Classification of random circle homeomorphisms up to topological conjugacy

TL;DR

This paper classifies random orientation-preserving homeomorphisms of the circle up to topological conjugacy, providing a comprehensive framework for understanding their dynamics under certain weak conditions.

Contribution

It offers a complete classification of random circle homeomorphisms up to topological conjugacy under broad conditions, extending previous results.

Findings

Classification covers all such homeomorphisms with connected Polish noise space

Applicable under an extremely weak additional condition

Framework for understanding random circle dynamics

Abstract

We provide a classification of random orientation-preserving homeomorphisms of $\mathbb{S}^1$, up to topological conjugacy of the random dynamical systems generated by i.i.d. iterates of the random homeomorphism. This classification covers all random circle homeomorphisms for which the noise space is a connected Polish space and an additional extremely weak condition is satisfied.