 3$ .

From now on, we fix the sections $\tilde {\Sigma }$ with fixed associated constants $\alpha ^{*}, \tau _*, \tau ^{*}$ given by Lemma 3.7; in particular, $\text {diam} \:\tilde {\Sigma }_i \leq \alpha ^{*}$ and its associated return time $\tilde {\tau }$ is bounded below and above by the fixed constants $\tau _*$ and $\tau ^{*}$ , respectively. By Lemma 3.7, we will be able to choose subsections $\Sigma $ such that $\text {diam}\:\Sigma _i= \alpha $ is arbitrarily small, while its associated return time $\tau $ preserves the same bounds.

Definition 3.8. Let $i, j \in I$ . We say $i \to j$ is a simple transition if there exists $\gamma \in \Sigma _i$ such that $\psi (\gamma ) \in \Sigma _j$ . Let

$$ \begin{align*}\text{dom}(\tilde{\psi}_{ij})=\text{dom}(\tilde{\tau}_{ij}) := \{\gamma \in \tilde{\Sigma}_{i}: g_t(\gamma) \in \tilde{\Sigma}_{j} \text{ for some } t \in (0, \tau^{*}) \}.\end{align*} $$

We define the extended first return time $\tilde {\tau }_{ij}$ and first return map $\tilde {\psi }_{ij}$ for the simple transition $i \to j$ as follows: for $\gamma \in \text {dom}(\tilde {\psi }_{ij})=\text {dom}(\tilde {\tau }_{ij})$ ,

$$ \begin{align*}\tilde{\tau}_{ij}(\gamma) = \inf\{t \in (0, \tau^{*}): g_t\gamma \in \tilde{\Sigma}_{j}\},\end{align*} $$

$$ \begin{align*}\tilde{\psi}_{ij}(\gamma) = g_{\tilde{\tau}_{ij}(\gamma)}\gamma.\end{align*} $$

In particular, using Lemma 3.7, we choose $\alpha $ to be small enough that $\Sigma _i \subseteq \text {dom}(\tilde {\psi }_{ij})$ and $\Sigma _j \subseteq \text {range}(\tilde {\psi }_{ij})$ for any simple transition $i \to j$ .

Note that for any canonical sequence $\theta \in \Theta (\Sigma )$ (see Definition 3.5), we have $\theta _{k} \to \theta _{k+1}$ is a simple transition for each $k \in \mathbb {Z}$ . Indeed, $\psi (\psi ^{k}(\pi (\theta )) ) = \psi ^{k+1}(\pi (\theta )) \in \Sigma _{\theta _{k+1}}$ . We now extend the canonical coding space $\Theta (\Sigma )$ to allow for double-sided sequences of simple transitions.

Definition 3.9. A pseudo-orbit is a double-sided sequence of simple transitions. Let $\Omega $ be the extended coding space of pseudo-orbits,

$$ \begin{align*}\Omega = \{\omega = (\ldots, \omega_{-1} \mid \omega_{0}, \omega_1, \ldots): \: \omega_k \to \omega_{k+1} \text{ is a simple transition for all } k\}.\end{align*} $$

It is immediate from its definition that $\Omega $ is also a sub-shift of finite type, and we denote by $\sigma : \Omega \to \Omega $ the associated left shift map. Note that $\Theta \subset \Omega $ . For each $\omega \in \Omega $ , define

$$ \begin{align*}\tilde{\psi}_\omega = \tilde{\psi}_{\omega_0 \omega_1}: \Sigma_{\omega_0} \to \tilde{\Sigma}_{\omega_1},\end{align*} $$

$$ \begin{align*}\tilde{\tau}_\omega = \tilde{\tau}_{\omega_0 \omega_1},\end{align*} $$

and, on the domain where the composition is well defined,

$$ \begin{align*}\tilde{\psi}_\omega^k = \tilde{\psi}_{\sigma^{k-1}(\omega)}\circ \cdots \circ \tilde{\psi}_{\omega}.\end{align*} $$

As before, we define the local stable and unstable sets of a pseudo-orbit $\omega \in \Omega $ . These are where the composition $\tilde {\psi }^k_\omega $ is well defined for all positive (respectively, negative) k.

Definition 3.10. For any $\omega \in \Omega $ , we define the local stable and unstable sets of $\omega $ to be

$$ \begin{align*} W_{\mathrm{loc}}^s(\omega) & := \{\gamma \in \text{Int}_\phi(\Sigma_{\omega_0}): \text{ for all }k\geq 0, \tilde{\psi}_{\omega}^k(\gamma) \in \Sigma_{\omega_k}\} ,\\ W_{\mathrm{loc}}^u(\omega) & := \{\gamma \in \text{Int}_\phi(\Sigma_{\omega_0}): \text{ for all }k\geq 0, \tilde{\psi}_{\omega}^{-k}(\gamma) \in \Sigma_{\omega_{-k}}\}. \end{align*} $$

For any $\omega , \omega ' \in \Omega $ with $\omega _0 = \omega _0'$ , $W_{\mathrm {loc}}^s(\omega )$ intersects $W_{\mathrm {loc}}^u(\omega ')$ at a unique point. This is due to the fact that $\alpha $ can be made arbitrarily small and Bowen’s shadowing lemma: consider the sequence formed by concatenating the past of $\omega '$ with the future of $\omega $ : $\zeta = (\ldots , \omega ^{\prime }_{-2}, \omega ^{\prime }_{-1} \mid \omega _0, \omega _1, \ldots )$ . By definition, for each j, $\zeta _j \to \zeta _{j+1},$ so there exists $\gamma _j \in \Sigma _{\zeta _j}$ such that $\psi _\zeta (\gamma _j) \in \Sigma _{\zeta _{j+1}}$ . This $(\gamma _j)_{j \in \mathbb {Z}}$ is a pseudo-orbit and by Bowen’s shadowing lemma, there exists a unique element $\gamma $ whose true orbit $(\psi _\zeta (\gamma ))_{j \in \mathbb {Z}}$ shadows $(\gamma _j)_{j \in \mathbb {Z}}$ . Therefore, $\{\gamma \} = W_{\mathrm {loc}}^s(\omega ) \cap W_{\mathrm {loc}}^u(\omega ')$ .

We now demonstrate that $\tilde {\psi }_\omega $ satisfies a property that is related to and resembles the Markov property (Definition 3.3).

Lemma 3.12. (Markov property for $\tilde {\psi }_\omega $ )

$\tilde {\psi }_\omega $ stabilizes the local stable set:

$$ \begin{align*}\tilde{\psi}_\omega(W^s_{\mathrm{loc}}(\omega)) \subset W^s_{\mathrm{loc}}(\sigma(\omega)),\end{align*} $$

and $\tilde {\psi }^{-1}_\omega $ stabilizes the local unstable set:

$$ \begin{align*}\tilde{\psi}_\omega^{-1}(W^u_{\mathrm{loc}}(\omega)) \subset W^u_{\mathrm{loc}}(\sigma^{-1}(\omega)).\end{align*} $$

Consequently, $\tilde {\psi }_\omega (\pi (\omega )) = \pi (\sigma (\omega ))$ .

Proof. Let $\gamma \in W^s_{\mathrm {loc}}(\omega )$ . Then, for any $k \geq 0$ ,

$$ \begin{align*}\tilde{\psi}_{\sigma(\omega)}^k(\tilde{\psi}_\omega(\gamma)) = \tilde{\psi}_\omega^{k+1}(\gamma) \in \Sigma_{\omega_{k+1}} = \Sigma_{(\sigma(\omega))_{k}},\end{align*} $$

so $\tilde {\psi }_\omega (\gamma ) \in W^s_{\mathrm {loc}}(\sigma (\omega ))$ . Similarly, let $\gamma \in W^u_{\mathrm {loc}}(\omega )$ . Then, for any $k \geq 0$ ,

$$ \begin{align*}\tilde{\psi}_{\sigma^{-1}(\omega)}^{-k}(\tilde{\psi}^{-1}_\omega(\gamma)) = \tilde{\psi}_\omega^{-(k+1)}(\gamma) \in \Sigma_{\omega_{-(k+1)}} = \Sigma_{(\sigma^{-1}(\omega))_{-k}},\end{align*} $$

so $\tilde {\psi }^{-1}_\omega (\gamma ) \in W^u_{\mathrm {loc}}(\sigma ^{-1}(\omega ))$ .

3.4 Exponential contraction

Exponential contraction along $W^s_{\mathrm {loc}}(\theta )$ under the return map $\psi $ for any $\theta \in \Theta $ plays an essential role in later proofs. We will prove our contraction result (an analog for our situation of [Reference Lopes and ThieullenLT05, Lemma 8(iii)]) for the simple first return map $\psi $ associated to the collection $\Sigma $ if sections constructed above. We note that the result holds for the ‘return map with instructions’ $\tilde {\psi }_\omega $ since $\tilde {\psi }_\omega $ is at any point $\psi ^k$ for an appropriate k.

Lemma 3.13. Let $\gamma , \gamma '\in W^s_{\mathrm {loc}}(\omega )$ for some $\omega \in \Omega $ and let $\psi $ be the first return map for the collection of sections $\Sigma $ . Then, there exists a constant $C>0$ (independent of $\gamma ,\gamma ',\omega $ ) such that for all $k>0$ ,

$$ \begin{align*}d_{GX}(\psi^k \gamma, \psi^k \gamma') < Ce^{-k\tau_*} d_{GX}(\gamma,\gamma').\end{align*} $$

Similarly, if $\gamma ,\gamma '\in W^u_{\mathrm {loc}}(\omega )$ , for all $k>0$ ,

$$ \begin{align*}d_{GX}(\psi^{-k} \gamma, \psi^{-k} \gamma') < Ce^{-k\tau_*} d_{GX}(\gamma,\gamma').\end{align*} $$

For this proof, we will need a sublemma. Consider the following setup (for the stable half of the proof). Suppose that $\gamma , \gamma '\in W^s_{\mathrm {loc}}(\omega ) \subset \Sigma _i$ and that $\psi ^k\gamma , \psi ^k \gamma ' \in \Sigma _j$ . Since $\gamma , \gamma '\in W^s_{\mathrm {loc}}(\omega )$ , $\gamma '$ is on the weak-stable leaf through $\gamma $ and hence for some $r_1$ , $g_{r_1}\gamma '\in W^{ss}(\gamma )$ . Similarly, as $\psi ^k \gamma , \psi ^k \gamma '\in W^s_{\mathrm {loc}}(\sigma ^k\omega )$ , for some $r_2$ , $g_{r_2} \psi ^k \gamma ' \in W^{ss}(\psi ^k \gamma )$ . (See Figure 3.)

Figure 3

The geometric setup for Lemma 3.13.

Proof. We prove the sublemma for $r_1$ ; the proof for $r_2$ is the same.

Note, since the strong stable set $W^{ss}(\gamma )$ is determined by values of $B_{\gamma (0)} (-,\gamma (+\infty ))$ , $|r_1| = B_{\gamma (0)}(\gamma '(0),\gamma (+\infty ))$ . Busemann functions are 1-Lipschitz, so $B_{\gamma (0)}(\gamma '(0), \gamma (+\infty )) \leq d_X(\gamma (0), \gamma '(0))$ . Therefore (using Lemma 2.3),

$$ \begin{align*}|r_1|\leq d_X(\gamma(0), \gamma'(0)) \leq 2d_{GX}(\gamma,\gamma'),\end{align*} $$

as desired.

Proof of Lemma 3.13

We prove the result only for the local stable sets, the proof for the unstable sets being essentially the same. As usual, we lift to the CAT( $-1$ ) universal cover to make our geometric arguments.

We continue with the setup and notation illustrated in Figure 3. Let $\tau , \tau '$ be such that $g_\tau \gamma = \psi ^k \gamma $ and $g_{\tau '} \gamma ' = \psi ^k \gamma '$ . Note that by construction of $\Sigma $ , $\tau , \tau ' \geq k \tau _*$ . Since $g_\tau $ maps $W^{ss}(\gamma )$ to $W^{ss}(\psi ^k \gamma )$ , we see that $g_{r_2} \psi \gamma ' = g_\tau g_{r_1} \gamma '$ and so $\tau '=\tau +(r_1-r_2).$

By the triangle inequality,

$$ \begin{align*}d_{GX}(\gamma, g_{r_1}\gamma') \leq d_{GX}(\gamma,\gamma') + |r_1|,\end{align*} $$

$$ \begin{align*}d_{GX}(\psi^k\gamma, \psi^k\gamma') \leq d_{GX}(\psi^k \gamma, g_{r_2}\psi^k \gamma') +|r_2|.\end{align*} $$

Since $g_t$ is a metric Anosov flow with $\unicode{x3bb} =1$ , for some uniform $C_1>0$ ,

$$ \begin{align*}d_{GX}(\psi^k \gamma, g_{r_2}\psi^k \gamma') \leq C_1 e^{-k \tau_*} d_{GX}(\gamma, g_{r_1}\gamma').\end{align*} $$

Combining these inequalities gives

$$ \begin{align*} d_{GX}(\psi^k \gamma, \psi^k \gamma') &\leq d_{GX}(\psi^k \gamma, g_{r_2}\psi^k \gamma') + |r_2| \\ & \leq C_1 e^{-k\tau_*} d_{GX}(\gamma, g_{r_1}\gamma')+|r_2| \\ & \leq C_1 e^{-k\tau_*} [d_{GX}(\gamma,\gamma')+|r_1|]+|r_2|. \end{align*} $$

Then, using Sublemma 3.15,

$$ \begin{align*} d_{GX}(\psi^k \gamma, \psi^k \gamma') &\leq C_1 e^{-k \tau_*}[d_{GX}(\gamma,\gamma')+ 2 d_{GX}(\gamma,\gamma')]+2d_{GX}(\psi^k \gamma, g_{r_2}\psi^k \gamma') \\ & \leq 3C_1 e^{-k\tau_*} d_{GX}(\gamma,\gamma') + 2C_1 e^{-k\unicode{x3bb}\tau_*} d_{GX}(\gamma, g_{r_1}\gamma') \\ & \leq 3C_1 e^{-k\tau_*} d_{GX}(\gamma,\gamma') + 2C_1 e^{-k\tau_*}[d_{GX}(\gamma,\gamma')+r_1] \\ & \leq 3C_1 e^{-k\tau_*} d_{GX}(\gamma,\gamma') + 2C_1 e^{-k\tau_*}[d_{GX}(\gamma,\gamma') + 2d_{GX}(\gamma,\gamma')] \\ & = 9C_1 e^{-k\tau_*} d_{GX}(\gamma,\gamma'), \end{align*} $$

as desired.

We also have the following analog of [Reference Lopes and ThieullenLT05, Lemma 8(iv)] which gives very rough upper and lower bounds on the distance between two points under iterates of our return map $\tilde \psi $ .

Lemma 3.16. There exist constants $K^{*}$ and $\Lambda _*^s < 0 < \Lambda _*^u$ such that the following holds. Suppose that $x,y\in \Sigma _i$ and that $i=i_0 \to \cdots \to i_n$ is a chain of simple transitions so that $\tilde \psi ^n:=\tilde \psi _{i_{n-1}i_n}\circ \cdots \circ \tilde \psi _{i_0i_1}$ exists at x and y. Then,

$$ \begin{align*}(K^{*})^{-1}\exp(n\Lambda_*^s)d_{GX}(x,y) \leq d_{GX}(\tilde \psi^n x,\tilde \psi^n y) \leq K^{*} \exp(n\Lambda_*^u)d_{GX}(x,y).\end{align*} $$

Proof. Say $\tilde \psi ^n x = g_tx$ and $\tilde \psi ^n y = g_{t'}y$ . Note that $t\leq n\tau ^{*}$ and that $|t-t'|$ is the difference in first return times for x and y and under flow from $\Sigma _{i}$ to $\Sigma _{i_n}$ .

By triangle inequality,

$$ \begin{align*}d_{GX}(\tilde \psi^n x, \tilde \psi^n y) \leq d_{GX}(g_tx, g_ty)+ |t-t'|.\end{align*} $$

By Lemma 2.4 and the Lipschitz behavior of return times given by Proposition 3.4,

$$ \begin{align*} d_{GX}(\tilde \psi^n x, \tilde \psi^n y) &\leq e^{2t}d_{GX}(x, y)+ Kd_{GX}(x, y) \\ & \leq (K+1) \exp((2\tau^{*})n)d_{GX}(x,y). \end{align*} $$

To prove the other bound, we can simply reverse the direction of the flow: $x = g_{-t}\tilde \psi x$ and $y = g_{-t'}\tilde \psi y$ . Applying the same argument, we get

$$ \begin{align*}d_{GX}(x,y)\leq (K+1) \exp((2\tau^{*})n)d_{GX}(\tilde \psi^nx, \tilde \psi^ny),\end{align*} $$

which completes the proof.

We now prove two lemmas that will be useful in proving that the discretized subaction is Hölder in §4.

Lemma 3.17. There exists $\delta> 0$ depending only on $\Sigma $ such that for any $i \in I$ and any $\gamma , \gamma ' \in \Sigma _i$ , if $d_{GX}(\gamma , \gamma ') < \delta $ , then there exists a simple transition $i \to j$ and $m, n \geq 1$ such that

$$ \begin{align*}\psi^m(\gamma) \in \Sigma_j,\: \psi^n(\gamma') \in \Sigma_j.\end{align*} $$

Furthermore, the first return times from $\Sigma _i$ to $\Sigma _j$ satisfy

$$ \begin{align*}\sum_{k=0}^{m-1} \tau\circ \psi^k(\gamma) < \tau^{*}, \ \ \ \sum_{k=0}^{n-1} \tau\circ \psi^k(\gamma') < \tau^{*}.\end{align*} $$

Proof. Let $U_i = g_{(0, \tau ^{*})} \Sigma _i$ . For any simple transition $i \to j$ , let

$$ \begin{align*}\Delta_{ij} = \{ \eta \in \Sigma_i: g_t(\eta)\in\Sigma_j \mbox{ for some } t\in (0,\tau^{*})\}\end{align*} $$

and let $t_j:\Delta _{ij}\to (0,\tau ^{*})$ be the time such that $g_{t_j(\eta )}\eta \in \Sigma _j$ . Note that when ${\eta \in \Delta _{ij}}$ , $\phi ^n(\eta )\in \Sigma _j$ for some $n\geq 1$ and $t_j(\eta ) = \sum _{k=0}^{n-1}\tau \circ \psi ^k(\eta ).$ Therefore, showing that $\gamma ,\gamma '\in \Delta _{ij}$ will prove the lemma.

Since $\tau ^{*}$ is the maximum return time, the sets $\{\Delta _{ij}\}_j$ cover $\Sigma _i$ , which is compact. Therefore, we can find a Lebesgue number $\delta> 0$ for the cover $\{\Delta _{ij}\}_j$ . If $\gamma '$ is contained in the $\delta $ -ball around $\gamma $ , then there exists $j\in I$ such that the ball is contained in $\Delta _{ij}$ . In particular, $\gamma , \gamma ' \in \Delta _{ij}$ .

In the following, $\Lambda _*^u$ and $K^{*}$ are constants given by Lemma 3.16.

Proof. Let $\gamma _0= \gamma $ and $\gamma ^{\prime }_0= \gamma ' \in \Sigma _i$ . By Lemma 3.17, since $d_{GX}(\gamma _0, \gamma ^{\prime }_0) < \delta $ , there exists $m_1, n_1$ and $i_1 \in I$ such that

$$ \begin{align*}\gamma_1 = \psi^{m_1}(\gamma_0), \quad \gamma^{\prime}_1 = \psi^{n_1}(\gamma^{\prime}_0) \in \Sigma_{i_1}.\end{align*} $$

By Lemma 3.16,

$$ \begin{align*} d_{GX}(\gamma_1, \gamma^{\prime}_1) \leq K^{*} e^{\Lambda_*^u} d_{GX}(\gamma_0, \gamma^{\prime}_0) < K^{*} e^{\Lambda_*^u} \frac{\delta}{K^{*}} e^{-N\Lambda_*^u} \: = \delta e^{-(N-1)\Lambda_*^u}\:<\delta, \end{align*} $$

so we can apply Lemma 3.17 again. We repeat this construction N times to find two sequences $(m_1, m_2, \ldots , m_N)$ and $(n_1, \ldots , n_N)$ such that for each $0 \leq l \leq N$ ,

$$ \begin{align*}\gamma_l = \psi^{m_1 + \cdots + m_l}(\gamma) \quad \text{and}\quad \gamma^{\prime}_l = \psi^{n_1 + \cdots + n_l}(\gamma')\end{align*} $$

belong to the same section $\Sigma _{i_l}$ with $d_{GX}(\gamma _l, \gamma ^{\prime }_l) < \delta $ .

Let $\theta ^\gamma , \theta ^{\gamma '}\in \Theta (\Sigma )$ be the canonical sequences associated to $\gamma $ and $\gamma '$ , respectively, and let

$$ \begin{align*}m = m_1 + \cdots + m_N, \quad n= n_1 + \cdots + n_N.\end{align*} $$

Define

$$ \begin{align*} \omega & = (\ldots, \theta_{-2}^{\gamma}, \theta_{-1}^{\gamma} \mid i_0, i_1, \ldots, i_N, \theta_{m+1}^{\gamma}, \theta_{m+2}^{\gamma}, \ldots), \\ \omega' & = (\ldots, \theta_{-2}^{\gamma'}, \theta_{-1}^{\gamma'} \mid i_0, i_1, \ldots, i_N, \theta_{n+1}^{\gamma'}, \theta_{n+2}^{\gamma'}, \ldots). \end{align*} $$

By construction, $\tilde {\psi }_{\omega }^k(\gamma ) \in \Sigma _{\omega _k}$ and $\tilde {\psi }_{\omega '}^k(\gamma ') \in \Sigma _{\omega ^{\prime }_k}$ for all $k \in \mathbb {Z}$ , so $\gamma = \pi (\omega )$ and ${\gamma ' = \pi (\omega ')}$ , and they coincide in the first $N+1$ times, as desired.

5.1 Constructing subsections

To prove Proposition 5.1, following the ideas of [Reference Lopes and ThieullenLT05], we will construct a nested sequence of Poincaré sections. The following lemma is a basic tool necessary in these arguments.

Below, let $\Sigma '$ be a subsection of $\Sigma $ as in Lemma 5.2. Let $\tau '$ and $\psi '$ be the corresponding first-return time and first-return map. For $\gamma \in \Sigma '$ , define

$$ \begin{align*}\mathcal{H}'(\gamma) : = \int_0^{\tau'(\gamma)}(A-m(A))\circ g_t\gamma \,dt - (\mathcal{V}\circ \psi'(\gamma)-\mathcal{V}(\gamma)).\end{align*} $$

By Proposition 4.1, $\mathcal {H}'\geq 0.$ To prove Proposition 5.1, we need to extend $\mathcal {H}'$ to a function $H'$ which is defined on all of $GX$ .

5.2 A smoothing function

A key element in the proof of Proposition 5.1 is the ‘smoothing function’ h provided by Lemma 5.5, which is stated later in this section. This function will allow us to take the values of $\mathcal {H}$ , currently concentrated on the section $\Sigma $ and smooth them out over orbits of the geodesic flow.

We first prove the key lemma necessary for the proof of Lemma 5.5.

We now specify our input functions.

Let $\mathcal {U}$ be a cover for a metric space X. Recall that a cover $\mathcal {U}$ has multiplicity at most $k \geq 0$ if any $x \in X$ belongs to at most k members of $\mathcal {U}$ . With these definitions in mind, we consider the following proposition.

Proposition 5.4. [Reference Dadarlat and GuentnerDG07, Proposition 4.1]

Let $\mathcal {U}$ be a cover of a metric space X with multiplicity at most $k + 1$ (where $k \geq 0$ ) and Lebesgue number $L> 0$ . For $U \in \mathcal {U}$ , define

$$ \begin{align*}\varphi_U(x) = \frac{d(x, X \setminus U)}{\sum\limits_{V \in \mathcal{U}} d(x, X \setminus V)}.\end{align*} $$

Then, $\{\varphi _U\}_{U \in \mathcal {U}}$ is a partition of unity on X subordinated to the cover $\mathcal {U}$ . Moreover, each $\varphi _U$ satisfies, for all $x, y \in X$ ,

$$ \begin{align*}\lvert \varphi_U(x) - \varphi_U(y) \rvert \leq \frac{2k + 3}{L}\,d(x, y).\end{align*} $$

Furthermore, the family $(\varphi _U)_{U \in \mathcal {U}}$ satisfies, for all $x, y \in X$ ,

$$ \begin{align*}\sum\limits_{U \in \mathcal{U}}\lvert \varphi_U(x) - \varphi_U(y) \rvert \leq \frac{(2k + 2)(2k + 3)}{L}\,d(x, y).\end{align*} $$

Proof of Lemma 5.3

Let $\psi : \mathbb {R} \rightarrow \mathbb {R}$ be a bump function supported on the interval $(-\epsilon , \epsilon )$ with the property that $\int _{-\epsilon }^{\epsilon } \psi (\gamma )\,dx = 1$ , where $\epsilon << \delta $ . Let $W_i = g_{(-\alpha , 0)} \text {Int} B_i \cap (\bigcup _{j = 1}^n g_{(-\delta - \epsilon , \delta + \epsilon )} B_j)^c$ . Let us denote $\varphi _{W_i}$ from Proposition 5.4 as $\varphi _i$ . Given $\gamma \in GX$ , we define the function,

$$ \begin{align*}h_i(\gamma) := ((\varphi_i \circ g_t)(\gamma) \ast \psi)(0) = \int_{\mathbb{R}} (\varphi_i \circ g_{- t})(\gamma)\psi(t)\,dt = \int_{\mathbb{R}} (\varphi_i \circ g_{t})(\gamma)\psi(- t)\,dt.\end{align*} $$

Note that the two integrals are equal by symmetry of convolution.

(1) Support of $\mathbf {h_i}$ . First, observe that since supp $(\psi ) = (-\epsilon , \epsilon )$ , it follows that

$$ \begin{align*}h_i(\gamma) = \int_{\mathbb{R}} (\varphi_i \circ g_{-t})(\gamma)\psi(t)\,dt = \int_{-\epsilon}^{\epsilon} (\varphi_i \circ g_{t})(\gamma)\psi(-t)\,dt.\end{align*} $$

Let $\gamma \in U_i$ . Then, there exists some union of non-empty open intervals $\mathcal {U}$ that includes an open interval around $0$ such that $(\varphi _i \circ g_{-t})(\gamma ) \neq 0$ for all $t \in \mathcal {U}$ . As a result,

$$ \begin{align*}h_i(\gamma) = \int_{\mathcal{U} \cap (-\epsilon, \epsilon)} (\varphi_i \circ g_{-t})(\gamma)\psi(t)\,dt> 0.\end{align*} $$

This shows that $\gamma \in \text {supp}(h_i)$ , so $U_i \subseteq \text {supp}(h_i)$ .

If $\gamma \in \bigcup \nolimits _{j = 1}^{n} g_{(-\delta , \delta )} B_j$ , then for $t \in (-\epsilon , \epsilon )$ , $g_t(\gamma ) \in \bigcup \nolimits _{j = 1}^{n} g_{(-\delta - \epsilon , \delta + \epsilon )} B_j$ , which is not in the support of $\varphi _i$ . Thus,

$$ \begin{align*} h_i(\gamma) = \int_{-\epsilon}^{\epsilon} (\varphi_i \circ g_{-t})(\gamma)\psi(t)\,dt = 0.\end{align*} $$

(2) Lipschitz continuity. Let $\gamma _1, \gamma _2 \in GX$ . Assuming that $\gamma _1$ and $\gamma _2$ are chosen so that supp $(\psi ) \cap \text {supp}((\varphi _i \circ g_{-t})(\gamma _1) - (\varphi _i \circ g_{-t})(\gamma _2))$ is non-empty (otherwise, the inequality is trivial), then for some $T> \epsilon $ ,

$$ \begin{align*} \lvert h_i(\gamma_1) - h_i(\gamma_2) \rvert &= \bigg\lvert \int_{\mathbb{R}} (\varphi_i \circ g_{-t})(\gamma_1)\psi(t) - (\varphi_i \circ g_{-t})(\gamma_2)\psi(t) \,dt \bigg\rvert\\&= \bigg\lvert\! \int_{\mathbb{R}} (\varphi_i \circ g_{t})(\gamma_1)\psi(-t) - (\varphi_i \circ g_{t})(\gamma_2)\psi(-t) \,dt \bigg\rvert\\&\leq \int_{\mathbb{R}} \lvert ((\varphi_i \circ g_{t})(\gamma_1) - (\varphi_i \circ g_{t})(\gamma_2))\psi(-t)\rvert \,dt \\&\leq \int_{\mathbb{R}} \bigg(\frac{2K + 3}{L}\bigg)d_{GX}(g_{t}(\gamma_1), g_{t}(\gamma_2))\lvert \psi(-t) \rvert \,dt &\text{(Proposition }5.4)\\&\leq \bigg(\frac{2K + 3}{L}\bigg)e^{2T}d_{GX}(\gamma_1, \gamma_2) \int_{\mathbb{R}} \lvert \psi(-t)\rvert \,dt &\text{(Lemma }2.4)\\&= \bigg(\frac{2K + 3}{L}\bigg)e^{2T}d_{GX}(\gamma_1, \gamma_2). \end{align*} $$

We now explain our choice of $T> 0$ . Notice that since we are integrating over a subset of supp $(\psi ) = (-\epsilon , \epsilon )$ , it follows that any $T> \epsilon $ will suffice.

(3) Smoothness along the flow direction. First, we show that $h_i$ is smooth along the flow direction. To do so, we show the infinite differentiability of the function (for any $s \in \mathbb {R}$ , not just $s = 0$ ):

$$ \begin{align*}((\varphi_i \circ g_t)(\gamma) \ast \psi)(s) = \int_{\mathbb{R}} (\varphi_i \circ g_{t})(\gamma)\psi(s - t)\,dt.\end{align*} $$

The proof proceeds similarly to the classical case. Recall that

$$ \begin{align*} \frac{\partial}{\partial s} ((\varphi_i \circ g_t)(\gamma) \ast \psi)(s) &= \lim\limits_{h \rightarrow 0} \dfrac{((\varphi_i \circ g_t) \ast \psi)(\gamma, s + h) - ((\varphi_i \circ g_t) \ast \psi)(\gamma, s)}{h} \\ &= \lim\limits_{h \rightarrow 0} \int_{\mathbb{R}} (\varphi_i \circ g_t)(\gamma)\bigg(\frac{\psi(s + h - t) - \psi(s - t)}{h}\bigg)\,dt. \end{align*} $$

The next step is to apply the dominated convergence theorem. Since $\psi $ is smooth with compact support, all the derivatives of $\psi $ are bounded, so we can set $M> 0$ to be some number such that ${d}/{dt} \lvert \psi (t) \rvert \leq M$ . We claim that $M(\varphi _i \circ g_t)(\gamma )$ is an appropriate dominating function. Note that it suffices to show $(\varphi _i \circ g_t)(\gamma )$ is integrable. Suppose $\gamma \notin W_i$ . Then,

$$ \begin{align*} \int_{-\infty}^{\infty} \lvert (\varphi_i \circ g_t)(\gamma) \rvert \,dt &\leq \int_{-\alpha}^{\alpha} \lvert (\varphi_i \circ g_t)(\gamma) - \underbrace{\varphi_i(\gamma)}_{0} \rvert \,dt \\ & \leq \int_{-\alpha}^{\alpha} \frac{2K + 3}{L} d_{G\widetilde{X}}(g_t(\gamma), \gamma)\,dt & \text{(Proposition~}5.4) \\&= \int_{-\alpha}^{\alpha} \frac{(2K + 3)t}{L} \,dt < \infty. \end{align*} $$

Otherwise, if $\gamma \in W_i$ , then simply replace $\varphi _i(\gamma )$ with $(\varphi _i \circ g_{-2\alpha })(\gamma ) = 0$ , for example, so t in the last line will be replaced with $t + 2\alpha $ , in which case the integral is still finite.

This allows us to finish showing the derivative exists:

$$ \begin{align*} \frac{\partial}{\partial s} ((\varphi_i \circ g_t)(\gamma) \ast \psi)(s) &= \int_{\mathbb{R}} (\varphi_i \circ g_t)(\gamma)\psi'(s - t)\,dt. \end{align*} $$

By induction, we have that

$$ \begin{align*}\frac{\partial^n ((\varphi_i \circ g_t)(\gamma) \ast \psi)}{\partial s}(s) = \int_{\mathbb{R}} (\varphi_i \circ g_t)(\gamma)\frac{\partial^n}{\partial s}\psi(s - t)\,dt.\end{align*} $$

Thus, for any $s \in \mathbb {R}$ , $((\varphi _i \circ g_t) \ast \psi )(\gamma , s)$ is smooth.

(4) Boundedness of the integral. It remains to bound $\int _0^{\tau (\gamma )} (h_i \circ g_t)(\gamma )\,dt$ below under the condition that $B_r(\gamma ) \subset U_i=W_i$ .

Recall that $\mathcal {U} = \{ g_{(-\alpha , 0)}B_i \setminus (\bigcup _{j = 1}^n g_{(-\delta - \epsilon , \delta + \epsilon )}B_j) \}_{i = 1}^n$ . Note that $D_0 := \max _{\gamma \in GX} \sum \nolimits _{U \in \mathcal {U}} d(\gamma , GX \setminus U)$ exists as the sum is finite and the diameter of $GX$ is finite.

Then, we have

$$ \begin{align*} &\int_{0}^{\tau(\gamma)} h_i(\gamma)\,ds \\&\quad= \int_{0}^{\tau(\gamma)} \int_{\mathbb{R}} (\varphi_i \circ g_{-t} \circ g_s)(\gamma)\psi(t)\,dt\,ds\\&\quad= \int_{0}^{\tau(\gamma)} \int_{\mathbb{R}} (\varphi_i \circ g_{s - t})(\gamma)\psi(t)\,dt\,ds = \int_{0}^{\tau(\gamma)} \int_{\mathbb{R}} (\varphi_i \circ g_{t})(\gamma)\psi(s - t)\,dt\,ds\\&\quad\geq \int_{0}^{\tau_{\ast}} (\varphi_i \circ g_{t})(\gamma)\int_{t - \epsilon}^{t + \epsilon} \psi(s - t)\,dt\,ds \geq \int_{0}^{\epsilon} (\varphi_i \circ g_t)(\gamma)\int_{t - \epsilon}^{t + \epsilon} \psi(s - t)\,dt\,ds \\&\quad\geq \int_{0}^{\epsilon} (\varphi_i \circ g_t)(\gamma)\int_{-\epsilon}^{\epsilon} \psi(u)\,du\,dt = \int_{0}^{\epsilon} (\varphi_i \circ g_t)(\gamma)\,dt \\&\quad\geq \int_{0}^{\epsilon} \frac{d_{G\gamma}(g_t(\gamma), GX \setminus W_i)}{D_0}\,dt. \end{align*} $$

To complete the argument, we simply need to bound $\int _0^\epsilon d_{GX}(g_t(\gamma ), GX\setminus W_i)$ below. Since $B_r(\gamma )\in W_i$ , and the geodesic flow is unit speed for $d_{GX}$ ,

$$ \begin{align*}d(g_t \gamma, GX\setminus W_i) \geq r-|t| \mbox{ for } t\in [-r,r].\end{align*} $$

Consider two cases.

Case I: $r>\epsilon $ .

$$ \begin{align*} \int_0^\epsilon d_{GX}(g_t \gamma, GX\setminus W_i) \,dt \geq \int_0^\epsilon r-|t|\,dt = r\epsilon - \frac{\epsilon^2}{2}> \frac{\epsilon^2}{2}. \end{align*} $$

Case II: $r\leq \epsilon $ .

$$ \begin{align*} \int_0^\epsilon d_{GX}(g_t \gamma, GX\setminus W_i) \,dt \geq \int_0^r r-|t|\,dt = \frac{r^2}{2}. \end{align*} $$

Letting $C(r) = \min \{ {\epsilon ^2}/{2}, {r^2}/{2}\}>0$ , we have the desired result.

We are now ready to prove Lemma 5.5.

Lemma 5.5. There exists a globally Lipschitz continuous, smooth along orbits, non-negative function $h: GX \rightarrow \mathbb {R}_{\geq 0}$ that is null in a neighborhood of $\bigcup \nolimits _{j = 1}^n \Sigma _j$ such that for all $\gamma \in GX$ ,

$$ \begin{align*}\int_0^{\tau(\gamma)} h \circ g_t(\gamma)\,dt \geq C\end{align*} $$

for some constant $C> 0$ .

Proof. We prove the lemma assuming Lemma 5.3. Let $N = GX \setminus (\bigcup \nolimits _{j = 1}^n g_{(-\delta , \delta )} \Sigma _j)$ , where $\delta \ll \tau _{\ast }$ . From Definition 3.3, we know

$$ \begin{align*} N &= \bigg(\bigcup\limits_{i = 1}^n g_{(-\alpha, 0)}\text{Int}_{g}(\Sigma_i)\bigg) \cap \bigg(\bigcup\limits_{j = 1}^n g_{(-\delta, \delta)} \Sigma_j\bigg)^c \\ &= \bigcup\limits_{i = 1}^n \bigg(g_{(-\alpha, 0)}\text{Int}_{g}(\Sigma_i) \cap \bigg(\bigcup_{j = 1}^n g_{(-\delta, \delta)} (\Sigma_j)\bigg)^c\bigg). \end{align*} $$

Set $U_i =: g_{(-\alpha , 0)}\text {Int}_{g}(\Sigma _i) \cap \big (\bigcup _{j = 1}^n g_{(-\delta , \delta )} (\Sigma _j)\big )^c$ . By Lemma 5.3, there exists some $h_i: GX \rightarrow \mathbb {R}_{\geq 0}$ whose support contains $U_i$ and is contained in N that is smooth in the flow direction and Lipschitz continuous. Let $h=\sum _{i=1}^n h_i$ . It is smooth in the flow direction, Lipschitz continuous, and null on a neighborhood of $\bigcup _{j=1}^n \Sigma _j.$

Note that $\mathcal {U} = \{ U_i\}$ forms an open cover of the compact space $GX$ . Let $\rho>0$ be a Lebesgue number for this cover. Then, for every $\gamma \in GX$ , there is some i such that $B_\rho (\gamma )\subset U_i$ . Applying part (4) of Lemma 5.3, we find that

$$ \begin{align*}\int_0^{\tau(\gamma)} h \circ g_t(\gamma)\,dt \geq \int_0^{\tau(\gamma)} h_i \circ g_t(\gamma)\,dt \geq C(\rho)>0.\end{align*} $$

Here, $C(\rho )$ depends only on $\rho $ , and hence only on the geometry of the sections, not on $\gamma $ , so it can serve as the constant C.

5.3 Inductive extension

To extend the discretized sub-action, we will need to augment the concept of simple transitions.

Lopes and Thieullen demonstrated (see [Reference Lopes and ThieullenLT05, Lemma 17]) that if we can make the diameter $\alpha $ of the subsections $\Sigma \subset \tilde {\Sigma }$ sufficiently small—as permitted by Lemma 3.7—then the rank of any multiple transition is bounded above by ${2\tau ^{*}}/{\tau _*}$ . This allows us to sensibly define N as the maximum rank of any multiple transition for $\{\Sigma _i\}.$

To extend $\mathcal {H}$ to the function $H'$ specified in Proposition 5.1, we follow the inductive scheme of [Reference Lopes and ThieullenLT05]. The reason for this inductive argument is ensuring the regularity of $H'$ . Extending $\mathcal {H}$ to a flow box based on an individual $\Sigma _i$ is straightforward (see Lemma 5.7 below). However, when two of these flow boxes overlap, ensuring regularity of the resulting extension requires care.

We begin by defining sets used in the construction. Let $i\Rightarrow j$ be a rank n transition. Using Lemma 5.2, let $\{ \Sigma _i^k\}_i$ for $k=0, \ldots , N$ be a sequence of Poincaré sections such that for all i, $\Sigma _i^0 = \Sigma _i$ and $\Sigma _i^{k+1}\subset \bar \Sigma _i^{k+1} \subset \Sigma _i^k$ for $k=0, \ldots N-1$ . Let $\Sigma _{ij}^k = \{ \gamma \in \Sigma _i^k : \psi _{ij}(\gamma ) \in \Sigma _j^k\}.$ The flow box of rank n and size k for $i\Rightarrow j$ is

$$ \begin{align*}B^k_{ij} = \{ g_t\gamma : \gamma\in \Sigma_{ij}^k, 0\leq t\leq \tau_{ij}(\gamma)\}.\end{align*} $$

Let

$$ \begin{align*}\Sigma_{ijk}^n = \{\gamma\in \Sigma_{ij}^n : g_t\gamma \in \Sigma_k^n \quad\mbox{for some } 0\leq t \leq \tau_{ij}(\gamma) \}\end{align*} $$

and

$$ \begin{align*}U_{ijk}^n = \{ g_t\gamma : \gamma\in \Sigma_{ijk}^n, 0\leq t \leq \tau_{ij}(n)\}.\end{align*} $$

These are the points in $\Sigma _{ij}^k$ (and corresponding partial flow box) which hit $\Sigma _k^n$ between $\Sigma _i$ and $\Sigma _j$ (see Figure 5). Note that

$$ \begin{align*} U_{ijk}^n &= \{g_t\gamma: \gamma\in \Sigma_{ijk}^n, 0\leq t \leq \tau_{ik}(\gamma) \} \\ &\quad \cup \{g_t\gamma: \gamma\in \Sigma_{ijk}^n, \tau_{ik}(\gamma)\leq t \leq \tau_{ij}(\gamma) \} \end{align*} $$

is the union of two partial flow boxes each of rank less than n.

Figure 5

An illustration of the flow boxes $U_{ijk}^1$ and $B_{ij}^1$ .

Let

$$ \begin{align*}\mathcal{U}^n = \bigcup_{\text{rank}(i\Rightarrow j)\leq n} B_{ij}^n\end{align*} $$

be the union of all flow boxes of size n and rank $\leq n$ . Note that $\mathcal {U}^N = GX.$

We now inductively build $H^n$ on $\mathcal {U}^n$ ; $H^N$ will be the $H'$ asked for in Proposition 5.1. For $\gamma \in \Sigma ^0_{ij}$ , let

$$ \begin{align*}\mathcal{H}_{ij}(\gamma) := \int_0^{\tau_{ij}^0(\gamma)}(A-m(A))\circ g_t \gamma \,dt - (\mathcal{V}\circ \psi^0_{ij}(\gamma)-\mathcal{V}(\gamma)).\end{align*} $$

Since $\mathcal {H}_{ij}$ is a sum of $\mathcal {H}\circ \psi ^k$ , it is non-negative. Let $i \Rightarrow j$ be a multiple transition of any rank. On the flow box $B_{ij}^0$ , define $H_{ij}^0$ by

$$ \begin{align*}H_{ij}^0 \circ g_t (\gamma) = \mathcal{H}_{ij}(\gamma) \frac{h\circ g_t (\gamma)}{\int_0^{\tau_{ij}(\gamma)} h\circ g_t(\gamma) \,dt}\end{align*} $$

for all $\gamma \in \Sigma _{ij}^0$ , where h is the function provided by Lemma 5.5.

Lemma 5.7. $H_{ij}^0$ is Lipschitz, smooth in the flow direction, null in a neighborhood of $\bigcup _{j=1}^n \Sigma _j$ , and satisfies

$$ \begin{align*}\mathcal{H}_{ij}(\gamma) = \int_0^{\tau_{ij}(\gamma)} H^0_{ij} \circ g_t (\gamma) \,dt\end{align*} $$

for all $\gamma \in \Sigma _{ij}^0$ .

The core idea of the construction of $H'$ is in the definition of $H_{ij}^0$ . However, $\{H^0_{ij}\}$ do not jointly define a well-defined function, as there is no reason for them to agree on the overlaps of the $\{B^0_{ij}\}$ . Even if this issue is fixed, at the transitions between flow boxes, there is no reason for a function patched together from $\{H_{ij}^0\}$ to satisfy the necessary regularity conditions. The inductive construction of [Reference Lopes and ThieullenLT05] is designed to fix these issues.

The only overlaps between the sets $\{B^1_{ij}: \text {rank}(i\to j)=1\}$ occur on $\bigcup _i\Sigma _i$ . Since h is zero in a neighborhood of this set, $H^1$ is well defined. It satisfies the integrability condition and regularity conditions thanks to Lemma 5.7.

Now suppose that $H^n: \mathcal {U}^n \to \mathbb {R}$ , satisfying the integrability and regularity conditions have been defined. Write $H_{ij}^n$ for the restriction of $H^n$ to $B_{ij}^n$ .

To define $H^{n+1}$ , suppose $\text {rank}(i\Rightarrow j)=n+1$ . For all k such that $\Sigma _{ijk}^n \neq \emptyset $ , as noted above, $U_{ijk}^n$ is a union of two partial flow boxes, each of rank $\leq n$ . Hence, $H^n$ is already defined on these partial flow boxes. Therefore, on the partial flow box $\{g_t \gamma : \gamma \in \bigcup _k \Sigma _{ijk}^n, 0\leq t \leq \tau _{ij}(\gamma )\}$ , $H^n$ provides a well-defined function $H_{ij}^n$ satisfying the integrability and regularity conditions.

On $\{g_t \gamma : \gamma \in \Sigma _{ij}^n \setminus \bigcup _k \Sigma _{ijk}^n, 0\leq t \leq \tau _{ij}(\gamma )\}$ , we have only $H_{ij}^0$ . We glue these two functions together with a partition of unity. Let $p,q:\Sigma _i^n \to [0,1]$ be Lipschitz functions such that $p+q=1$ on $\overline {\Sigma }_{ij}^{n+1}$ and so that

$$ \begin{align*}\textrm{supp}(p) \subset \bigcup_k \Sigma_{ijk}^{n+1}\end{align*} $$

$$ \begin{align*}\textrm{supp}(q) \subset \Sigma_{ij}^n \setminus \bigcup_k \Sigma_{ijk}^{n+1}.\end{align*} $$

Define $H_{ij}^{n+1}$ on $B_{ij}^{n+1}$ by

$$ \begin{align*}H_{ij}^{n+1}(g_t\gamma) = p(\gamma) H_{ij}^n(g_t\gamma) + q(\gamma)H_{ij}^0(g_t\gamma).\end{align*} $$

Then, $H^{n+1}:\mathcal {U}^{n+1}\to \mathbb {R}$ is well defined by setting $H^{n+1}|_{B_{ij}^{n+1}} = H_{ij}^{n+1}$ . It is straightforward to check that it satisfies the integrability condition given that $H_{ij}^n$ , $H_{ij}^0$ do and that $p+q=1$ on $\Sigma _{ij}^{n+1}$ . The regularity follows from the regularity of $H_{ij}^n$ and $H_{ij}^0$ , and the fact that p and q are Lipschitz. This finishes the proof of Proposition 5.1.

6.1 Surface amalgams

The following definition is adapted from [Reference LafontLaf07, Definition 2.3], which instead uses the terminology two-dimensional P-manifolds.

If Y forms a totally geodesic subspace of X consisting of disjoint simple closed curves, we say that X is simple. If each connected component of Y (gluing curve) is attached to at least three distinct boundary components of chambers, then we say X is thick. Like Lafont in [Reference LafontLaf07], we will only be considering simple, thick, negatively curved surface amalgams, as doing so ensures the surface amalgam is locally CAT( $-1$ ) (see Figure 6).

Figure 6

An example of a simple, thick surface amalgam with four chambers.

We will equip a simple, thick negatively curved surface amalgam X with a metric in a class we denote as $\mathcal {M}_{\leq }$ , following the notation from [Reference Constantine and LafontCL19]. Roughly speaking, metrics in $\mathcal {M}_{\leq }$ are piecewise Riemannian metrics with an additional condition that limits pathological behavior around the gluing curves of X. More precisely, we say $g \in \mathcal {M}_{\leq }$ if g satisfies the following properties:

1. (1) each chamber of $C \subset X$ is equipped with a negatively curved Riemannian metric with sectional curvature bounded above by $-1$ so that C has geodesic boundary components;

2. (2) the restrictions of g to the chambers of X are ‘compatible’ in the sense that if two boundary components $b_1$ and $b_2$ of two (possibly the same) chambers $C_1$ and $C_2$ are both attached to a gluing curve $\gamma \subset X$ , then the gluing maps $b_1 \hookrightarrow \gamma $ and $b_2 \hookrightarrow \gamma $ are isometries (in particular, we do not allow circle maps of degree two;

3. (3) for any two boundary components $b_1 \in C_1$ and $b_2 \in C_2$ , the restriction of g to $N_{b_1} \bigcup \nolimits _{b_1 \sim b_2} N_{b_2}$ is a negatively curved smooth Riemannian metric with sectional curvature bounded above by $-1$ , where $N_{b_1}$ and $N_{b_2}$ are $\epsilon $ -neighborhoods around $b_1$ and $b_2$ , respectively, for some $\epsilon> 0$ .

We impose the third condition to ensure that we can exploit previous marked length spectrum rigidity results for surfaces which, in particular, require Riemannian negatively curved metrics with at most a finite number of cone singularities. We now discuss some properties of $\mathcal {M}_{\leq }$ that will be useful in the proof of Theorem 6.6.

Indeed, suppose X is equipped with a metric $g \in \mathcal {M}_{\leq }$ and $C \subset X$ is a chamber in X. Recall a generalization of the Cartan–Hadamard theorem which states that a smooth Riemannian manifold M has sectional curvature $\leq \kappa $ if and only if M is locally CAT( $\kappa $ ) (see [Reference Bridson and HaefligerBH99, Theorem 1A.6]). As a result, the restriction of g to C is locally CAT( $-1$ ) since C is endowed with a negatively curved metric with sectional curvature bounded above by $-1$ . If $\kappa \in \mathbb {R}$ , and $X_1$ and $X_2$ are locally CAT( $\kappa $ ) spaces glued isometrically along a convex, complete metric subspace $A \subset X_1 \cap X_2$ , then $X_1 \sqcup _A X_2$ is locally CAT( $\kappa $ ) (see [Reference Bridson and HaefligerBH99, Theorem 2.11.1]). As a result, a negatively curved surface amalgam $(X, g)$ with locally CAT( $-1$ ) chambers will also be locally CAT( $-1$ ), as claimed.

6.2 Volume rigidity

Let $\mathscr {G}\widetilde X$ be the set of unparameterized, unoriented geodesics in $\widetilde X$ . A geodesic current on $(X, g)$ is a $\pi _1$ -invariant Radon measure on $\mathscr {G}\widetilde {X}$ .

There are two especially important examples of geodesic currents. We will assume, for simplicity, that $(X, g)$ is locally CAT( $-1$ ). Furthermore, we assume there is a well-defined notion of transversality in $\mathscr {G}\widetilde X$ . If $(X, g)$ is a negatively curved surface, there is a natural such notion since $\partial ^\infty \widetilde {X}$ is a circle. For other settings, transversality must be defined with more care; for examples, see [Reference Constantine and LafontCL19, §6] and [Reference WuWu23, §2.4.1].

First, given a homotopy class $[\alpha ] \in \pi _1(X)$ with geodesic representative $\alpha $ , there is a counting current associated with $\alpha $ that assigns to each Borel set $E \subset \mathscr {G}\widetilde {X}$ the measure

$$ \begin{align*}\mu_{\alpha}(E) = \lvert E \cap \{\widetilde{\alpha}\} \rvert,\end{align*} $$

where $\{\widetilde {\alpha }\}$ denotes the set of lifts of $\alpha $ to $\widetilde {X}$ . We follow convention and, with abuse of notation, write $\alpha := \mu _\alpha $ , which makes clear the fact that the collection of closed geodesics of $(X, g)$ embeds naturally into the space of geodesic currents.

Second, there is a Liouville current which, roughly speaking, captures lengths of closed geodesics. There is no general definition of a Liouville current; rather, the Liouville current should capture important properties, which we state in Assumption 6.3. Before stating the assumption, we need to introduce the notion of intersection numbers.

In the case of negatively curved Riemannian surfaces, the geometric intersection number of two closed geodesics (viewed as geodesic currents) extends to a symmetric, bilinear form (see [Reference BonahonBon86, Proposition 4.5]), the intersection number of two geodesic currents. Explicitly, given two geodesic currents $\mu , \nu $ , one can define the intersection number of $\mu $ and $\nu $ as