A formula for the local metric pressure

TL;DR

This paper introduces a new formula for local metric pressure, extending the Brin-Katok result for metric entropy, and demonstrates that non-atomic weak-Gibbs measures are equilibrium states.

Contribution

It generalizes the concept of metric entropy to local metric pressure and provides a simple proof that certain measures are equilibrium states.

Findings

The formula extends Brin-Katok's result to local metric pressure.

Non-atomic weak-Gibbs measures are shown to be equilibrium states.

Provides a straightforward proof for the equilibrium state property.

Abstract

In this note we present a formula for the local metric pressure that generalizes Brin-Katok result for the metric entropy. As an application, we give a straightforward proof of the fact that non-atomic weak-Gibbs invariant probability measures are equilibrium states.

Equations

$$P_{\mu}(f,\varphi)=\sup_{\{\nu\colon\,f_{*}(\nu)\,=\,\nu\}}\,\big{\{}\,P_{\nu}(f,\varphi)\,\big{\}}$$

$$\int P_{\mu }(x,f,\phi )d\mu =P_{\mu }(f,\phi ).$$

$$\underset{n\to +\infty }{\lim }\frac{\log {\delta }_n(\epsilon ,x)}{n}=0$$

$$\delta_{n}(\varepsilon,x)^{-1}\leqslant\frac{\mu\big{(}B^{f}_{n}(x,\varepsilon)\big{)}}{\exp\big{(}-P_{\text{top}}(\varphi,f)\,n+S^{f}_{n}\varphi(x)\big{)}}\leqslant\delta_{n}(\varepsilon,x)$$

$$B^{f}_{n}(x,\varepsilon)=\Big{\{}y\in X\colon\,d(f^{i}(x),f^{i}(y))<\varepsilon,\quad\forall\,0\leqslant i\leqslant n-1\Big{\}}$$

$$h_{\mu}(x,f)\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\lim_{\varepsilon\to 0}\limsup_{n\to+\infty}\frac{-\log\mu\big{(}B^{f}_{n}(x,\varepsilon)\big{)}}{n}=\lim_{\varepsilon\to 0}\liminf_{n\to+\infty}\frac{-\log\mu\big{(}B^{f}_{n}(x,\varepsilon)\big{)}}{n}$$

$$P_{\mu }(x,f,\phi )$$

$$x\in {\mathcal{B}}_{\phi }\mapsto \tilde{\phi }(x)=\underset{n\to +\infty }{\lim }\frac{1}{n}\sum _{i=0}^{n-1}\phi (f^i(x))$$

$$\begin{split}&\liminf_{n\,\to\,+\infty}\,\frac{-\log\mu\big{(}B^{f}_{n}(x,\varepsilon)\big{)}}{n}+\widetilde{\varphi}(x)=\liminf_{n\,\to\,+\infty}\,\frac{-\log\mu\big{(}B^{f}_{n}(x,\varepsilon)\big{)}+S^{f}_{n}\varphi(x)}{n}\leqslant\\ &\leqslant\limsup_{n\,\to\,+\infty}\,\frac{-\log\mu\big{(}B^{f}_{n}(x,\varepsilon)\big{)}+S^{f}_{n}\varphi(x)}{n}=\limsup_{n\,\to\,+\infty}\,\frac{-\log\mu\big{(}B^{f}_{n}(x,\varepsilon)\big{)}}{n}+\widetilde{\varphi}(x)\end{split}$$

$$P_{\mu }(x,f,\phi )=h_{\mu }(x,f)+\tilde{\phi }(x)$$

$$P_{\text{top}}(f,\varphi)-\frac{\log\delta_{n}(\varepsilon,x)}{n}\leqslant\frac{-\log\mu\big{(}B^{f}_{n}(x,\varepsilon)\big{)}+S^{f}_{n}\varphi(x)}{n}\leqslant P_{\text{top}}(\varphi,f)-\frac{\log\delta_{n}(\varepsilon,x)^{-1}}{n}.$$

$$P_{\mu }(x,f,\phi )=P_{\text{top}}(f,\phi )\text{mbox}at\mu -a.e.x\in X.$$

$$h_{\mu }(f)+\int \phi d\mu =\int P_{\mu }(x,f,\phi )d\mu =\int P_{\text{top}}(f,\phi )d\mu =P_{\text{top}}(f,\phi ).$$

Full text

1. Introduction

Let $(X,d)$ be a compact metric space, $f\colon X\to X$ a continuous transformation and $\varphi\colon X\to\mathbb{R}$ a continuous potential. The topological pressure $P_{\text{top}}(f,\varphi)$ of $f$ and $\varphi$ is a topological invariant that generalizes the notion of topological entropy of $f$, one denotes by $h_{\text{top}}(f)$, in the sense that $P_{\text{top}}(f,\varphi)=h_{\text{top}}(f)$ whenever $\varphi\equiv 0$. We refer the reader to [3] for precise definitions and properties of these notions. A Borel $f$-invariant probability measure $\mu$ is said to be an equilibrium state for $f$ and the potential $\varphi$ if

[TABLE]

where the $P_{\nu}(f,\varphi)$ stands for the sum $h_{\nu}(f)+\int\varphi\,d\nu$ and the supremum is taken over all the Borel $f$-invariant probability measures. According to the Variational Principle ([3, Theorem 9.10]), the previous least upper bound coincides with the supremum evaluated on the set of ergodic probability measures, and is equal to $P_{\text{top}}(f,\varphi)$. We will show how to estimate the metric pressure of any continuous potential, thereby generalizing Brin-Katok formula for the metric entropy [1].

Theorem A**.**

Let $\mu$ be a Borel non-atomic $f$-invariant probability measure. Then there exists a $\mu$-integrable map $\,x\,\in\,X\,\,\mapsto\,\,P_{\mu}(x,f,\varphi)$ which is $f$-invariant and satisfies

[TABLE]

If, in addition, $\mu$ is ergodic, then $P_{\mu}(x,f,\varphi)=P_{\mu}(f,\varphi)$ for $\mu$ almost every $x\in X$.

We remark that, when $\varphi\equiv 0$, the map $\,x\,\in\,X\,\,\mapsto\,\,P_{\mu}(x,f,\varphi)$ is the local entropy as defined by Brin and Katok.

A Borel probability measure $\mu$ is said to be weak-Gibbs for the dynamical system $f$ with respect to a potential $\varphi$ if there exists $\varepsilon_{0}>0$ and a subset $\Lambda\subset X$ with full $\mu$-measure such that, for every $0<\varepsilon<\varepsilon_{0}$ and every $x\in\Lambda$, there is a sequence of positive constants \big{(}\delta_{n}(\varepsilon,x)\big{)}_{n\,\in\,\mathbb{N}} satisfying

[TABLE]

and, for every $n\in\mathbb{N}$,

[TABLE]

where

[TABLE]

is the $n$th dynamical ball of $f$ at $x$ with radius $\varepsilon$ and $S^{f}_{n}\varphi(x)$ stands for the $n$th Birkhoff’s sum $\sum_{i=0}^{n-1}\,\varphi(f^{i}(x))$ at $x$ associated to the dynamics $f$ and the fixed potential $\varphi$. We say that a weak-Gibbs measure $\mu$ for $f$ with respect to $\varphi$ is Gibbs if the sequence \big{(}\delta_{n}(\varepsilon,x)\big{)}_{n\,\in\,\mathbb{N}} is independent of $n$ and $x$.

Corollary I**.**

Let $f\colon X\to X$ be a continuous map on a compact metric space $(X,d)$ whose topological entropy $h_{\text{top}}(f)$ is finite and which preserves a Borel non-atomic probability measure $\mu$. Consider a continuous potential $\varphi\colon X\to\mathbb{R}$. If $\mu$ is a weak-Gibbs measure for $f$ with respect to $\varphi$, then $\mu$ is an equilibrium state for $f$ and $\varphi$.

2. Proof of Theorem A

In this section we will extend Brin-Katok local entropy formula to general continuous potentials (another generalization may by found in [4]). Brin-Katok’s result asserts that, given a compact metric space $X$, a continuous map $f:X\to X$ and Borel non-atomic $f$-invariant probability measure $\mu$, there exists a full $\mu$-measure set $\mathcal{BK}\subset X$ such that:

• (a)

For every $x\in\mathcal{BK}$,

[TABLE]

is well defined.

• (b)

The map $\,x\in\mathcal{BK}\,\mapsto\,h_{\mu}(x,f)\,\,$ is $f$-invariant.

• (c)

$\int h_{\mu}(x,f)\,d\mu=h_{\mu}(f)$.

Having fixed a continuous potential $\varphi$ whose pressure $P_{\text{top}}(f,\varphi)$ finite, consider the Birkhoff’s sums $S^{f}_{n}\varphi$ and, for $x\in X$, define the local pressure of $\varphi$ at $x$ by

[TABLE]

if these limits exist and are equal.

Lemma 2.1**.**

The following properties are valid for $\mu$:

• (a)

$P_{\mu}(x,f,\varphi)$* is well defined at $\mu$ almost every $x\in X$.*

• (b)

The map $\,x\,\mapsto\,P_{\mu}(x,f,\varphi)$ is $f$-invariant.

• (c)

$\int P_{\mu}(x,f,\varphi)\,d\mu=h_{\mu}(f)+\int\varphi\,d\mu$.

• (d)

If, in addition, $\mu$ is ergodic, then $P_{\mu}(x,f,\varphi)=h_{\mu}(f)+\int\varphi\,d\mu\,\,$ at $\mu$ almost every $x\in X$.

Proof.

Birkhoff’s Theorem provides a full $\mu$-measure set $\mathcal{B}_{\varphi}$ and an $f$-invariant map

[TABLE]

satisfying $\int\widetilde{\varphi}\,d\mu=\int\varphi\,d\mu$. Therefore, for every $x\in\mathcal{BK}\cap\mathcal{B}_{\varphi}$, we have

[TABLE]

Taking the limit when $\varepsilon$ goes to [math] at the last inequality, we conclude that

[TABLE]

exists for every $x\in\mathcal{BK}\cap\mathcal{B}_{\varphi}$. Items (b), (c) and (d) are immediate after (a). ∎

3. Proof of Corollary I

Firstly recall that, given a compact metric space $X$ and a continuous map $f:X\to X$, the pressure map $P_{\text{top}}(f,.)\colon\,C^{0}(X,\mathbb{R})\,\to\,\mathbb{R}\cup\{+\infty\}$, defined on the space $C^{0}(X,\mathbb{R})$ of continuous potentials, is either finite valued or constantly $+\infty$ (cf. [3, §9.2]).

Consider a Gibbs measure $\mu$ for the dynamics $f$ and a continuous potential $\varphi$, and gather the corresponding $\varepsilon_{0}$, $\Lambda$ and \big{(}\delta_{n}(\varepsilon,x)\big{)}_{n\,\in\,\mathbb{N}} satisfying equations (1) and (2) for every $x\in\Lambda$ and every $n\in\mathbb{N}$. As we are assuming that $h_{\text{top}}(f)<+\infty$, we know that $P_{\text{top}}(f,\varphi)$ is finite. Rewriting (2), we obtain, for every $x\in\mathcal{BK}\cap\mathcal{B}_{\varphi}\cap\Lambda$,

[TABLE]

Taking $\limsup_{n\to+\infty}$ (or $\liminf_{n\to+\infty}$) and afterwards the limit as $\varepsilon$ goes to [math], we get

[TABLE]

Thus, applying Lemma 2.1, we conclude that

[TABLE]

Therefore $\mu$ is an equilibrium state for $f$ with respect to $\varphi$.

4. Open questions

Is a Gibbs measure for a dynamical system $f$ with respect to a potential $\varphi$ always $f$-invariant? We may also wonder whether the existence of a Gibbs measure for $f$ and $\varphi$ prompts the existence of an equilibrium state for $\varphi$. Or we may ask under what additional conditions, other than $f$-invariance, is a Gibbs measure for $f$ and $\varphi$ an equilibrium state for these dynamics and potential, or else an equilibrium state for another natural potential somehow related to $\varphi$. For instance, under a stronger definition of the Gibbs property and assuming that $f$ is a homeomorphism satisfying expansiveness and specification, the answer is positive (cf. [2]). As far as we know, these are still open questions.

Acknowledgements

MC and SP were partially supported by CMUP (UID/MAT/00144/2019), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. SP also acknowledges financial support from a postdoctoral grant of the project PTDC/MAT-CAL/3884/2014