Quenched linear response for smooth expanding on average cocycles

TL;DR

This paper proves a quenched linear response theory for certain random dynamical systems, including non-uniformly mixing cocycles, using advanced ergodic theory techniques.

Contribution

It introduces a novel quenched linear response result applicable to non-uniform decay systems, expanding the scope of existing dynamical systems theory.

Findings

Established quenched linear response for non-uniform decay systems

Applied theory to smooth expanding on average cocycles on the circle

Utilized infinite-dimensional ergodic theory and Oseledets spaces

Abstract

We establish an abstract quenched linear response result for random dynamical systems, which we then apply to the case of smooth expanding on average cocycles on the unit circle. In sharp contrast to the existing results in the literature, we deal with the class of random dynamics that does not necessarily exhibit uniform decay of correlations. Our techniques rely on the infinite-dimensional ergodic theory and in particular, on the study of the top Oseledets space of a parametrized transfer operator cocycle.