Fiber bunching and cohomology for Banach cocycles over hyperbolic systems

Victoria Sadovskaya

doi:10.3934/dcds.2017213

Article Contents

2017, Volume 37, Issue 9: 4959-4972. Doi: 10.3934/dcds.2017213

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Fiber bunching and cohomology for Banach cocycles over hyperbolic systems

Victoria Sadovskaya

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received: December 2016

Revised: April 2017

Published: October 2017

The author is supported in part by NSF grant DMS-1301693

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Abstract

We consider Hölder 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 Hölder continuously cohomologous if and only if they have Hölder 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 Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and one with values in a set which is compact in strong operator topology.

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Mathematics Subject Classification: Primary: 37D20, 37C15, 37H05.

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