Unstable entropies and dimension theory of partially hyperbolic systems

Abstract In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory–Pesin dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in Bowen (1973 Trans. Am. Math. Soc. 184 125–36); Pfister and Sullivan (2007 Ergod. Theor. Dynam. Syst. 27 929–56). Our results give new insights to the multifractal analysis for partially hyperbolic systems.

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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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Unstable entropy and unstable pressure for random partially hyperbolic dynamical systems

Stochastics and Dynamics ◽

10.1142/s0219493721500210 ◽

2020 ◽

pp. 2150021

Author(s):

Xinsheng Wang ◽

Weisheng Wu ◽

Yujun Zhu

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Variational Principle ◽

Topological Entropy ◽

Equilibrium States ◽

Metric Entropy ◽

Unstable Foliation ◽

Hyperbolic Dynamical Systems ◽

Partially Hyperbolic

Let [Formula: see text] be a [Formula: see text] random partially hyperbolic dynamical system. For the unstable foliation, the corresponding unstable metric entropy, unstable topological entropy and unstable pressure via the dynamics of [Formula: see text] on the unstable foliation are introduced and investigated. A version of Shannon–McMillan–Breiman Theorem for unstable metric entropy is given, and a variational principle for unstable pressure (and hence for unstable entropy) is obtained. Moreover, as an application of the variational principle, equilibrium states for the unstable pressure including Gibbs [Formula: see text]-states are investigated.

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Diffeomorphism cocycles over partially hyperbolic systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.131 ◽

2020 ◽

pp. 1-24

Author(s):

VICTORIA SADOVSKAYA

Keyword(s):

Hyperbolic System ◽

Compact Manifold ◽

Hyperbolic Systems ◽

Riemannian Metrics ◽

Hölder Continuous ◽

Group Of Diffeomorphisms ◽

Small Loss ◽

Partially Hyperbolic ◽

Loss Of Regularity

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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VARIATIONAL PRINCIPLE FOR SUBADDITIVE SEQUENCE OF POTENTIALS IN BUNDLE RANDOM DYNAMICAL SYSTEMS

Stochastics and Dynamics ◽

10.1142/s0219493709002634 ◽

2009 ◽

Vol 09 (02) ◽

pp. 205-215 ◽

Cited By ~ 2

Author(s):

XIANFENG MA ◽

ERCAI CHEN

Keyword(s):

Dynamical Systems ◽

Variational Principle ◽

Multifractal Analysis ◽

Weak Condition ◽

Random Dynamical Systems ◽

Topological Pressure ◽

Set Up

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

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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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Topological entropy and partially hyperbolic diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385707000405 ◽

2008 ◽

Vol 28 (3) ◽

pp. 843-862 ◽

Cited By ~ 12

Author(s):

YONGXIA HUA ◽

RADU SAGHIN ◽

ZHIHONG XIA

Keyword(s):

Topological Entropy ◽

Topological Invariant ◽

Compact Manifolds ◽

Two Dimensional ◽

One Dimensional ◽

Hyperbolic Diffeomorphisms ◽

Lower Semi ◽

Partially Hyperbolic ◽

Partially Hyperbolic Diffeomorphisms

AbstractWe consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all $C^{\infty }$ diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

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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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