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Partially hyperbolic diffeomorphisms with a uniformly compact center foliation: the quotient dynamics

Published online by Cambridge University Press:  11 February 2015

DORIS BOHNET  and
CHRISTIAN BONATTI
DORIS BOHNET
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université de Bourgogne, Dijon, France email Doris.Bohnet@u-bourgogne.fr, bonatti@u-bourgogne.fr
CHRISTIAN BONATTI
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université de Bourgogne, Dijon, France email Doris.Bohnet@u-bourgogne.fr, bonatti@u-bourgogne.fr
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Abstract

We show that a partially hyperbolic $C^{1}$-diffeomorphism $f:M\rightarrow M$ with a uniformly compact $f$-invariant center foliation ${\mathcal{F}}^{c}$ is dynamically coherent. Further, the induced homeomorphism $F:M/{\mathcal{F}}^{c}\rightarrow M/{\mathcal{F}}^{c}$ on the quotient space of the center foliation has the shadowing property, i.e. for every ${\it\epsilon}>0$ there exists ${\it\delta}>0$ such that every ${\it\delta}$-pseudo-orbit of center leaves is ${\it\epsilon}$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics. Other interesting properties of the quotient dynamics are also discussed.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 4 , June 2016 , pp. 1067 - 1105
Copyright
© Cambridge University Press, 2015 

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