Cohomology of Lipschitz-valued cocycles

TL;DR

This paper studies the cohomology of Lipschitz-valued cocycles over a shift space, proving conditions under which cocycles are cohomologous with H"older continuous conjugacies, extending understanding of cocycle classification.

Contribution

It establishes that dominated cocycles coinciding on periodic points are cohomologous via H"older conjugacies and links measurable conjugacy to H"older conjugacy under additional conditions.

Findings

Dominated cocycles coinciding on periodic points are cohomologous.

Existence of a measurable conjugacy implies a H"older conjugacy under certain conditions.

Provides a classification framework for Lipschitz-valued cocycles.

Abstract

We consider the set of H\"older continuous cocycles over a finite shift acting on a group of Lipschitz homeomorphisms Lip(G), where G is a metrisable compact topological group. We establish that two dominated cocycles that coincide over periodic points of the base are cohomologous, with the conjugacy being H\"older continuous. Moreover, we prove that, under an additional condition, the existence of a measurable conjugacy implies the existence of a H\"older continuous conjugacy between the cocycles.