Pressure and Equilibrium States in Ergodic Theory

TL;DR

This paper explores the fundamental concepts of Gibbs and equilibrium states in ergodic theory, highlighting their roles in symbolic dynamical systems and their connections to statistical mechanics and information theory.

Contribution

It provides an overview of key results on invariant measures in symbolic systems and their applications to differentiable dynamical systems, emphasizing interdisciplinary links.

Findings

Characterization of Gibbs and equilibrium states in symbolic dynamics

Illustration of their applications to differentiable systems

Highlighting the relationship between ergodic theory, statistical mechanics, and information theory

Abstract

Our goal is to present the basic results on one-dimensional Gibbs and equilibrium states viewed as special invariant measures on symbolic dynamical systems, and then to describe without technicalities a sample of results they allowed to obtain for certain differentiable dynamical systems. We hope that this contribution will illustrate the symbiotic relationship between ergodic theory and statistical mechanics, and also information theory.