Weak Gibbs measures and large deviations

TL;DR

This paper establishes large deviations estimates for weak Gibbs measures in dynamical systems, providing insights into the probability of deviations from typical behavior.

Contribution

It introduces large deviations results specifically for weak Gibbs measures, extending the understanding of statistical properties in dynamical systems.

Findings

Large deviations estimates are proved for weak Gibbs measures.

The results apply to systems with weak Gibbs measures, broadening the scope of statistical analysis.

The paper advances the theoretical framework connecting Gibbs measures and large deviations.

Abstract

Let (X,T) be a dynamical system, where X is a compact metric space and T a continuous onto map. For weak Gibbs measures we prove large deviations estimates.

Equations

$${\mathcal{E}}_n(x):=\frac{1}{n}\sum _{k=0}^{n-1}{\delta }_{T^{k}x}.$$

$$-\delta \le \frac{1}{m}\ln \nu (B_m(x,\epsilon ))-\int \phi d{\mathcal{E}}_m(x)\le \delta .$$

$$-\frac{K_{\epsilon }}{m}\le \frac{1}{m}\ln \nu (B_m(x,\epsilon ))-\int \phi d{\mathcal{E}}_m(x)\le \frac{K_{\epsilon }}{m}.$$

$$0$$

$$\lim_{\varepsilon\downarrow 0}\liminf_{m}\inf_{x\in X}\Big{(}\frac{1}{m}\ln\nu(B_{m}(x,\varepsilon))+\int e\,d{\mathcal{E}}_{m}(x)\Big{)}\geq 0\,.$$

$$\lim_{\varepsilon\downarrow 0}\limsup_{m}\sup_{x\in X}\Big{(}\frac{1}{m}\ln\nu(B_{m}(x,\varepsilon))+\int e\,d{\mathcal{E}}_{m}(x)\Big{)}\leq 0\,.$$

$$-\delta$$

$$-\delta \le \frac{1}{m}\ln \nu (B_m(x,\epsilon ))-\int \phi d{\mathcal{E}}_m(x)\le \delta .$$

$$\underset{m}{\mathrm{liminf}}\frac{1}{m}\ln \nu ({\mathcal{E}}_m\in G)\ge h(T,\rho )+\int \phi d\rho .$$

$$\limsup_{m}\frac{1}{m}\ln\nu({\mathcal{E}}_{m}\in F)\leq\sup_{\rho\in F\cap M_{1}(X,T)}\big{(}h(T,\rho)+\int\varphi\,d\rho\big{)}\,.$$

$$P(\varphi)=\sup_{\rho\,{\rm ergodic}}\big{(}h(T,\rho)+\int\varphi\,d\rho\big{)}\leq 0\leq\sup_{\rho\in M_{1}(X,T)}\big{(}h(T,\rho)+\int\varphi\,d\rho\big{)}=P(\varphi)\,.$$

$$\limsup_{m}\frac{1}{m}\ln\nu({\mathcal{E}}_{m}\in F)\leq\!\!\!\!\sup_{\rho\in F\cap M_{1}(X,T)}\big{(}h(T,\rho)+\int\varphi\,d\rho\big{)}\,.$$

$$\liminf_{m}\frac{1}{m}\ln\nu({\mathcal{E}}_{m}\in G)\geq\!\!\!\!\sup_{\rho\in G\cap M_{1}(X,T)}\big{(}h(T,\rho)+\int\varphi\,d\rho\big{)}\,.$$

$$P(f)=\sup\Big{\{}h(T,\rho)+\int f\,d\rho\,{:}\;\rho\in M_{1}(X,T)\Big{\}}=h(T,\mu)+\int f\,d\mu\,.$$

$$\underset{m}{\mathrm{limsup}}\int \phi d{\mathcal{E}}_m(x)$$

$$\underset{m}{\mathrm{liminf}}\int \phi d{\mathcal{E}}_m(x)$$

$$\underset{m}{\lim }\int \phi d{\mathcal{E}}_m(x)={\phi }^{\ast }(x)\nu -a.s.$$

$$\int \phi d\nu =\int {\phi }^{\ast }d\nu .$$

$$h(T,\nu)\leq-\int\varphi(x)\,d\nu(x)=\int\big{(}-\lim_{\varepsilon\downarrow 0}\limsup_{m}\frac{1}{m}\ln\nu(B_{m}(x,\varepsilon))\big{)}\,d\nu(x)\,.$$

$$h_{\mathcal{P}}(x):=\underset{n}{\lim }-\frac{1}{n}\ln \nu ({\mathcal{P}}^n(x))\nu -a.s.$$

$$\int h_{\mathcal{P}}(x)d\nu (x)=h_{\mathcal{P}}(T,\nu ),$$

$$h_{\mathcal{P}}(T,\nu)=\lim_{n}\Big{(}-\frac{1}{n}\sum_{B\in{\mathcal{P}}^{n}}\nu(B)\ln\nu(B)\Big{)}\leq h(T,\nu)\,.$$

$$\int\big{(}-\limsup_{m}\frac{1}{m}\ln\nu(B_{m}(x,\varepsilon))\big{)}\,d\nu(x)\leq\int h_{\mathcal{P}}(x)\,d\nu(x)\leq h(T,\nu)\,,$$

Full text

Weak Gibbs measures and large deviations

C.-E. Pfister111E-mail: charles.pfister@epfl.ch

Section of Mathematics, Faculty of Basic Sciences, EPFL

CH-1015 Lausanne, Switzerland

  

W.G. Sullivan222E-mail: wgs@maths.ucd.ie

Department of Mathematics, UCD,

Belfield, Dublin 4, Ireland

**Abstract: ** Let $(X,T)$ be a dynamical system, where $X$ is a compact metric space and $T\,{:}\;X\rightarrow X$ a continuous onto map. For weak Gibbs measures we prove large deviations estimates.

1 Introduction

In [PS1] a general method for proving large deviations estimates for dynamical systems (X,T) is developed. In this note we make the connection with the main results of [PS1] and the notion of weak Gibbs measures, which was not explicit in the original paper.

Let $X$ be a compact metric space and $T\,{:}\;X\rightarrow X$ a continuous map which is onto. $M_{1}(X)$ is the set of Borel probability measures on $X$ (with weak convergence topology) and $M_{1}(X,T)$ the subset of $T$-invariant probability measures. Let $x\in X$ and

[TABLE]

The metric entropy of $\nu\in M_{1}(X,T)$ is denoted $h(T,\nu)$ and $B_{m}(x,\varepsilon)$ is the dynamical ball $\{y\in X\,{:}\;d(T^{k}x,T^{k}y)\leq\varepsilon\,,\;k=0,\ldots,m-1\}$. There are several variants in the literature for the definition of weak Gibbs measures (see e.g. [BV] and [Yu]). In this paper a weak Gibbs measure is defined as follows.

Definition 1**.**

Let $\varphi\in C(X)$. A probability measure $\nu$ is a weak Gibbs measure for $\varphi$ if $\forall\delta>0$ $\exists\varepsilon_{\delta}>0$ such that for $0<\varepsilon\leq\varepsilon_{\delta}$ $\exists N_{\delta,\varepsilon}<\infty$, $\forall m\geq N_{\delta,\varepsilon}$, $\forall x\in X$,

[TABLE]

The set of weak Gibbs measures for a given $\varphi$ is convex (possibly empty). Gibbs measures as defined in [Bo] (see [Bo], theorem 1.2) and quasi-Gibbs measures (see [HR], proposition 2.1) are examples of weak Gibbs measures since these measures satisfy the stronger inequalities: there exists $0<\varepsilon_{0}<\infty$ such that for $0<\varepsilon\leq\varepsilon_{0}$ $\exists K_{\varepsilon}<\infty$, $\forall m$, $\forall x\in X$,

[TABLE]

2 Results

If $\nu$ is a weak Gibbs measure, then

[TABLE]

that is, $-\varphi$ is a lower, respectively upper, energy function for $\nu$ in the sense of [PS1] (definitions 3.2 and 3.4). Indeed, in [PS1] a function $e$ on $X$ is called a lower energy function for $\nu$ if it is upper semi-continuous and

[TABLE]

It is called an upper energy function for $\nu$ if it is lower semi-continuous, bounded and

[TABLE]

The terminology used in [PS1] comes from statistical mechanics.

Proposition 1**.**

If the continuous function $e$ verifies (2.1) and (2.2), then $\nu$ is a weak Gibbs measure for $\varphi=-e$.

Proof. For any $\delta>0$, if $\varepsilon$ is small enough and $m$ large enough,

[TABLE]

so that for $\forall m\geq N_{\delta,\varepsilon}$ and $\forall x\in X$

[TABLE]

For any dynamical system $(X,T)$ and any weak Gibbs measure the following large deviations estimates are true.

Theorem 1**.**

Let $\nu$ be a weak Gibbs measure for $\varphi$.

1. If $G\subset M_{1}(X)$ is open, then for any ergodic probability measure $\rho\in G$

[TABLE]

2. If $F\subset M_{1}(X)$ is convex and closed, then

[TABLE]

Proof. Proposition 3.1 and theorem 3.2 in [PS1].

Proposition 2**.**

If $\nu$ is a weak Gibbs measure for $\varphi$, then the topological pressure $P(\varphi)=0$.

Proof. This is an immediate consequence from theorem 1, theorem 9.10 and corollary 9.10.1 in [Wa]. Let $G=F=M(X)$. Then

[TABLE]

The following hypothesis about the entropy-map $h(T,\cdot)$ and the dynamical system $(X,T)$ are sufficient to obtain a full large deviations principle.

Theorem 2**.**

Let $\nu$ be a weak Gibbs measure for $\varphi$. If the entropy map $h(T,\cdot)$ is upper semi-continuous, then for $F\subset M_{1}(X)$ closed

[TABLE]

If the ergodic measures are entropy dense, then for $G\subset M_{1}(X)$ open

[TABLE]

Proof. Theorems 3.1 and 3.2 in [PS1].

Entropy density of the ergodic measures means ([PS1]): for any $\mu\in M_{1}(X,T)$, any neighbourhood $N$ of $\mu$ and any $h^{*}<h(T,\mu)$, there exists an ergodic measure $\rho\in N$ such that $h(T,\rho)\geq h^{*}$. Entropy density is true under various types of specifications properties for the dynamical system $(X,T)$, see e.g. [PS1], [PS2], [CTY], [KLO] and [GK].

Proposition 3**.**

If $\nu\in M_{1}(X,T)$ is a weak Gibbs measure for $\varphi$, then it is an equilibrium measure for $\varphi$.

Proof. By definition an equilibrium measure $\mu\in M_{1}(X,T)$ for a continuous function $f$ satisfies the variational principle

[TABLE]

Since $P(\varphi)=0$, $h(T,\nu)\leq-\int\varphi\,d\nu$. Since $\nu$ is a weak Gibbs measure for $\varphi$,

[TABLE]

By the ergodic theorem there exists an integrable function $\varphi^{*}$ such that

[TABLE]

and

[TABLE]

Therefore

[TABLE]

Let ${\mathcal{P}}=\{A_{1},\ldots,A_{p}\}$ be a finite measurable partition of $X$, $\max_{i}{\rm diam}A_{i}<\varepsilon$. For $x\in X$, let ${\mathcal{P}}^{n}(x)$ be the element of the partition ${\mathcal{P}}^{n}={\mathcal{P}}\vee T^{-1}{\mathcal{P}}\vee\cdots\vee T^{-n+1}{\mathcal{P}}$ containing $x$. By the McMillan-Breiman theorem

[TABLE]

and

[TABLE]

where

[TABLE]

Since $B_{n}(x,\varepsilon)\supset{\mathcal{P}}^{n}(x)$, for any $\varepsilon>0$,

[TABLE]

so that -$\int\varphi\,d\nu\leq h(T,\nu)$.

**Concluding remark ** The results in [PS1] are proven for continuous ${\bf Z}_{+}^{d}$-actions or ${\bf Z}^{d}$-actions on $X$. The results of this note are also true for these cases. The empirical measure ${\mathcal{E}}_{n}(x)$ and the dynamical ball $B_{n}(x,\varepsilon)$ are defined as in [PS1].