The geometric approach for constructing Sinai-Ruelle-Bowen measures

TL;DR

This paper surveys geometric methods, especially the push-forward approach, for constructing Sinai-Ruelle-Bowen measures in hyperbolic dynamical systems, including recent extensions to non-uniform hyperbolicity.

Contribution

It provides a comprehensive review of the geometric push-forward technique for SRB measures and discusses recent advancements extending these methods to non-uniform hyperbolic systems.

Findings

Detailed description of the push-forward approach in uniformly hyperbolic systems

Extension of geometric techniques to non-uniform hyperbolic systems

Insights into properties and construction of SRB measures

Abstract

An important class of `physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric `push-forward' approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems.