Genericity of nonuniform hyperbolicity in dimension 3

Jana Rodriguez Hertz

doi:10.3934/jmd.2012.6.121

Article Contents

2012, Volume 6, Issue 1: 121-138. Doi: 10.3934/jmd.2012.6.121

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Genericity of nonuniform hyperbolicity in dimension 3

Jana Rodriguez Hertz^1,

1.
IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo

Received: March 2012

Published: April 2012

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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.

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References

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