Gibbs measures for hyperbolic attractors defined by densities

TL;DR

This paper introduces a new, simplified method for constructing Gibbs measures for hyperbolic attractors, generalizing classical SRB measure constructions and applicable in broader contexts.

Contribution

It presents a novel approach to constructing Gibbs measures by modifying densities along unstable manifolds, extending previous methods and simplifying proofs.

Findings

New construction of Gibbs measures for hyperbolic attractors

Method generalizes classical SRB measure construction

Proof relies on growth rate and entropy estimates

Abstract

In this article we will describe a new construction for Gibbs measures for hyperbolic attractors generalizing the original construction of Sinai, Bowen and Ruelle of SRB measures. The classical construction of the SRB measure is based on pushing forward the normalized volume on a piece of unstable manifold. By modifying the density at each step appropriately we show that the resulting measure is a prescribed Gibbs measure. This contrasts with, and complements, the construction of Climenhaga-Pesin-Zelerowicz who replace the volume on the unstable manifold by a fixed reference measure. Moreover, the simplicity of our proof, which uses only explicit properties on the growth rate of unstable manifold and entropy estimates, has the additional advantage that it applies in more general settings.