The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited

Christian Bonatti; Sylvain Crovisier; Katsutoshi Shinohara

doi:10.3934/jmd.2013.7.605

Article Contents

2013, Volume 7, Issue 4: 605-618. Doi: 10.3934/jmd.2013.7.605

This issue Previous Article Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation Next Article Regularity and convergence rates for the Lyapunov exponents of linear cocycles

The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited

1.
Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004
2.
Laboratoire de Mathématiques d’Orsay CNRS - UMR 8628 Université Paris-Sud 11 Orsay 91405
3.
FIRST, Aihara Innovative Mathematical Modelling Project, JST, Institute of Industrial Science, University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8505

Received: May 31, 2013

Published: February 2014

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Abstract

We show that for every compact $3$-manifold $M$ there exists an open subset of $Diff^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures that are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher-regularity case, where Pesin Theory gives stable and unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity.

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Mathematics Subject Classification: Primary: 37C20, 37D25, 37D30, 37G30; Secondary: 37C29, 37C40, 37G15, 37G25.

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