Journal of Modern Dynamics (JMD)

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

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

       Abstract        References        Full Text (869.6K)       Related Articles       

Rafael Potrie - Centro de Matemática, Facultad de Ciencias, Universidad de la República, Igua 4225, Montevideo, 11400, Uruguay (email)

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.

Keywords:  Partial hyperbolicity (pointwise), dynamical coherence, global product structure, codimension-one foliations.
Mathematics Subject Classification:  Primary: 37C05, 37C20; Secondary: 37C25, 37C29, 37D30, 57R30.

Received: January 2013;      Revised: June 2014;      Available Online: June 2015.