Globally coupled Anosov diffeomorphisms: Statistical properties

TL;DR

This paper investigates the statistical properties of infinite systems of weakly coupled Anosov diffeomorphisms, establishing existence, uniqueness, and exponential convergence to equilibrium of their invariant states.

Contribution

It introduces a rigorous analysis of coupled Anosov systems using transfer operators on anisotropic Banach spaces, proving key properties of their invariant measures.

Findings

Existence and uniqueness of the physical invariant state $h_psilon$.

Exponential convergence to equilibrium for certain distributions.

Lipschitz continuity of the invariant state with respect to coupling strength psilon.

Abstract

We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, $h_\varepsilon$. Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map $\varepsilon\mapsto h_\varepsilon$ is Lipschitz continuous.