# American Institute of Mathematical Sciences

October  2013, 7(4): 565-604. doi: 10.3934/jmd.2013.7.565

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

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

Received  March 2013 Revised  November 2013 Published  March 2014

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).
Citation: Doris Bohnet. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. Journal of Modern Dynamics, 2013, 7 (4) : 565-604. doi: 10.3934/jmd.2013.7.565
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