A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics

TL;DR

This paper develops a spectral method to analyze how physical measures in smooth random dynamical systems respond to perturbations, extending deterministic spectral techniques to the random setting.

Contribution

It adapts spectral perturbation theory to random systems with good mixing, providing new linear and higher-order response results without infinite-dimensional ergodic theory.

Findings

Response results for random Anosov diffeomorphisms

Response results for random U(1) extensions of expanding maps

Applicable to systems over general ergodic base maps

Abstract

For smooth random dynamical systems we consider the quenched linear and higher-order response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gou\"ezel, Keller, and Liverani, which has been applied to deterministic systems with great success, may be adapted to study random systems that possess good mixing properties. As a consequence, we obtain general linear and higher-order response results for random dynamical systems that we then apply to random Anosov diffeomorphisms and random U(1) extensions of expanding maps. We emphasise that our results apply to random dynamical systems over a general ergodic base map, and are obtained without resorting to infinite dimensional multiplicative ergodic theory.