Statistical stability and linear response for random hyperbolic dynamics

TL;DR

This paper investigates the statistical stability and linear response of random hyperbolic dynamical systems, specifically families of Anosov diffeomorphisms, and establishes differentiability of variance in the CLT under certain conditions.

Contribution

It introduces a novel analysis of the top Oseledets space of a parametrized transfer operator cocycle for random hyperbolic systems, proving stability and response results.

Findings

Statistical stability holds for families of random Anosov diffeomorphisms.

Linear response is established for the associated measures.

Variance in the quenched CLT depends differentiably on parameters under certain conditions.

Abstract

We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis rely on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.