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Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves

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

Published online by Cambridge University Press:  31 May 2021

A. AVILA ,
MARCELO VIANA Open the ORCID record for MARCELO VIANA [Opens in a new window]  and
A. WILKINSON Open the ORCID record for A. WILKINSON [Opens in a new window]
A. AVILA
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057Zürich, Switzerland IMPA—Estrada D. Castorina 110, Jardim Botânico, 22460-320Rio de Janeiro, Brazil (e-mail: artur@math.sunysb.edu)
MARCELO VIANA*
Affiliation:
IMPA—Estrada D. Castorina 110, Jardim Botânico, 22460-320Rio de Janeiro, Brazil (e-mail: artur@math.sunysb.edu)
A. WILKINSON
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois60637, USA (e-mail: wilkinson@math.uchicago.edu)
*
e-mail: viana@impa.br
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Abstract

We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact.

Keywords

Lyapunov exponentrigidityabsolute continuityinvariance principlepartial hyperbolicity

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37C75: Stability theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 437 - 490
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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