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Smoothness of stable holonomies inside center-stable manifolds

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

Published online by Cambridge University Press:  18 October 2021

AARON BROWN
AARON BROWN*
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL60208, USA
*
(e-mail: awb@northwestern.edu)
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Abstract

Under a suitable bunching condition, we establish that stable holonomies inside center-stable manifolds for $C^{1+\beta }$ diffeomorphisms are uniformly bi-Lipschitz and, in fact, $C^{1+\mathrm {H}\ddot{\rm o}\mathrm {lder}}$ . This verifies the ergodicity of suitably center-bunched, essentially accessible, partially hyperbolic $C^{1+\beta }$ diffeomorphisms and verifies that the Ledrappier–Young entropy formula holds for $C^{1+\beta }$ diffeomorphisms of compact manifolds.

Keywords

holonomiespartial hyperbolicityergodicityentropy

MSC classification

Secondary: 37C40: Smooth ergodic theory, invariant measures 37D30: Partially hyperbolic systems and dominated splittings
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 12 , December 2022 , pp. 3593 - 3618
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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