Equilibrium states in dynamical systems via geometric measure theory

TL;DR

This paper extends geometric measure theory methods to construct and analyze equilibrium states in hyperbolic dynamical systems, broadening applicability beyond traditional Markov partition techniques.

Contribution

It introduces a new geometric construction for equilibrium states using Carathéodory dimension, avoiding reliance on Markov partitions or the specification property.

Findings

Provides a new proof of existence and uniqueness of equilibrium states.

Applies to systems lacking Markov partitions or specification property.

Generalizes the geometric approach to a wider class of dynamical systems.

Abstract

Given a dynamical system with a uniformly hyperbolic (`chaotic') attractor, the physically relevant Sinai-Ruelle-Bowen (SRB) measure can be obtained as the limit of the dynamical evolution of the leaf volume along local unstable manifolds. We extend this geometric construction to the substantially broader class of equilibrium states corresponding to H\"older continuous potentials; these states arise naturally in statistical physics and play a crucial role in studying stochastic behavior of dynamical systems. The key step in our construction is to replace leaf volume with a reference measure that is obtained from a Carath\'eodory dimension structure via an analogue of the construction of Hausdorff measure. In particular, we give a new proof of existence and uniqueness of equilibrium states that does not use standard techniques based on Markov partitions or the specification property; our approach can be applied to systems that do not have Markov partitions and do not satisfy the specification property.