Spectral methods for limit theorems for random expanding transformations

TL;DR

This paper extends spectral methods to prove limit theorems for random non-uniformly expanding systems, achieving broader applicability and improved rates, with effective results for smooth systems.

Contribution

It introduces a generalized spectral approach for limit theorems in complex random dynamical systems, surpassing previous methods in scope and effectiveness.

Findings

Established CLT and MDP for a wide class of systems

Rates improve as non-uniformity decreases

Effective rates for smooth systems

Abstract

We extend the spectral method for proving limit theorems to random non-uniformly expanding dynamical systems. This yields the CLT and moderate deviations principles (MDP). We show that as the amount of non-uniformity decreases the CLT rates and the speed in the MDP become closer to the optimal ones. For smooth systems the rates are effective. Compared to recent progress on the subject [34] we are able to consider much more general maps, Gibbs measures and observables. However, our main results are new even in the setup of [34].