Journal of Modern Dynamics (JMD)

Genericity of nonuniform hyperbolicity in dimension 3

Pages: 121 - 138, Issue 1, January 2012      doi:10.3934/jmd.2012.6.121

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Jana Rodriguez Hertz - IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay (email)

Abstract: For a generic conservative diffeomorphism of a closed connected 3-manifold $M$, the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is nonuniformly hyperbolic and ergodic.
    This is the 3-dimensional version of the well-known result by Mañé-Bochi [14, 4], stating that a generic conservative surface diffeomorphism is either Anosov or all Lyapunov exponents vanish almost everywhere. This result was inspired by and answers in the positive in dimension 3 a conjecture by Avila-Bochi [2].

Keywords:  Nonuniform hyperbolicity, domination of the Oseledets splitting, partially hyperbolic sets with positive measure.
Mathematics Subject Classification:  Primary: 37D25, 37C20; Secondary: 37C20.

Received: March 2012;      Available Online: May 2012.