1 Introduction
The study of the statistical properties of dynamical systems is one of the main pillars of ergodic theory. In particular, one of the principal lines of investigation is to try and obtain quantitative information on the long-term behaviour of orbits (such as return and hitting times, dynamical extremal indices or logarithm laws).
In a metric space
$(X,d)$
, the problem of the shortest distance between two orbits of a dynamical system
$T:X\to X$
, with an ergodic measure
$\mu $
, was introduced in [Reference Burns, Masur, Matheus and WilkinsonBLR]. That is, for
$n\in \mathbb {N}$
and
$x, y\in X$
, they studied
$$ \begin{align*}\mathbb{M}_n(x, y)= \mathbb{M}_{T, n}(x, y):=\min_{0\le i,j\le n-1} d(T^i(x), T^j(y))\end{align*} $$
and showed that the decay of
$\mathbb {M}_n$
depends on the correlation dimension.
The lower correlation dimension of
$\mu $
is defined by
$$ \begin{align*}\underline{C}_\mu:=\liminf_{r\to 0} \frac{\log\int\mu(B(x, r))~d\mu(x)}{\log r},\end{align*} $$
and the upper correlation dimension
$\overline {C}_\mu $
is analogously defined via the limsup. If these are equal, then this is
$C_\mu $
, the correlation dimension of
$\mu $
. This dimension plays an important role in the description of the fractal structure of invariant sets in dynamical systems and has been widely studied from different points of view: numerical estimates (e.g. [Reference Badii and BroggiBB, Reference Bessis, Paladin, Turchetti and VaientiBPTV, Reference Sprott and RowlandsSR]), existence and relations with other fractal dimension (e.g. [Reference Barbaroux, Germinet and TcheremchantsevBGT, Reference PesinP]), and relations with other dynamical quantities (e.g. [Reference Faranda and VaientiFV, Reference ManticaM]).
It is worth mentioning that the problem of the shortest distance between orbits is a generalisation of the longest common substring problem for random sequences, a key feature in bioinformatics and computer science (see e.g. [Reference WatermanW]).
In [Reference Burns, Masur, Matheus and WilkinsonBLR, Theorem 1], under the assumption
$\underline {C}_\mu>0$
, a general lower bound for
$\mathbb {M}_n$
was obtained, as follows.
Theorem 1.1. For a dynamical system
$(X,T,\mu )$
, we have
$$ \begin{align*} \limsup_{n}\frac{\log \mathbb{M}_{T, n}(x, y)}{-\log n} \le \frac{2}{\underline{C}_\mu}, \quad \mu\times\mu\text{-almost every (a.e.) } x, y. \end{align*} $$
To replace the inequality above with equality, in [Reference Burns, Masur, Matheus and WilkinsonBLR, Theorems 3 and 6], the authors assumed that
$C_\mu $
exists and proved
$$ \begin{align} \liminf_{n}\frac{\log \mathbb{M}_{T,n}(x, y)}{-\log n} \ge \frac{2}{C_\mu}, \quad \mu\times\mu\text{-a.e. } x, y, \end{align} $$
using some exponential mixing conditions on the system.
One could naturally wonder if this mixing condition could be relaxed or even dropped. In [Reference Burns, Masur, Matheus and WilkinsonBLR], a partial answer was given and it was proved that for irrational rotation (which are not mixing), the inequality in Theorem 1.1 could be strict.
In this paper, we extend the above results in equation (1.1) to discrete systems with no requirement on mixing conditions. The main tool in proving our positive results is inducing: the idea in the discrete case is to first take advantage of the fact that Theorem 1.1 holds in great generality (including, as we note later, to higher-dimensional hyperbolic cases) and then to show that if there is an induced version of the system satisfying equation (1.1), then this inequality passes to the original system.
Moreover, we also extend the results of [Reference Burns, Masur, Matheus and WilkinsonBLR] to flows. Thus, we first have to prove an analogue of Theorem 1.1 and observe that in the continuous setting, the correct scaling is
$C_\mu -1$
. Then, using inducing via Poincaré sections, we also obtain an analogue of equation (1.1).
We will give examples for all of these results both in the discrete and continuous setting. We also give a class of examples in §5 where the conclusions of [Reference Burns, Masur, Matheus and WilkinsonBLR] fail to hold. This class is slowly mixing and also does not admit an induced version.
Finally, we emphasise that one of the obstacles to even wider application is proving that the correlation dimension
$C_\mu $
exists, see §3.1 for some discussion and results. For suspension flows, under some natural assumptions, we will show in §4.2 that if the correlation dimension of the base exists, then the correlation dimension of the invariant measure of the flow also exists.
2 Main results and proofs for orbits closeness in the discrete case
2.1 The main theorem in the non-uniformly expanding case
We will suppose that given
$(X, T, \mu )$
, there is a subset
$Y=\overline {\bigcup _iY_i}\subset X$
and an inducing time
$\tau :Y\to \mathbb {N}\cup \{\infty \}$
, constant on each
$Y_i$
and denoted
$\tau _i$
, so that our induced map is
$F=T^\tau : Y\to Y$
. We suppose that there is an F-invariant probability measure
$\mu _F$
with
$\int \tau ~d\mu _F<\infty $
which projects to
$\mu $
by the following rule:
$$ \begin{align} \mu(A)=\frac{1}{\int\tau~d\mu_F} \sum_i\sum_{k=0}^{\tau_i-1}\mu_F(Y_i\cap T^{-k}(A)). \end{align} $$
We call
$(Y, F, \mu _F)$
an inducing scheme, or an induced system for
$(X, T, \mu )$
. For systems which admit an inducing scheme, we have our main theorem, as follows.
Theorem 2.1. Assume that the inducing scheme
$(Y, F, \mu _F)$
satisfies equation (1.1) and that
$C_{\mu _F}=C_{\mu }$
. Then,
$$ \begin{align*} \lim_{n\to \infty}\frac{\log \mathbb{M}_{T, n}(x, y)}{-\log n} = \frac{2}{C_{\mu}}, \quad \mu\times\mu\text{-a.e. } x, y.\end{align*} $$
In §5, we give an example of a class of mixing systems where the conclusion of this theorem fails. These systems do not have good inducing schemes, see Remark 5.2 below.
Remark 2.2. As can be seen from the proof of this theorem, as well as related results in this paper, in fact what we prove is that if there is an induced system satisfying
$$ \begin{align*}\liminf_{n}\frac{\log \mathbb{M}_{F,n}(x, y)}{-\log n} \ge \frac{2}{\overline{C}_{\mu_F}}, \quad \mu_F\times\mu_F\text{-a.e. } x, y,\end{align*} $$
then
$$ \begin{align*}\liminf_{n\to \infty}\frac{\log \mathbb{M}_{T, n}(x, y)}{-\log n} \ge \frac{2}{\overline{C}_{\mu_F}}, \quad \mu\times\mu\text{-a.e. } x, y,\end{align*} $$
with the analogous statements for flows in §4.
In §3, we will give examples of systems where
$\{Y_i\}_i$
is countable,
$\mu $
and
$\mu _F$
are absolutely continuous with respect to Lebesgue, and
$C_{\mu _F}=C_{\mu }$
.
Proof of Theorem 2.1
The main observation here is that it is sufficient to prove that
$\lim _{k\to \infty } ({\log \mathbb {M}_{T,n_k}(x, y)}/({-\log n_k})) \ge {2}/{C_{\mu }}$
along a subsequence
$(n_k)_k$
which scales linearly with k.
For
$x\in Y$
, define
$\tau _n(x):=\sum _{k=0}^{n-1} \tau (F^k(x))$
. Given
$\varepsilon>0$
and
$N\in \mathbb {N}$
, set
$$ \begin{align*}U_{\varepsilon, N}:=\{x\in Y:|\tau_n(x)-n\bar\tau|\le \varepsilon n\text{ for all } n\ge N\}.\end{align*} $$
These are nested sets and by Birkhoff’s ergodic theorem, we have
$\lim _{N\to \infty }\mu _F (U_{\varepsilon , N})=1$
. In particular, by equation (2.1),
$\mu (U_{\varepsilon , N})>0$
for N sufficiently large and hence
$$ \begin{align*}\lim_{n\to \infty}\mu\bigg(\bigcup_{i=0}^{\lfloor \varepsilon N\rfloor}T^{-i}(U_{\varepsilon, N})\bigg)=1.\end{align*} $$
So for
$\mu \times \mu $
-typical
$(x, y)\in X\times X$
, there is
$N\in \mathbb {N}$
such that
$x, y\in \bigcup _{i=0}^{\lfloor \varepsilon N\rfloor }T^{-i}(U_{\varepsilon , N})$
. Set
$i, j\le \lfloor \varepsilon N\rfloor $
minimal such that
$T^i(x), T^{j}(y)\in U_{\varepsilon , N}\subset Y$
. Then [Reference Burns, Masur, Matheus and WilkinsonBLR, Theorem 3] implies that for any
$\eta>0$
and sufficiently large n,
$$ \begin{align*}\frac{\log \mathbb{M}_{F,n}(T^i(x), T^{j}(y))}{-\log n} \ge \frac{2}{C_{\mu_F}}-\eta=\frac{2}{C_{\mu}}-\eta.\end{align*} $$
Putting together the facts that the n-orbit by F of
$T^i(x)$
(respectively
$T^{j}(y)$
) is a subset of the
$\tau _n(x)$
- (respectively
$\tau _n(y)$
-) orbit by T of
$T^i(x)$
(respectively
$T^{j}(y)$
) and that
$i, j, |\tau _n(T^i(x))-n\bar \tau |, |\tau _n(T^{j}(y))-n\bar \tau | \le n\varepsilon $
for
$n\ge N$
, we obtain
$$ \begin{align*}\mathbb{M}_{T,n\lceil \bar\tau+2\varepsilon\rceil}(x, y)\leq \mathbb{M}_{T,n\lceil \bar\tau+\varepsilon\rceil}(T^i(x), T^{j}(y))\leq \mathbb{M}_{F,n}(T^i(x), T^{j}(y))\end{align*} $$
and thus
$$ \begin{align*} \frac{\log \mathbb{M}_{T,n\lceil \bar\tau+2\varepsilon\rceil}(x, y)}{-\log n} \ge \frac{2}{C_{\mu}}-\eta.\end{align*} $$
Observing that
$\lim _{n\to \infty }({\log n\lceil \bar \tau +2\varepsilon \rceil }/{\log n})=1$
and taking limit in the previous equation, we deduce that
$\lim _{n\to \infty }({\log \mathbb {M}_{T,n}(x, y)}/({-\log n})) \ge {2}/{C_{\mu }}-\eta $
. Since
$\eta $
can be choose arbitrary small, the theorem is proved.
2.2 The main theorem in the non-uniformly hyperbolic case
We next consider systems
$T:X\to X$
with invariant measure
$\mu $
which are non-uniformly hyperbolic in the sense of Young, see [Reference YoungY1]. Then there is some
$Y\subset X$
and an inducing time
$\tau $
defining
$F=T^\tau :Y\to Y$
, with measure
$\mu _F$
, which is uniformly expanding modulo uniformly contracting directions. We can quotient out these contracting directions to obtain a system
$\bar F: \bar Y\to \bar Y$
, which has an invariant measure
$\mu _{\bar F}$
.
Theorem 2.3. Assume that the induced system
$(\bar Y, \bar F, \mu _{\bar F})$
satisfies equation (1.1) and that
$C_{\mu }=C_{\mu _F}$
. Then,
$$ \begin{align*} \lim_{n\to \infty}\frac{\log \mathbb{M}_{T, n}(x, y)}{-\log n} = \frac{2}{C_{\mu}}, \quad \mu\times\mu\text{-a.e. } x, y.\end{align*} $$
The proof is directly analogous to that of Theorem 2.1.
2.3 Requirements on the induced system
In [Reference Burns, Masur, Matheus and WilkinsonBLR], the main requirement for equation (1.1) to hold is that the system has some Banach space
${\mathcal C}$
of functions from X to
$\mathbb {R}$
,
$\theta \in (0, 1)$
and
$C_1\geq 0$
such that for all
$\varphi , \psi \in {\mathcal C}$
and
$n\in \mathbb {N}$
,
$$ \begin{align*}\bigg|\int \psi\cdot\varphi\circ F^n~d\mu_F-\int\psi~d\mu_F\int\varphi~d\mu_F\bigg|\le C_1\|\varphi\|_{{\mathcal C}} \|\psi\|_{{\mathcal C}}\theta^n.\end{align*} $$
Some regularity conditions on the norms of characteristics on balls and the measures were also required, as well as a topological condition on our metric space (always satisfied for subset of
$\mathbb {R}^n$
with the Euclidean metric and subset of a Riemannian manifold of bounded curvature), but we leave the details to [Reference Burns, Masur, Matheus and WilkinsonBLR]. We can also remark that for Lipschitz maps on a compact metric space with
${\mathcal C} = \mathrm {Lip}$
, these regularity conditions can be dropped [Reference Galatolo, Rousseau and SaussolGoRS].
In [Reference Burns, Masur, Matheus and WilkinsonBLR, Theorem 3], the main application was to systems where
${\mathcal C} = BV$
so, for example, we have a Rychlik interval map, and in [BLR, Theorem 6], the main application was to Hölder observables, so that the induced system is Gibbs–Markov, see for example [Reference AlvesA, §3].
3 Examples in the discrete setting
Examples of our theory require an inducing scheme and, ideally, well-understood correlation dimensions. In [Reference Pesin and WeissPW], correlation dimension is dealt with in the Gibbs–Markov setting in the case
$\{Y_i\}_i$
is a finite collection of sets, but under inducing, we usually expect this collection to be infinite (in which case much less is known), so this is not directly relevant here.
The simplest case in the context of our results is when the invariant probability measure
$\mu $
for the system is d-dimensional Lebesgue, or is absolutely continuous with respect to Lebesgue (an acip) with a regular density, since in these cases, the correlation dimension for both
$\mu $
and the corresponding measure for the system is d.
3.1 Existence of the correlation dimension
First of all, we will give a result which implies that the correlation dimension for regular acips exists.
Proposition 3.1. Let
$X\subset \mathbb {R}^d$
. If
$\mu $
is a probability measure on X which is absolutely continuous with respect to the d-dimensional Lebesgue measure such that its density
$\rho $
is in
$L^2$
, then
$$ \begin{align*}C_\mu=d.\end{align*}$$
Proof. The fact that
$\overline {C}_\mu \le d$
follows, for example, from [Reference Fan, Lau and RaoFLR, Theorem 1.4].
To prove a lower bound, we start by defining the Hardy–Littlewood maximal function (see e.g. [Reference Stein and ShakarchiSS, Ch. 2.4]) of
$\rho $
:
$$ \begin{align*}M\rho(x) = \sup_{r>0} \frac1{\mathrm{Leb}(B(x, r))}\int_{B(x, r)}\rho(x)~dx.\end{align*} $$
Moreover, by Hardy–Littlewood maximal inequality,
$M\rho \in L^2$
and there exists
$c_1>0$
(depending only on d) such that
$$ \begin{align*}\|M\rho\|_2\leq c_1 \|\rho\|_2.\end{align*}$$
Thus, using the Cauchy–Schwarz inequality, we have
$$ \begin{align*} \int \mu(B(x, r))~d\mu(x)&\le \int M\rho(x)\cdot \mathrm{Leb}(B(x,r))\cdot\rho(x)~dx \\ &\le \mathrm{Leb}(B(x,r))\bigg(\int\rho^2~dx\bigg)^{1/2} \bigg(\int(M\rho)^2~dx\bigg)^{1/2}\le Kr^d \end{align*} $$
for some
$K>0$
. Hence,
$\underline {C}_\mu \ge d$
and thus
$C_\mu =d$
.
If the density of the acip is not sufficiently regular, the correlation dimension may differ from the correlation dimension of the Lebesgue measure, as in the following case.
Proposition 3.2. Let
$\alpha \in (1/2, 1)$
. Assume that
$\mu $
is supported on
$[0, 1]$
and
$d\mu = \rho \,dx$
with
$\rho (x) = x^{-\alpha }$
. Then, we have
$$ \begin{align*} C_\mu = 2(1-\alpha).\end{align*} $$
Proof. We write
$\int \mu (B(x, r))~d\mu (x) = \int _0^{2r} \mu (B(x, r))~d\mu (x)+ \int _{2r}^1 \mu (B(x, r))~d\mu (x)$
. We estimate the first term from above by
$$ \begin{align*}\int_0^{2r}\mu((0, 3r))~d\mu(x)= \mu((0, 2r))\mu((0, 3r))\asymp r^{2(1-\alpha)}.\end{align*} $$
For the second term, we split the sum into
$\int _{nr}^{(n+1)r}x^{-\alpha }\int _{x-r}^{x+r}t^{-\alpha }~dt~dx$
for
$n=2, \ldots , \lceil 1/r\rceil $
. This yields
$$ \begin{align*}\int_{nr}^{(n+1)r}x^{-\alpha}\int_{x-r}^{x+r}t^{-\alpha}~dt~dx \le \int_{nr}^{(n+1)r} x^{-\alpha}((n-1)r)^{-\alpha}2r \lesssim r^2(nr)^{-2\alpha}.\end{align*} $$
Since
$(1/n^{2\alpha })_n$
is a summable sequence, we estimate
$\int \mu (B(x, r))~d\mu (x)$
from above by
$r^{2(1-\alpha )}$
. Therefore,
$\underline {C}_\mu \ge 2(1-\alpha )$
.
However, since
$$ \begin{align*} \int_0^{r}\mu(B(x, r))~d\mu(x) & \asymp \int_0^{r}(x+r)^{1-\alpha} x^{-\alpha}~dx \ge \int_0^{r} (x+r)^{1-2\alpha}~dx \asymp r^{2(1-\alpha)}, \end{align*} $$
we obtain
$\overline {C}_\mu \le 2(1-\alpha )$
.
From now on, suppose that we are dealing with
$X=\mathbb {R}^d$
and
$\mu $
,
$\mu _F$
being acips with m denoting normalized Lebesgue measure. We assume
$\overline {\bigcup _iY_i}= Y$
. First notice that if F has bounded distortion (in the one-dimensional case, it is sufficient that F is
$C^{1+\alpha }$
with uniform constants),
${d\mu _F}/{dm}$
is uniformly bounded away from 0 and 1, so
$C_{\mu _F} = C_{m}=d$
.
For
$C_{\mu }$
, we assume that
${d\mu }/{dm} = \rho $
. Moreover, we assume there is
$C>0$
with
$\rho (x)\ge C$
for any
$x\in X$
and that
$\rho \in L^2$
. Thus,
$C_\mu =d$
.
3.2 Manneville–Pomeau maps
For
$\alpha \in (0,1)$
, define the Manneville–Pomeau map by
$$ \begin{align*} f=f_\alpha:x\mapsto \begin{cases} x(1+2^\alpha x^\alpha) & \text{ if } x\in [0, 1/2),\\ 2x-1 & \text{ if } x\in [1/2, 1].\end{cases} \end{align*} $$
(This is the simpler form given by Liverani, Saussol and Vaienti, often referred to as LSV maps.) This map has an acip
$\mu $
. The standard procedure is to induce on
$Y=[1/2, 1]$
, letting
$\tau $
be the first return time to Y. Then by [Reference Liverani, Saussol and VaientiLSV, Lemma 2.3],
$\rho \in L^2$
if
$\alpha \in (0, 1/2)$
. As in for example [Reference AlvesA, Lemma 3.60], the map
$f^\tau $
is Gibbs–Markov, so [Reference Burns, Masur, Matheus and WilkinsonBLR, Theorem 6] implies equation (1.1). Thus, we can apply Theorem 2.1 to our system whenever
$\alpha \in (0, 1/2)$
.
In the case
$\alpha \in (1/2, 1)$
, then the density is similar to that in Proposition 3.2 and a similar proof gives
$C_\mu =2(1-\alpha ) <1 =C_{\mu _F}$
, so our upper and lower bounds on the behaviour of
$M_n$
do not coincide.
3.3 Multimodal and other interval maps
Our results apply to a wide range of interval maps with equilibrium states, for example, many of those considered in [Reference Dobbs and ToddDT], which guarantees the existence of inducing schemes under mild conditions. Here we will focus on
$C^3$
interval maps
$f:I\to I$
(where
$I=[0,1]$
) with critical points with order in
$(1, 2)$
, that is, for c with
$Df(c)=0$
, there is a diffeomorphism
${\varphi }:U\to \mathbb {R}$
with U a neighbourhood of 0, such that if x is close to c, then
$f(x) = f(c) \pm {\varphi }(x-c)^{\ell _c}$
for
$\ell _c\in (1,2)$
. Moreover, we assume that for each critical point c,
$|Df^n(c)|\to \infty $
and that for any open set
$V\subset I$
, there exists
$n\in \mathbb {N}$
such that
$f^n(V)=I$
. Then, as in the main theorem in [Reference Barreira and SaussolBRSS], the system has an acip and the density is
$L^2$
, and hence Theorem 2.1 applies.
3.4 Higher dimensional examples
We will not go into details here, but there is a large amount of literature on non-uniformly expanding systems in higher dimensions which have acips and which have inducing schemes with tails which decay faster than polynomially. A standard class of examples of this are the maps derived from expanding maps given in [Reference Alves, Bonatti and VianaABV].
4 Orbits closeness for flows
In this section, we will extend our study to flows. First of all, as in Theorem 1.1, we will prove that an upper bound (related to the correlation dimension of the invariant measure) can be obtained in a general setting. Then, under some mixing assumptions, we will give an equivalent of Theorem 2.1 for flows. We will prove the abstract results before giving specific examples.
Let
$(X, \Psi _t, \nu )$
be a measure-preserving flow on a manifold. We will study the shortest distance between two orbits of the flow, defined by
$$ \begin{align*}\mathbb{M}_t(x, y)= \mathbb{M}_{\Psi, t}(x, y):=\min_{0\le t_1,t_2< t} d(\Psi_{t_1}(x), \Psi_{t_2}(y)).\end{align*} $$
We assume that the flow has bounded speed: there exists
$K\ge 0$
such that for
$T>0$
,
$d(\Psi _t(x),\Psi _{t+T}(x))\le KT$
.
We will also assume that the flow is Lipschitz: there exists
$L>0$
such that
$d(\Psi _t(x),\Psi _t(y))\leq L^t d(x,y)$
, and then prove an analogue of Theorem 1.1.
Theorem 4.1. For
$(X, \Psi _t, \nu )$
, a measure-preserving Lipschitz flow with bounded speed, we have
$$ \begin{align*} \limsup_{t\rightarrow+\infty}\frac{\log \mathbb{M}_{\Psi, t}(x, y)}{-\log t} \le \frac{2}{\underline{C}_\nu-1}, \quad \nu\times\nu\text{-a.e. } x, y. \end{align*} $$
Proof. We define
Observe that for
$t>1>r$
,
$$ \begin{align*} \{(x,y): \mathbb{M}_t(x,y)< r\}\subset \{(x,y):S_{2t,K_0r}(x,y)\geq r\},\end{align*} $$
where
$K_0=K+\max \{1,L\}$
.
Indeed, for
$(x,y)$
such that
$\mathbb {M}_t(x,y)< r$
, there exist
$0\le \bar t_1,\bar t_2<t$
such that
$d(\Psi _{\bar t_1}(x), \Psi _{\bar t_2}(y))<r$
. Thus, for any
$s\in [0,1]$
and
$q\in [0,r]$
, we have
$$ \begin{align*} d(\Psi_{\bar t_1+s}(x),\Psi_{\bar t_2+s+q}(y))&\leq d(\Psi_{\bar t_1+s}(x),\Psi_{\bar t_2+s}(y))+d(\Psi_{\bar t_2+s}(y),\Psi_{\bar t_2+s+q}(y))\\ &\leq L^sd(\Psi_{\bar t_1}(x),\Psi_{\bar t_2}(y))+Kq < K_0r \end{align*} $$
and we obtain
Then, using Markov’s inequality and the invariance of
$\nu $
,
For
$\varepsilon>0$
, let us define
$$ \begin{align*}r_t=(t^{2}\log t )^{-1/(\underline{C}_\nu-1-\varepsilon)}.\end{align*} $$
By the definition of the lower correlation dimension, for t large enough, we have
$$ \begin{align*}\nu\otimes\nu ((x,y):\mathbb{M}_t(x,y)< r_t)\leq (r_t)^{-1}(2t)^2(K_0r_t)^{\underline{C}_\nu-\varepsilon}=\frac{c}{\log t}\end{align*} $$
with
$c=4K_0^{\underline {C}_\nu -\varepsilon }$
. Therefore, choosing a subsequence
$t_\ell =\lceil e^{\ell ^2}\rceil $
, we have
$$ \begin{align*}\nu\otimes\nu ((x,y):\mathbb{M}_{t_\ell}(x,y)< r_{t_\ell})\leq \frac{c}{\ell^2}.\end{align*} $$
Thus, by the Borel–Cantelli lemma, for
$\nu \otimes \nu $
-a.e.
$(x,y)\in X\times X$
, if
$\ell $
is large enough, then
$$ \begin{align*}\mathbb{M}_{t_\ell}(x,y)\geq r_{t_\ell}\end{align*} $$
and
$$ \begin{align*}\frac{\log \mathbb{M}_{t_\ell}(x,y)}{-\log t_\ell}\leq \frac{1}{\underline{C}_\nu-1-\varepsilon}\bigg(2+\frac{\log\log t_\ell}{\log t_\ell}\bigg).\end{align*} $$
Finally, taking the limsup in the previous equation and observing that
$(t_\ell )_\ell $
is increasing,
$(\mathbb {M}_t)_t$
is decreasing and
$\lim _{\ell \rightarrow +\infty }({\log t_\ell }/{\log t_{\ell +1}})=1$
, we have
$$ \begin{align*} \underset{t\rightarrow+\infty}{\limsup}\frac{\log \mathbb{M}_t(x,y)}{-\log t}=\underset{\ell\rightarrow+\infty}{\limsup}\frac{\log \mathbb{M}_{t_\ell}(x,y)}{-\log t_\ell}\leq \frac{2}{\underline{C}_\nu-1-\varepsilon}.\end{align*} $$
Then the theorem is proved since
$\varepsilon $
can be chosen arbitrarily small.
To obtain the lower bound, we will assume the existence of a Poincaré section Y transverse to the direction of the flow. We denote by
$\tau (x)$
the first hitting time of x in Y, and obtain
$F=\Psi _{\tau }$
on Y, the Poincaré map and
$\mu $
the measure induced on Y.
Theorem 4.2. Let
$(X, \Psi _t, \nu )$
be a measure-preserving Lipschitz flow with bounded speed. We assume that there exists a Poincaré section Y transverse to the direction of the flow such that the Poincaré map
$(Y,F,\mu )$
, or the relevant quotiented version
$(\bar Y, \bar F, \bar \mu )$
, satisfies equation (1.1). If
$C_\mu $
exists and satisfies
$C_\nu =C_\mu +1$
, then
$$ \begin{align*} \lim_{t\to +\infty}\frac{\log \mathbb{M}_{\Psi, t}(x, y)}{-\log t} =\frac{2}{{C}_\nu-1}=\frac{2}{{C}_\mu}, \quad \nu\times\nu\text{-a.e. } x, y.\end{align*} $$
Proof. One can mimic the proof of Theorem 2.1 to prove that
$$ \begin{align*} \liminf_{t\to +\infty}\frac{\log \mathbb{M}_{\Psi, t}(x, y)}{-\log t} \geq \frac{2}{C_{\mu}}=\frac{2}{C_{\nu}-1}, \quad \nu\times\nu\text{-a.e. } x, y.\end{align*} $$
And the result is proved using Theorem 4.1.
We note that we are not aware of cases where
$C_\nu $
and
$C_\mu $
are well defined, but the condition
$C_\nu =C_\mu +1$
above fails. We give various examples in the remainder of this section of cases where these conditions hold.
4.1 Examples of flows
Examples where
$C_\nu $
exists and there is a Poincaré section, as in Theorem 4.2, with a measure
$\mu $
such that
$C_\mu $
exists include Teichmüller flows [Reference Avila, Goüezel and YoccozAGY] and a large class of geodesic flows with negative curvature, see [Reference Bowen and WaltersBMMW], a classic example being the geodesic flow on the modular surface. In these cases, the relevant measure for (the tangent bundle on) the flow is Lebesgue, and the measure on the Poincaré section is an acip.
In the case of conformal Axiom A flows, the conditions of Theorem 4.2 hold for equilibrium states of Hölder potentials, see the proof of [Reference Pesin and SadovskayaPS, Theorem 5.2].
4.2 Suspension flows
For Theorem 4.2, we assume that
$C_\nu =C_\mu +1$
. Obtaining this equality in a general setting is an open and challenging problem. In this section, we will prove that, under some natural assumptions, for suspension flows, this equality holds.
Let
$T:X\rightarrow X$
be a bi-Lipschitz transformation on the separable metric space
$(X,d)$
.
Let
${\varphi }:X\rightarrow (0,+\infty )$
be a Lipschitz function. We define the space
$$ \begin{align*}Y:=\{(u,s)\in X\times\mathbb{R} : 0\leq s\leq {\varphi}(u)\}, \end{align*} $$
where
$(u,{\varphi }(u))$
and
$(Tu,0)$
are identified for all
$u\in X$
. The suspension flow or the special flow over T with height function
${\varphi }$
is the flow
$\Psi $
which acts on Y by the following transformation:
$$ \begin{align*}\Psi_t(u,s)=(u,s+t).\end{align*} $$
The metric on Y is the Bowen–Walters distance, see [Reference Bruin, Rivera-Letelier, Shen and van StrienBW]. First, we recall the definition of the Bowen–Walters distance
$d_1$
on Y when
${\varphi }(x)=1$
for every
$x\in X$
. Let
$x,y\in X$
and
$t\in [0,1]$
, so the length of the horizontal segment
$[(x,t),(y,t)]$
is defined by
$$ \begin{align*}\alpha_h((x,t),(y,t))=(1-t)d(x,y)+td(Tx,Ty).\end{align*} $$
Let
$(x,t),(y,s)\in Y$
be on the same orbit, so the length of the vertical segment
$[(x,t),(y,s)]$
is defined by
$$ \begin{align*}\alpha_v((x,t),(y,s))=\inf\{|r|:\Psi_r(x,t)=(y,s)\textrm{ and }r\in\mathbb{R}\}.\end{align*} $$
Let
$(x,t),(y,s)\in Y$
, so the distance
$d_1((x,t),(y,s))$
is defined as the infimum of the lengths of paths between
$(x,t)$
and
$(y,s)$
composed by a finite number of horizontal and vertical segments. When
${\varphi }$
is arbitrary, the Bowen–Walters distance on Y is given by
$$ \begin{align*}d_Y((x,t),(y,s))=d_1\bigg(\bigg(x,\frac{t}{{\varphi}(x)}\bigg),\bigg(y,\frac{s}{{\varphi}(y)}\bigg)\bigg).\end{align*} $$
For more details on the Bowen–Walters distance, one can see [Reference Barros, Liao and RousseauBS, Appendix A].
Let
$\mu $
be a T-invariant Borel probability measure in X. We recall that the measure
$\nu $
on Y is invariant for the flow
$\Psi $
where
$$ \begin{align*} \int_Y g d\nu=\frac{\int_X\int_0^{{\varphi}(x)}g(x,s)\,ds \,d\mu(x)}{\int_X{\varphi} \,d\mu} \end{align*} $$
for every continuous function
$g:Y\rightarrow \mathbb {R}$
. Moreover, any
$\Psi $
-invariant measure is of this form. For an account of equilibrium states for suspension flows, see for example [Reference Iommi, Jordan and ToddIJT].
Theorem 4.3. Let X be a compact space and
$T:X\rightarrow X$
a bi-Lipschitz transformation. We assume that for the invariant measure
$\mu $
, the correlation dimension exists. If
$\Psi $
is a suspension flow over T as above, then
$$ \begin{align*}C_\nu=C_\mu+1\end{align*} $$
with respect to the Bowen–Walters distance.
Remark 4.4. Under the same assumptions, one can observe that if
$C_\mu $
does not exist, then we have
$\underline C_\nu =1+\underline C_\mu $
and
$\overline C_\nu =1+\overline C_\mu $
.
Before proving the theorem, we will recall some properties of the Bowen–Walters distance. First of all, for
$(x,s)$
and
$(y,t)\in Y$
, we define
$$ \begin{align*}d_\pi((x,s),(y,t))=\min\left\{\begin{array}{@{}l@{}}d(x,y)+|s-t| \\ d(Tx,y)+{\varphi}(x)-s+t \\ d(x, Ty)+{\varphi}(y)-t+s\end{array}\right\}\!.\end{align*} $$
Proposition 4.5. [Reference Barros, Liao and RousseauBS, Proposition 17]
There exists a constant
$c>1$
such that for each
$(x,s)$
and
$(y,t)\in Y$
,
$$ \begin{align*} c^{-1}d_\pi((x,s),(y,t))\leq d_Y((x,s),(y,t))\leq c d_\pi((x,s),(y,t)). \end{align*} $$
Proof of Theorem 4.3
We will denote by L a constant which is simultaneously a Lipschitz constant for T,
$T^{-1}$
and
${\varphi }$
.
Let
$0<\varepsilon <{\min \{{\varphi }(x)\}}/{2}$
. We define
$$ \begin{align*}Y_\varepsilon=\{(x,s)\in Y: \varepsilon<s<{\varphi}(x)-\varepsilon\}.\end{align*} $$
We will prove that for all
$(x,s)\in Y_\varepsilon $
and all
$0<r<\min \{c\varepsilon , {c\varepsilon }/{L}\}$
:
-
(a)
$B(x,{r}/{2c})\times (s-{r}/{2c},s+{r}/{2c})\subset Y$
; -
(b)
$B(x,{r}/{2c})\times (s-{r}/{2c},s+{r}/{2c})\subset B_Y((x,s),r)$
,
where
$B_Y((x,s),r)$
denotes the ball centred in
$(x,s)$
and of radius r with respect to the distance
$d_Y$
.
Let
$(y,t)\in B(x,{r}/{2c})\times (s-{r}/{2c},s+{r}/{2c})$
.
Since
$s>\varepsilon $
and
${r}/{c}<\varepsilon $
, we have
$t>s-{r}/{2c}>{\varepsilon }/{2}>0$
.
Since
${\varphi }$
is L-Lipschitz, we have
$|{\varphi }(x)-{\varphi }(y)|\leq L d(x,y)<{Lr}/{2c}$
. Moreover, since
$s<{\varphi }(x)-\varepsilon $
, we obtain
$$ \begin{align*} t&< s+\frac{r}{2c} <{\varphi}(x)-\varepsilon+\frac{\varepsilon}{2}\\ &< {\varphi}(y)+\frac{Lr}{2c}-\frac{\varepsilon}{2}< {\varphi}(y). \end{align*} $$
Thus,
$(y,t)\in Y$
and item (a) is proved.
For
$(y,t)\in B(x,{r}/{2c})\times (s-{r}/{2c},s+{r}/{2c})$
, we can use Proposition 4.5 to obtain
$$ \begin{align*} d_Y((x,s),(y,t))&\leq c d_\pi((x,s),(y,t))\\ &\leq c(d(x,y)+|s-t|)\\ &< c\bigg(\frac{r}{2c}+\frac{r}{2c}\bigg)=r \end{align*} $$
and item (b) is proved.
We can now use items (a) and (b) to obtain an upper bound for
$C_\nu $
. For
$0<r<\min \{c\varepsilon , {c\varepsilon }/{L}\}$
, we have
with
$C_1=({1}/{\int _X{\varphi }\,d\mu })^2\min ({\varphi }(x)-2\varepsilon ){1}/{c}>0$
. We conclude that
$$ \begin{align} &\limsup_{r\rightarrow0}\frac{\log\int_Y\nu(B_Y((x,s),r))\,d\nu(x,s)}{\log r}\nonumber\\ &\quad\leq \lim_{r\rightarrow0}\frac{\log C_1 r\int_X\mu(B(x,{r}/{2c}))\,d\mu(x)}{\log r}=1+C_\mu. \end{align} $$
To prove the lower bound, we define, for
$(x,s)\in Y$
, the sets
$$ \begin{align*} B_1&= B(x,cr)\times(s-rc,s+rc),\\ B_2 &= B(Tx,cr)\times[0,rc),\\ B_3 &= \{(y,t)\in Y:y\in B(T^{-1}x,Lrc)\textrm{ and }{\varphi}(y)-rc<t\leq {\varphi}(y)\}. \end{align*} $$
We have
$$ \begin{align*} B_Y((x,s),r)\subset (B_1\cup B_2\cup B_3)\cap Y. \end{align*} $$
Indeed, if
$(y,t)\in B_Y((x,s),r)$
, then, using Proposition 4.5, we have
$d_\pi ((x,s),(y,t))\leq c d_Y((x,s),(y,t))<cr$
. Thus, by definition of
$d_\pi $
, there are three possibilities:
-
• if
$d_\pi ((x,s),(y,t))=d(x,y)+|s-t|$
, then
$d(x,y)<cr$
and
$|s-t|<cr$
, and thus,
$(y,t)\in B_1$
; -
• if
$d_\pi ((x,s),(y,t))=d(Tx,y)+{\varphi }(x)-s+t$
, then
$d(Tx,y)<cr$
and
$0\leq t<cr$
(since
${\varphi }(x)-s\geq 0$
and
$(y,t)\in Y$
), and thus
$(y,t)\in B_2$
; -
• if
$d_\pi ((x,s),\hspace{-0.5pt}(y,t))\hspace{-0.5pt}=\hspace{-0.5pt}d(x,Ty)\hspace{-0.5pt}+\hspace{-0.5pt}{\varphi }(y)\hspace{-0.5pt}-\hspace{-0.5pt}t\hspace{-0.5pt}+\hspace{-0.5pt}s$
, then
$d(T^{-1}x,y)\hspace{-0.5pt}\leq\hspace{-0.5pt} L d(x,Ty)\hspace{-0.5pt}<\hspace{-0.5pt}Lcr$
. Since
$s\geq 0$
, we have
$\psi (y)-t<cr$
and since
$(y,t)\in Y$
, we have
$t\leq {\varphi }(y)$
, and thus
$(y,t)\in B_3$
.
Using the definition of
$\nu $
, we have
$$ \begin{align*} \nu(B_1\cap Y)&\leq \frac{1}{\int_X{\varphi} d\mu}2rc\mu(B(x,cr)),\\ \nu(B_2\cap Y)&\leq \frac{1}{\int_X{\varphi} d\mu}rc\mu(B(Tx,cr)),\\ \nu(B_2\cap Y)&\leq \frac{1}{\int_X{\varphi} d\mu}rc\mu(B(T^{-1}x,Lcr)). \end{align*} $$
Denoting
$c_1=\max \{c,Lc\}$
, we have
$$ \begin{align*} &\int_Y\nu(B_Y((x,s),r))\,d\nu(x,s)\leq \int_Y\nu(B_1\cap Y)+\nu(B_2\cap Y)+\nu(B_3\cap Y)\,d\nu(x,s)\\ &\quad\leq \frac{2c}{\int_X{\varphi} d\mu}r\bigg(\int_X \mu(B(x,c_1r))\,d\mu+\int_X \mu(B(Tx,c_1r))\,d\mu+\int_X \mu(B(T^{-1}x,c_1r))\,d\mu\bigg)\\ &\quad=\frac{6c}{\int_X{\varphi} d\mu}r\int_X \mu(B(x,c_1r))\,d\mu, \end{align*} $$
since
$\mu $
is T-invariant and
$T^{-1}$
-invariant.
Finally, we obtain
$$ \begin{align} &\liminf_{r\rightarrow0}\frac{\log\int_Y\nu(B_Y((x,s),r))\,d\nu(x,s)}{\log r}\nonumber\\&\quad\geq \lim_{r\rightarrow0}\frac{\log \frac{6c}{\int_X{\varphi} d\mu}r\int_X \mu(B(x,c_1r))\,d\mu}{\log r}=1+C_\mu. \end{align} $$
5 A class of examples with orbits remoteness
In this section, we give an example of a class of mixing systems where equation (1.1) fails to hold, see Remark 5.2 for the relation to the other results in this paper. This family of systems was defined in [Reference Gouëzel, Rousseau and StadlbauerGRS] and its mixing and recurrence/hitting time properties were studied.
We will consider a class of systems constructed as follows. The base is a measure-preserving system
$(\Omega ,T,\mu )$
. We assume that T is a piecewise expanding Markov map on a finite-dimensional Riemannian manifold
$ \Omega $
.
-
• There exists some constant
$\beta>1$
such that
$\Vert D_{x}T^{-1}\Vert \leq \beta ^{-1}$
for every
$x\in \Omega $
. -
• There exists a collection
$\mathcal {J}=\{J_1,\ldots ,J_p\}$
such that each
$J_i$
is a closed proper set and:-
(M1) T is a
$C^{1+\eta }$
diffeomorphism from
$\operatorname {\mathrm {int}} J_i$
onto its image; -
(M2)
$\Omega =\bigcup _i J_i$
and
$\operatorname {\mathrm {int}} J_i\cap \operatorname {\mathrm {int}} J_j=\emptyset $
unless
$ i=j $
; -
(M3)
$T(J_i)\supset J_j$
whenever
$T(\operatorname {\mathrm {int}} J_i)\cap \operatorname {\mathrm {int}} J_j\neq \emptyset $
.
-
Here,
$\mathcal {J}$
is called a Markov partition. It is well known that such a Markov map is semi-conjugated to a subshift of finite type. Without loss of generality, we assume that T is topologically mixing, or equivalently that for each i, there exists
$n_i$
such that
$T^{n_i}J_i=\Omega $
. We assume that
$\mu $
is the equilibrium state of some potential
$\psi \colon \Omega \to \mathbb {R}$
, Hölder continuous in each interior of the
$J_i$
. The sets of the form
$J_{i_0,\ldots ,i_{q-1}}:=\bigcap _{n=0}^{q-1} T^{-n}J_{i_n}$
are called cylinders of size q and we denote their collection by
$\mathcal {J}_q$
.
In this setting, the correlation dimension of
$\mu $
exists as in [Reference Pesin and WeissPW, Theorem 1]. Note that we could arrange our system so that our
$\mu $
is acip: the density here will be bounded, so the correlation dimension is one.
The system is extended by a skew product to a system
$(M,S)$
where
$M=\Omega \times \mathbb {T}$
and
$S:M\rightarrow M$
is defined by
$$ \begin{align*} S(\omega ,t)=(T\omega ,t+\alpha\varphi (\omega )), \end{align*} $$
where
$\varphi =1_{I}$
is the characteristic function of a set
$I\subset \Omega $
which is a union of cylinders. In this system, the second coordinate is translated by
$\boldsymbol {\alpha }$
if the first coordinate belongs to I. We endow
$(M,S)$
with the invariant measure
$\nu =\mu \times \mathrm {Leb}$
(so
$C_\nu =C_\mu +1$
). On
$\Omega \times \mathbb {T}$
, we will consider the sup distance.
We make the standing assumption on our choice of
$\varphi $
that:
-
• (NA) for any
$u\in \lbrack -\pi ,\pi ]$
, the equation
$fe^{iu\varphi }=\unicode{x3bb} f\circ T$
, where f is Hölder (on the subshift) and
$\unicode{x3bb} \in S^{1}$
, has only the trivial solutions
$\unicode{x3bb} =1$
and f constant.
The simple case, where the I which defines
$\varphi $
is a non-empty union of size
$1$
cylinders such that both I and
$I^{c}$
contain a fixed point, fulfils this assumption.
Definition 5.1. Given an irrational number
$\alpha $
, we define the irrationality exponent of
$\alpha $
as the following (possibly infinite) number:
$$ \begin{align*} \gamma (\alpha )=\inf \Big\{\beta :\liminf_{q\rightarrow \infty }q^{\beta }\| q\alpha \|>0\Big\}, \end{align*} $$
where
$\| \cdot \|$
indicates the distance to the nearest integer number in
$\mathbb {R}$
.
First note that
$\gamma (\alpha )\ge 1$
for any irrational
$\alpha $
.
Remark 5.2. By [Reference Gouëzel, Rousseau and StadlbauerGRS, Theorem 19], if
$\gamma (\alpha )>d_\mu +1$
, then the hitting time statistics is typically degenerate. This is an indirect way of seeing that there cannot be an inducing scheme satisfying equation (1.1), otherwise [Reference Bruin, Saussol, Troubetzkoy and VaientiBSTV, Theorem 2.1] would be violated; it also suggests that the conclusions of Theorem 2.1 will not hold here, which we show below is indeed the case.
Theorem 5.3. For
$\nu \times \nu $
-a.e.
$x,y\in M$
, we have
$$ \begin{align*} \limsup_{n}\frac{\log \mathbb{M}_{S, n}(x, y)}{-\log n}\leq\min\bigg(\frac{2}{C_\nu},1\bigg)=\min\bigg(\frac{2}{C_\mu+1},1\bigg) \end{align*} $$
and
$$ \begin{align} \liminf_{n}\frac{\log \mathbb{M}_{S,n}(x, y)}{-\log n}\leq\min\bigg(\frac{2}{C_\nu},\frac{1}{\gamma(\alpha)}\bigg)=\min\bigg(\frac{2}{C_\mu+1},\frac{1}{\gamma(\alpha)}\bigg). \end{align} $$
Proof. First of all, applying Theorem 1.1 to S and since one can easily show that
$C_\nu =C_\mu +1$
, we obtain for
$\nu \times \nu $
-a.e.
$x,y\in M$
,
$$ \begin{align*} \liminf_{n}\frac{\log \mathbb{M}_{S,n}(x, y)}{-\log n}\leq\limsup_{n}\frac{\log \mathbb{M}_{S, n}(x, y)}{-\log n}\leq\frac{2}{C_\nu}=\frac{2}{C_\mu+1}. \end{align*} $$
Moreover, one can observe that for
$x=(\omega ,t)\in M$
and
$y=(\tilde \omega ,s)\in M$
,
$$ \begin{align*} &\mathbb{M}_{S,n}(x,y)= \min_{0\le i,j\le n-1} \max\Big(d(T^i(\omega), T^j(\tilde\omega)),\|(t-s)+\alpha(S_i\varphi(\omega)-S_j\varphi(\tilde\omega))\|\Big)\\&\quad\geq \max\Big(\min_{0\le i,j\le n-1} d(T^i(\omega), T^j(\tilde\omega)),\min_{0\le i,j\le n-1}\|(t-s)+\alpha(S_i\varphi(\omega)-S_j\varphi(\tilde\omega))\|\Big)\\&\quad\geq \max\Big(\min_{0\le i,j\le n-1} d(T^i(\omega), T^j(\tilde\omega)),\min_{-(n-1)\le i\le n-1}\|(t-s)+i\alpha)\|\Big)\\&\quad=\max \Big(\mathbb{M}_{T,n}(\omega,\tilde\omega),\mathbb{M}_{R,n}(t,s)\Big), \end{align*} $$
where
$R:\mathbb {T}\mapsto \mathbb {T}$
with
$R(s)=s+\alpha $
. Thus, by [Reference Burns, Masur, Matheus and WilkinsonBLR, Theorems 1 and 10], we obtain
$$ \begin{align*} \limsup_n\frac{\log \mathbb{M}_{S, n}(x, y)}{-\log n}&\leq \min\bigg(\limsup_n\frac{\log \mathbb{M}_{T,n}(\omega,\tilde\omega)}{-\log n},\limsup_n\frac{\log\mathbb{M}_{R,n}(t,s)}{-\log n}\bigg)\\ &\leq \min\bigg(\frac{2}{C_\mu},1\bigg) \end{align*} $$
and
$$ \begin{align*} \underset{n}{\liminf}\frac{\log \mathbb{M}_{S,n}(x, y)}{-\log n}&\leq \min\bigg(\underset{n}{\liminf}\frac{\log \mathbb{M}_{T,n}(\omega,\tilde\omega)}{-\log n},\underset{n}{\liminf}\frac{\log\mathbb{M}_{R,n}(t,s)}{-\log n}\bigg)\\ &\leq \min\bigg(\frac{2}{C_\mu},\frac{1}{\gamma(\alpha)}\bigg). \end{align*} $$
Finally, since
$C_\nu =C_\mu +1>C_\mu $
, the theorem is proved.
Finally, we prove that if
$\mu $
is a Bernoulli measure, then equation (5.1) is sharp.
Theorem 5.4. We assume that all the branches of the Markov map T are full, that is,
$ T(J_{i})=\Omega $
for all i, where
$\mu $
is a Bernoulli measure, that is,
$\mu ([a_{1}\ldots a_{n}])=\mu ([a_{1}])\cdots \mu ([a_{n}])$
, and I depends only on the first symbol, that is, I is an union of
$1$
-cylinders (recall that
$\varphi =1_{I}$
).
If
$\gamma (\alpha )> d_\mu +1$
, then
$$ \begin{align} \liminf_{n}\frac{\log \mathbb{M}_{S,n}(x, y)}{-\log n}=\frac{1}{\gamma(\alpha)}< \frac{2}{C_\nu}, \quad \nu\times\nu\text{-a.e. } x,y. \end{align} $$
Proof. First of all, we recall that
$C_\mu \leq d_\mu $
(see e.g. [Reference PesinP]), thus our assumption on
$\alpha $
implies that
$1/\gamma (\alpha )<2/C_\nu $
, so equation (5.1) implies
$$ \begin{align*}\liminf_{n}\frac{\log \mathbb{M}_{S,n}(x, y)}{-\log n}\le\frac{1}{\gamma(\alpha)}.\end{align*} $$
So it remains to show the reverse of the above inequality.
By [Reference Gouëzel, Rousseau and StadlbauerGRS, Proposition 21], for any y, for
$\nu $
-a.e. x, we have
$$ \begin{align} \limsup_{r\rightarrow0}\frac{\log W_r(x,y)}{-\log r}\leq \max ({d}_{\mu }+1,\gamma(\alpha)), \end{align} $$
where
$ W_r(x,y)=\inf \{k\geq 1, S^k(x)\in B(y,r)\}$
.
Let
$\epsilon>0$
and let
$x,y$
such that equation (5.3) holds. Since
$\gamma (\alpha ) \geq d_\mu +1$
, for any r small enough, we have
$$ \begin{align*} W_r(x,y)\leq r^{-(\gamma(\alpha)+\epsilon)}, \end{align*} $$
which implies that
$$ \begin{align*}\mathbb{M}_{S, \lceil r^{-(\gamma(\alpha)+\epsilon)}\rceil}(x,y)<r.\end{align*} $$
Thus, for any r small enough,
$$ \begin{align*} \frac{\log \mathbb{M}_{S, \lceil r^{-(\gamma(\alpha)+\epsilon)} \rceil}(x,y)}{-\log \lceil r^{-(\gamma(\alpha)+\epsilon)} \rceil}>\frac{1}{\gamma(\alpha)+\epsilon} \end{align*} $$
and then
$$ \begin{align*}\liminf_{n}\frac{\log \mathbb{M}_{S,n}(x, y)}{-\log n}\geq\frac{1}{\gamma(\alpha)+\epsilon}.\end{align*} $$
The theorem is proved taking
$\epsilon $
arbitrary small.
Acknowledgements
We would like to thank Thomas Jordan for helpful suggestions on correlation dimension. We also thank the referee(s) for useful comments. Both authors were partially supported by FCT projects PTDC/MAT-PUR/28177/2017 and by CMUP (UIDB/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. J.R. was also partially supported by CNPq and PTDC/MAT-PUR/4048/2021, and with national funds.
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