Partial hyperbolicity and foliations in $\mathbb{T}^3$

Rafael  Potrie

doi:10.3934/jmd.2015.9.81

Article Contents

2015, Volume 9: 81-121. Doi: 10.3934/jmd.2015.9.81

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Partial hyperbolicity and foliations in $\mathbb{T}^3$

Rafael Potrie^1,

1.
Centro de Matemática, Facultad de Ciencias, Universidad de la República, Igua 4225, Montevideo, 11400

Received: January 2013

Revised: June 2014

Published: May 2015

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Abstract

We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general results on codimension one foliations which may be of independent interest.

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Mathematics Subject Classification: Primary: 37C05, 37C20; Secondary: 37C25, 37C29, 37D30, 57R30.

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