Cohomology of fiber bunched cocycles over hyperbolic systems

TL;DR

This paper establishes conditions under which fiber bunched cocycles over hyperbolic systems are cohomologous, based on periodic data, and explores implications for smooth conjugacy and quasiconformal cocycles.

Contribution

It proves that fiber bunched cocycles with equal periodic data are Holder cohomologous and extends results to conjugate periodic data, with applications to smooth conjugacy of Anosov diffeomorphisms.

Findings

Cocycles with equal periodic data are Holder cohomologous.

Fiber bunching condition can be deduced from periodic data.

Results apply to cocycles over hyperbolic sets and in various group settings.

Abstract

We consider Holder continuous fiber bunched GL(d,R)-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Holder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of the diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question when an Anosov diffeomorphism is smoothly conjugate to a C^1-small perturbation. We also establish Holder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of GL(d,R), for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle.