Journal of Modern Dynamics (JMD)

Partially hyperbolic diffeomorphisms with compact center foliations

Pages: 747 - 769, Issue 4, October 2011      doi:10.3934/jmd.2011.5.747

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Andrey Gogolev - Department ofMathematical Sciences, The State University of New York, Binghamton, NY 13902, United States (email)

Abstract: Let $f\colon M\to M$ be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of $f$. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves.
    Also we show that a finite cover of $f$ fibers over an Anosov toral automorphism if one of the following conditions is met:

  1. 1. the center foliation of $f$ has codimension 2, or
  2. 2. the center leaves of $f$ are simply connected leaves and the unstable foliation of $f$ is one-dimensional.

Keywords:  Partially hyperbolic diffeomorphism, skew product, compact foliation, Reeb stability, Wada Lakes, accessible boundary, Anosov homeomorphism, holonomy.
Mathematics Subject Classification:  Primary: 37D30, 57R30; Secondary: 37D20.

Received: November 2011;      Revised: February 2012;      Available Online: March 2012.