Journal of Modern Dynamics (JMD)

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

Pages: 605 - 618, Issue 4, December 2013      doi:10.3934/jmd.2013.7.605

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Christian Bonatti - Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004, France (email)
Sylvain Crovisier - Laboratoire de Mathématiques d’Orsay CNRS - UMR 8628 Université Paris-Sud 11 Orsay 91405, France (email)
Katsutoshi Shinohara - FIRST, Aihara Innovative Mathematical Modelling Project, JST, Institute of Industrial Science, University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8505, Japan (email)

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.

Keywords:  Pesin theory, wild diffeomorphism, dominated splitting, Lyapunov exponents.
Mathematics Subject Classification:  Primary: 37C20, 37D25, 37D30, 37G30; Secondary: 37C29, 37C40, 37G15, 37G25.

Received: June 2013;      Available Online: March 2014.