On statistical properties for equilibrium states of partially hyperbolic horseshoes

Describing the behavior of the orbits of a dynamical system can be a challenging task, especially for systems that have a complicated topological and geometrical structure. A very useful way to obtain features of such systems is via invariant probability measures. For instance, by Birkhoff’s Ergodic Theorem, almost every initial condition in each ergodic component of an invariant measure has the same statistical distribution in space. When the system admits more than one invariant probability measure, an efficient way to chose an interesting one is to select those that have regular Jacobians, which are called equilibrium states. We formally define an equilibrium state with respect to a potential as follows.

By studying the decay of correlations of an equilibrium measure, one can obtain significant information regarding the system: how fast memory of the past is lost by the system as time evolves. In particular, this gives the speed at which the equilibrium is reached.

However, while standard counterexamples show that in general there is no specific rate at which this loss of memory occurs, it is sometimes possible to obtain specific rates of decay which depend only on the map $F$, as long as the observables belong to some appropriate space of functions.

Another way to characterize weak correlations of successive observations is given by a central limit theorem: the probability of a given deviation of the average values of an observable along an orbit from the asymptotic average is essentially given by a normal distribution.

In a pioneering work [7], Ferrero and Schmitt applied the theory of projective metrics, due to Birkhoff [1], to the transfer operator for expanding maps, thus obtaining spectral properties. For one dimensional piecewise expanding maps, an exponential decay of correlations was proved by Liverani [11] and a central limit theorem was proved by Keller [8]. In the context of volume preserving hyperbolic maps, Liverani [10] established exponential decay of correlations for the SRB measure. In the more general context of hyperbolic attractors, Viana [15] proved the exponential decay of correlations and a central limit theorem. The latter was inspired by the work of Dürr and Goldstein [6].

In the context of non-uniformly hyperbolic maps we may cite the independent works of Young [16] and Keller and Nowicki [9] that used towers extensions and cocycles to prove exponential decay of correlations for quadratic maps. In the same context Castro and Varandas [4] obtained statistical properties for the unique equilibrium state associated to a class of non-uniformly expanding maps. In this work they use the projective metrics approach.

For a class of partially hyperbolic systems semiconjugated to nonuniformly expanding maps Castro and Nascimento [3] proved exponential decay of correlations and a central limit theorem for the maximal entropy measure.

In this work we address the problem of studying statistical properties for the unique equilibrium state of partially hyperbolic horseshoes. The family of three dimensional horseshoes was introduced by Díaz, Horita, Rios and Sambarino in [5] and the uniqueness of equilibrium states associated to Hölder continuous potentials with small variation was proved by Rios and Siqueira in [12].

We start by studying a two dimensional abstract map obtained from the horseshoe by projecting its inverse on two center-stable leaves. We refer to this map as the projection map. We construct metrics with respect to which the Perron-Frobenius operator associated to the projection map is a contraction. Such a contraction allows us to obtain a spectral gap property on the space of Hölder continuous observables. From this we deduce exponential decay of correlations and a central limit theorem for the equilibrium state associated to the projection map. Finally we show that the equilibrium state of the horseshoe carries the same statistical properties.

The paper is organized as follows. In Section 2 we describe both the horseshoe and its projection map and we give a precise formulation of the statistical properties of its equilibrium. We also define the transfer operator associated to the projection map and state the spectral gap property. In Section 3 we give a brief review of the theory of projective metrics in cones. This will be used as a key tool to obtain the spectral gap theorem, which we prove in Section 4. In Section 5 we derive the exponential decay of correlations and a central limit theorem for the unique equilibrium of the projection map. Finally, in Section 6 we extend the results obtained for the projection map to the horseshoe.

We start describing the family of three dimensional horseshoes introduced by Díaz, Horita, Rios and Sambarino in [5]. Let $R=[0,1]\times[0,1]\times[0,1]\subset\mathbb{R}^{3}$ be the cube in $\mathbb{R}^{3}$ and consider the parallelepipeds:

The horseshoe map is defined on $\tilde{R}_{0}$ and $\tilde{R}_{1}$ as follows

where $0<\rho<{1/3}$, $\beta>6$ and $f(y)=\frac{1}{1-(1-\frac{1}{y})e^{-1}}.$

And

where $0<\sigma<{1/3}$ and $3<\beta_{1}<4$.

Then, for $X\in R$, we have

If $X\in R$ but does not belong to $\tilde{R}_{0}$ or $\tilde{R}_{1}$, then $X$ will be mapped injectively outside $R$.

We point out that besides we refer simply to $F$, we have described a family of maps that depends on the parameters $\rho,\beta,\beta_{1}$ and $\sigma$. We consider fixed parameters satisfying conditions above.

In figure 2 we see the steps of the construction of the horseshoe.

Let $\Omega$ be the maximal invariant set under $F$ of the union of the parallelepipeds $\tilde{R}_{0}$ and $\tilde{R}_{1}$:

In [5] it was shown that the maximal invariant set $\Omega$ is partially hyperbolic, with one dimensional central direction, parallel to the $y$-axis. The central direction presents contractive and expanded behavior. The horizontal direction is contractive while the vertical direction, parallel to the $z$-axis, is expanding.

The uniqueness of equilibrium states for the horseshoe $F$ associated to potentials with small variation was proved in [12]. The main goal of this work is to study the statistical behavior of this equilibrium. Here we state the result in [12]. Let $\omega=\frac{1+\sqrt{5}}{2}$.

We consider potentials $\phi$ as above and additionally we assume that the Hölder constant of $\exp(\phi)$ is small. The explicit condition will be stated in Section 4. We point out that this is an open condition which includes, for instance, constant potentials.

For the equilibrium state $\mu_{\phi}$ of the system $(F,\phi)$ we will establish exponential decay of correlations for Hölder continuous observables.

We also derive a central limit theorem for the equilibrium state of the horseshoe with respect to a potential $\phi$ as considered above.

Now we describe a map $G$ that was defined in [12] which is related to the projection of $F^{-1}$ on two center-stable planes. By an abuse of notation the map $G$ will be called the projection map. Besides the inherent interest in the dynamics of the map $G$, understanding the statistical behavior of its equilibrium is the crucial ingredient in the proofs of Theorem A and Theorem B.

We define as follows the rectangles $R_{1}$, $R_{2}$ and $R_{3}$:

with $\varepsilon>0$ close to zero.

The rectangles are inside two planes that we will call $P_{0}$ and $P_{1}$ (see figure 3). We consider an abstract space $\mathcal{Q}:=\bigcup_{i=1}^{3}{R_{i}}$ which is the union of the rectangles. Notice this is a metric space endowed with some natural metric $d$, say the one, induced by $\mathbb{R}^{3}$.

Let $g_{0},g_{1}:[0,1]\to\mathbb{R}$ be defined by $g_{0}(y)=f^{-1}(y)$ and $g_{1}(y)=1-\sigma^{-1}y$. Take $\gamma=\rho^{-1}$.

Consider the map $G:\mathcal{Q}\to P_{0}\cup P_{1}$ defined by its restrictions $G_{i}$ to each rectangle $R_{i}$ as follows:

Note that $R_{2}$ is uniform expanding while we have both, expanding and contracting, behaviors in $R_{1}$ and $R_{3}$. The map $G$ acts similarly on $R_{1}$ and $R_{3}$. In these rectangles, the points are sent from the right side to the left, except for the extreme points whose $x$ coordinates are fixed.

Let $\Lambda$ be the maximal invariant set under $G$ of the union of the rectangles $R_{1}$, $R_{2}$ and $R_{3}$:

and from now on we denote simply by $G$ the restriction of $G$ to $\Lambda$.

Let $\Sigma_{A}$ be the subshift of finite type

with transition matrix:

The following transitions are allowed:

Notice that $G$ is is not conjugated but only semi-conjugated to the shift $\sigma$. That is because the entire segment $[0,1]$ is associated to the constant sequence $(11111\cdots)$ on $\Sigma_{A}$.

We point out that the 3-rd iterate of any rectangle covers $\mathcal{Q}$. Moreover, $G$ is topologically mixing.

Note that points belonging to the rectangles 1 and 2 have two pre-images, while points in the rectangle 3 have just one pre-image.

Figure 5 and figure 5 show the first steps in the generation of the set $\Lambda$ . Since $\Lambda$ contains infinitely many line segments it is not a Cantor set.

The topological entropy of the subshift $\sigma$ is given by:

Since $G$ and $\sigma$ are semiconjugated we obtain $h_{top}(G)\geq\log\omega$.

In [12] it was shown the uniqueness of equilibrium states associated to Hölder continuous potentials $\phi_{\ast}:\mathcal{Q}\to\mathbb{R}$ satisfying $\sup\phi_{\ast}-\inf\phi_{\ast}<\frac{\log\omega}{2}$. In this work we consider potentials $\phi_{\ast}$ as above and assume an additional condition that will be stated in Section 4. For the system $(G,\phi_{\ast})$ we obtain some statistical properties of its equilibrium measure $\mu_{\ast}$. As mentioned before, these results will be used to derive the statistical properties of the equilibrium of the horseshoe announced above.

The following result states the exponential decay of correlations for Hölder continuous observables.

We also obtain a central limit theorem for the equilibrium.

The theory of projective metrics on convex cones and positive operators on a vector space is due to Birkhoff [1] and has been extensively studied (see [2] and [10]). Projective metrics associated to cones provide an elegant way to express spectral properties of the transfer operator.

In this section we will state some results regarding this theory in order to prove the spectral gap of the transfer operator.

Let $E$ be a Banach space. A subset $\mathcal{C}$ of $E\!-\!\{0\}$ is called a cone in $E$ if it is a convex space which satisfies:

1. 1)

   $\forall\lambda>0:\lambda\mathcal{C}\subset\mathcal{C};$

2. 2)

   $\mathcal{C}\cap\left(-\mathcal{C}\right)=\{\emptyset\}.$

We say that a cone $\mathcal{C}$ is closed if $\bar{\mathcal{C}}=\mathcal{C}\cup\{0\}$.

Let $\mathcal{C}$ be a closed cone and given $v,w\in\mathcal{C}$ define

We point out that $A(v,w)$ is finite, $B(v,w)$ is positive and $A(v,w)\leq B(v,w)$ for all $v,w\in\mathcal{C}$. We set

with $\Theta$ possibly infinity in the case $A=0$ or $B=+\infty$.

It is straightforward to check that $\Theta(v,w)$ is well-defined and takes values in $[0,+\infty]$. Since $\Theta(v,w)=0\Leftrightarrow v=tw\ \mbox{for some}\ t>0$ we have that $\Theta$ defines a pseudo-metric on $\mathcal{C}$. Then $\Theta$ induces a metric on a projective quotient space of $\mathcal{C}$ called the projective metric of $\mathcal{C}$.

Note that the projective metric depends in a monotone way on the cone: if $\mathcal{C}_{1}\subset\mathcal{C}_{2}$ are two cones in $E$, then we have

where $\Theta_{1}$ and $\Theta_{2}$ are the projective metrics in $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ respectively.

Moreover, if $L:{E}_{1}\to{E}_{2}$ is a linear operator and $\mathcal{C}_{1},\mathcal{C}_{2}$ are cones in ${E}_{1},{E}_{2}$ respectively, satisfying $L(\mathcal{C}_{1})\subset\mathcal{C}_{2}$ then

However $L$ is not necessarily a strict contraction, that will be the case for instance if $L(\mathcal{C}_{1})$ had finite diameter in $\mathcal{C}_{2}$. This will be stated in the following result which is a key tool to prove the spectral gap for the Ruelle-Perron-Frobenius operator.

For the proof of the last proposition see for example [[15], Proposition 2.3].

Next we will define a cone in the space of positive continuous functions. We start by recalling some definitions.

Let $\varphi$ be an $\alpha$-Hölder continuous function and denote by

the Hölder constant of $\varphi$.

Given $\delta>0$ we say that a function $\varphi$ is $(C,\alpha)$-Hölder continuous in balls of radius $\delta$ if for some constant $C>0$ we have $|\varphi(x)-\varphi(y)|\leq Cd(x,y)^{\alpha}$ for all $y\in B(x,\delta)$.

We will denote by $|\varphi|_{\alpha,\delta}$ the smallest Hölder constant of $\varphi$ in balls of radius $\delta>0$. We consider the space of $\alpha$-Hölder continuous observables endowed with the norm $\|\cdot\|:=|\cdot|_{0}+|\cdot|_{\alpha}$.

Consider $\mathcal{Q}$ the union of the rectangles $R_{1}$, $R_{2}$ and $R_{3}$ and fix $1/2\leq\delta\leq 3/4-2\rho$. Let $\varphi:\mathcal{Q}\to\mathbb{R}$ be a $(C,\alpha)$-Hölder continuous function in balls of radius $\delta$ . Then $\varphi$ is $(C(1+r^{\alpha}),\alpha)$-Hölder continuous in balls of radius $(1+r)\delta$ for each $0\leq r\leq 1.$

Indeed, fixing $r\in[0,1]$ and given $x,y\in\mathcal{Q}$ with $d(x,y)<(1+r)\delta$, there exists $z\in\mathcal{Q}$ such that $d(x,z)=\delta$ and $d(z,y)<rd(x,z)$. Hence,

The next result states that every locally Hölder continuous function defined on $\mathcal{Q}$ is Hölder continuous.

Now we consider the cone of locally Hölder continuous observables defined on $\mathcal{Q}$:

It follows by definition that $\mathcal{C}_{k_{1},\delta}\subset\mathcal{C}_{k_{2},\delta}$ if $k_{1}\leq k_{2}$.

Given an arbitrary $\varphi\in\mathcal{C}_{k,\delta}$ we have $\left|\varphi\right|_{\alpha,\delta}\leq k\cdot\inf\varphi$. Moreover, by Lemma 3.2, $\varphi$ is Hölder continuous with constant $m\cdot\left|\varphi\right|_{\alpha,\delta}$. Then

In the next lemma we give another expression for the projective metric on the cone $\mathcal{C}_{k,\delta}$, that we denote by $\Theta_{k}$ and use in further estimates.

Consider the system $(G,\phi_{\ast})$ where $G:\mathcal{Q}\to P_{0}\cup P_{1}$ is the projection map defined in Section 2 and $\phi_{\ast}:\mathcal{Q}\to\mathbb{R}$ is a Hölder continuous potential with variation smaller than $\frac{\log(\omega)}{2}$. We also assume that $\phi_{\ast}$ satisfies

Let $\mathcal{L}_{\phi_{\ast}}$ be the transfer operator of $G$ associated to the potential $\phi_{\ast}$. When there is no risk of confusion we will denote $\mathcal{L}_{\phi_{\ast}}$ simply by $\mathcal{L}$. Here we prove that the transfer operator $\mathcal{L}$ has the spectral gap property on the space of Hölder continuous observables.

As mentioned in Section 2, the map $G$ is not injective on $\mathcal{Q}$ but it is injective when restricted to each of the rectangles $R_{1}$, $R_{2}$ and $R_{3}$. Moreover, $G^{3}(R_{i})\supset\mathcal{Q}$ for each $i=1,2,3$. We will explore this property in order to construct a partition of $\mathcal{Q}$ such that the distance between pre-images under $G^{3}$ of points in the same element of the partition can be controlled.

From now on we fix $\delta>0$ as in Lemma 4.1. Given $k>0$ let $\mathcal{C}_{k,\delta}$ be the cone of locally Hölder continuous functions defined in equation 4. The next result states the strict invariance of the cone $\mathcal{C}_{k,\delta}$ under the operator $\mathcal{L}_{\phi_{\ast}}^{3}$ for $k$ large enough.