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

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 Gouëzel, Keller and Liverani [28, 33], 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, as well as the differentiability of the variance in quenched central limit theorems (CLTs), for random dynamical systems (RDSs) that we then apply to random Anosov diffeomorphisms and random U(1) extensions of expanding maps. We emphasize 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.

Almost sure invariance principle for random dynamical systems via Gouëzel's approach

“…For annealed linear response results (mostly dealing with the case when the maps T ω,ε are composed in an i.i.d fashion) which rely on techniques very similar to the ones for deterministic dynamics, we refer to [2,22,24,25]. On the other hand, the study of the quenched linear response was initiated by Rugh and Sedro [34] for random expanding dynamics, followed by the works by Dragičević and Sedro [20] and Crimmins and Nakano [14] for random (partially) hyperbolic dynamics. We emphasize that all three papers deal with cases of random dynamics which exhibit uniform decay of correlations (with respect to the random parameter ω ∈ Ω).…”

Linear response for intermittent maps with critical point

“…There has been deep investigation about response in the setting of unimodal maps [2,7,9,37,39], providing examples of systems where the SRB measure does not vary smoothly under perturbations. Recently, linear response has also been actively studied in the context of random and extended dynamical systems; see, for instance [13,17,19,25,41]. For extensive background information on linear response, we recommend the survey [8].…”