Journal of Modern Dynamics (JMD)

Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation

Pages: 565 - 604, Issue 4, December 2013      doi:10.3934/jmd.2013.7.565

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Doris Bohnet - Institut de Mathématiques de Bourgogne CNRS - URM 5584 Université de Bourgogne Dijon 21004, France (email)

Abstract: We consider a partially hyperbolic $C^1$-diffeomorphism $f\colon M \rightarrow M$ with a uniformly compact $f$-invariant center foliation $\mathcal{F}^c$. We show that if the unstable bundle is one-dimensional and oriented, then the holonomy of the center foliation vanishes everywhere, the quotient space $M/\mathcal{F}^c$ of the center foliation is a torus and $f$ induces a hyperbolic automorphism on it, in particular, $f$ is centrally transitive.
    We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set of the quotient dynamics, and we establish the existence of a holonomy-invariant family of measures on the unstable leaves (Margulis measure).

Keywords:  Partial hyperbolicity, center foliation, uniformly compact foliation, transitivity, Margulis measure.
Mathematics Subject Classification:  Primary: 37D30; Secondary: 37C15.

Received: March 2013;      Revised: November 2013;      Available Online: March 2014.