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

Doris Bohnet

doi:10.3934/jmd.2013.7.565

Article Contents

2013, Volume 7, Issue 4: 565-604. Doi: 10.3934/jmd.2013.7.565

This issue Previous Article A generic-dimensional property of the invariant measures for circle diffeomorphisms Next Article The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited

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

Doris Bohnet^1,

1.
Institut de Mathématiques de Bourgogne CNRS - URM 5584 Université de Bourgogne Dijon 21004

Received: March 2013

Revised: November 2013

Published: February 2014

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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).

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Mathematics Subject Classification: Primary: 37D30; Secondary: 37C15.

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