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Diffeomorphism cocycles over partially hyperbolic systems

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  28 December 2020

VICTORIA SADOVSKAYA
VICTORIA SADOVSKAYA*
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA16802, USA
*
e-mail:sadovskaya@psu.edu
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Abstract

We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$ . We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$ , we show that it has a continuous invariant family of $\gamma $ -Hölder Riemannian metrics on $\mathcal {M}$ . We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.

Keywords

Cocyclediffeomorphism grouppartially hyperbolic systemconjugacyisometry

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 263 - 286
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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