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Article Dans Une Revue Ergodic Theory and Dynamical Systems Année : 2016

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

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Doris Bohnet
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Christian Bonatti

#### Résumé

We show that a partially hyperbolic $C^1$-diffeomorphism $f : M \to M$ with a uniformly compact $f$-invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i. e. for every $\varepsilon> 0$ there exists $\delta > 0$ such that every $\delta$--pseudo-orbit of center leaves is $\varepsilon$-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.

#### Domaines

Mathématiques [math] Systèmes dynamiques [math.DS]

### Dates et versions

hal-01410903 , version 1 (06-12-2016)

### Identifiants

• HAL Id : hal-01410903 , version 1
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### Citer

Doris Bohnet, Christian Bonatti. Partially hyperbolic diffeomorphisms with a uniformly compact center foliation: the quotient dynamics. Ergodic Theory and Dynamical Systems, 2016, 36 (4), pp.1067-1105 ⟨10.1017/etds.2014.102⟩. ⟨hal-01410903⟩

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