Statistical properties of mostly contracting fast-slow partially hyperbolic systems

Abstract We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$ . We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$ , we show that it has a continuous invariant family of $\gamma $ -Hölder Riemannian metrics on $\mathcal {M}$ . We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.

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Unstable Topological Entropy in Mean u-Metrics for Partially Hyperbolic Systems

Dynamical Systems ◽

10.1080/14689367.2021.1923659 ◽

2021 ◽

pp. 1-18

Author(s):

Ping Huang ◽

Chenwei Wang ◽

Ercai Chen

Keyword(s):

Topological Entropy ◽

Hyperbolic Systems ◽

Partially Hyperbolic

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Quasi-shadowing for partially hyperbolic diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.126 ◽

2014 ◽

Vol 35 (2) ◽

pp. 412-430 ◽

Cited By ~ 8

Author(s):

HUYI HU ◽

YUNHUA ZHOU ◽

YUJUN ZHU

Keyword(s):

Spectral Decomposition ◽

Hyperbolic Systems ◽

Decomposition Theorem ◽

Closing Lemma ◽

Shadowing Property ◽

Hyperbolic Diffeomorphisms ◽

Uniformly Hyperbolic Systems ◽

Partially Hyperbolic ◽

Partially Hyperbolic Diffeomorphisms ◽

Partially Hyperbolic Diffeomorphism

AbstractA partially hyperbolic diffeomorphism $f$ has the quasi-shadowing property if for any pseudo orbit $\{x_{k}\}_{k\in \mathbb{Z}}$, there is a sequence of points $\{y_{k}\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_{k})$ by a motion ${\it\tau}$ along the center direction. We show that any partially hyperbolic diffeomorphism has the quasi-shadowing property, and if $f$ has a $C^{1}$ center foliation then we can require ${\it\tau}$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^{0}$-perturbation. When $f$ has a uniformly compact $C^{1}$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems which hold for uniformly hyperbolic systems, such as the Anosov closing lemma, the cloud lemma and the spectral decomposition theorem.

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Statistical properties of the maximal entropy measure for partially hyperbolic attractors

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.86 ◽

2016 ◽

Vol 37 (4) ◽

pp. 1060-1101 ◽

Cited By ~ 7

Author(s):

ARMANDO CASTRO ◽

TEÓFILO NASCIMENTO

Keyword(s):

Probability Measure ◽

Hyperbolic Systems ◽

Maximal Entropy ◽

Entropy Measure ◽

Hölder Continuous ◽

Hyperbolic Diffeomorphisms ◽

The Central Limit Theorem ◽

Exponential Decay Of Correlations ◽

Partially Hyperbolic ◽

Partially Hyperbolic Diffeomorphisms

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semiconjugate to non-uniformly expanding maps. Using the theory of projective metrics on cones, we then prove exponential decay of correlations for Hölder continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from a solenoid, we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partially hyperbolic systems derived from Anosov.

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