Analyticity of the SRB measure of a lattice of coupled Anosov diffeomorphisms of the torus

TL;DR

This paper proves that the SRB measure of a lattice of coupled hyperbolic toral diffeomorphisms is analytic with respect to coupling strength, using symbolic dynamics and Gibbs measure techniques.

Contribution

It introduces a novel approach to analyze the analyticity of SRB measures in coupled hyperbolic systems via symbolic dynamics and cluster expansion methods.

Findings

SRB measure is analytic in coupling strength

Mapping to Gibbs measure enables analysis of the system

Extension of cluster expansion techniques applied

Abstract

We consider the "thermodynamic limit"of a d-dimensional lattice of hyperbolic dynamical systems on the 2-torus, interacting via weak and nearest neighbor coupling. We prove that the SRB measure is analytic in the strength of the coupling. The proof is based on symbolic dynamics techniques that allow us to map the SRB measure into a Gibbs measure for a spin system on a (d+1)-dimensional lattice. This Gibbs measure can be studied by an extension (decimation) of the usual "cluster expansion" techniques.