Fiber bunching and cohomology for Banach cocycles over hyperbolic systems

TL;DR

This paper investigates the cohomology of fiber bunched Banach cocycles over hyperbolic systems, establishing conditions under which cocycles are Holder conjugate based on periodic data and demonstrating the Holder continuity of conjugacies.

Contribution

It proves that fiber bunched cocycles are Holder cohomologous if their periodic data are Holder conjugate, and shows Holder continuity of conjugacies in this setting.

Findings

Fiber bunched cocycles are Holder cohomologous if and only if their periodic data are Holder conjugate.

The fiber bunching condition can be verified using periodic data.

Holder continuity of measurable conjugacies is established under fiber bunching.

Abstract

We consider Holder continuous cocycles over hyperbolic dynamical systems with values in the group of invertible bounded linear operators on a Banach space. We show that two fiber bunched cocycles are Holder continuously cohomologous if and only if they have Holder conjugate periodic data. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base system. We show that this condition can be established based on the periodic data of a cocycle. We also establish Holder continuity of a measurable conjugacy between a fiber bunched cocycle and one with values in a set which is compact in strong operator topology.