A coupling approach to random circle maps expanding on the average

TL;DR

This paper develops a coupling method for random circle maps that are expanding on average, achieving exponential convergence and mixing without uniform bounds, and proves an almost sure invariance principle for vector-valued observables.

Contribution

It introduces a novel coupling scheme for nonuniform random circle maps, enabling exponential decay of correlations and invariance principles.

Findings

Exponential convergence of measures established

Achieved exponential mixing without uniform bounds

Proved an almost sure invariance principle

Abstract

We study random circle maps that are expanding on the average. Uniform bounds on neither expansion nor distortion are required. We construct a coupling scheme, which leads to exponential convergence of measures (memory loss) and exponential mixing. Leveraging from the structure of the associated correlation estimates, we prove an almost sure invariance principle for vector-valued observables. The motivation for our paper is to explore these methods in a nonuniform random setting.