Approximation of Bernoulli measures for non-uniformly hyperbolic systems

2018 ◽  
Vol 40 (1) ◽  
pp. 233-247
Author(s):  
GANG LIAO ◽  
WENXIANG SUN ◽  
EDSON VARGAS ◽  
SHIROU WANG
Keyword(s):  
Invariant Measure ◽  
Bernoulli Shift ◽  
Hyperbolic Systems ◽  
Weak Mixing ◽  
Bernoulli Measure ◽  
Uniformly Hyperbolic Systems ◽  
Bernoulli Measures

An invariant measure is called a Bernoulli measure if the corresponding dynamics is isomorphic to a Bernoulli shift. We prove that for$C^{1+\unicode[STIX]{x1D6FC}}$diffeomorphisms any weak mixing hyperbolic measure could be approximated by Bernoulli measures. This also holds true for$C^{1}$diffeomorphisms preserving a weak mixing hyperbolic measure with respect to which the Oseledets decomposition is dominated.

scholarly journals Quasi-shadowing for partially hyperbolic diffeomorphisms

2014 ◽  
Vol 35 (2) ◽  
pp. 412-430 ◽  
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.

scholarly journals Symbolic dynamics for non-uniformly hyperbolic systems

2020 ◽  
pp. 1-68
Author(s):  
YURI LIMA
Keyword(s):  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Markov Partition ◽  
Equilibrium Measures ◽  
Markov Partitions ◽  
Recent Advances ◽  
Closed Orbits ◽  
Symbolic Extension ◽  
Uniformly Hyperbolic Systems

Abstract This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

scholarly journals Extremal dichotomy for uniformly hyperbolic systems

2015 ◽  
Vol 30 (4) ◽  
pp. 383-403 ◽  
Author(s):  
Maria Carvalho ◽  
Ana Cristina Moreira Freitas ◽  
Jorge Milhazes Freitas ◽  
Mark Holland ◽  
Matthew Nicol
Keyword(s):  
Hyperbolic Systems ◽  
Uniformly Hyperbolic Systems

scholarly journals Equilibrium stability for non-uniformly hyperbolic systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2619-2642 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
VANESSA RAMOS ◽  
JAQUELINE SIQUEIRA
Keyword(s):  
Hyperbolic Systems ◽  
Equilibrium States ◽  
Topological Pressure ◽  
Equilibrium Stability ◽  
Hölder Continuous ◽  
Skew Products ◽  
Uniformly Hyperbolic Systems ◽  
Wide Family

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally, we show that these equilibrium states vary continuously in the $\text{weak}^{\ast }$ topology within such systems.

Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities

1992 ◽  
Vol 12 (3) ◽  
pp. 487-508 ◽  
Author(s):  
Tyll Krüger ◽  
Serge Troubetzkoy
Keyword(s):  
Large Class ◽  
Hyperbolic Systems ◽  
Markov Partitions ◽  
Uniformly Hyperbolic Systems ◽  
Dispersing Billiards

AbstractWe show the existence of countable Markov partitions for a large class of non-uniformly hyperbolic systems with singularities including dispersing billiards in any dimension.

scholarly journals POLYNOMIAL DECAY OF CORRELATIONS IN THE GENERALIZED BAKER'S TRANSFORMATION

2013 ◽  
Vol 23 (08) ◽  
pp. 1350130 ◽  
Author(s):  
CHRISTOPHER BOSE ◽  
RUA MURRAY
Keyword(s):  
Hyperbolic Systems ◽  
Decay Of Correlations ◽  
Polynomial Decay ◽  
Mixing Rate ◽  
New Techniques ◽  
Correlation Decay ◽  
Simple Geometry ◽  
Uniformly Hyperbolic Systems ◽  
Mixing Rates ◽  
Area Preserving

We introduce a family of area preserving generalized baker's transformations acting on the unit square and having sharp polynomial rates of mixing for Hölder data. The construction is geometric, relying on the graph of a single variable "cut function". Each baker's map B is nonuniformly hyperbolic and while the exact mixing rate depends on B, all polynomial rates can be attained. The analysis of mixing rates depends on building a suitable Young tower for an expanding factor. The mechanisms leading to a slow rate of correlation decay are especially transparent in our examples due to the simple geometry in the construction. For this reason, we propose this class of maps as an excellent testing ground for new techniques for the analysis of decay of correlations in non-uniformly hyperbolic systems. Finally, some of our examples can be seen to be extensions of certain 1D non-uniformly expanding maps that have appeared in the literature over the last twenty years, thereby providing a unified treatment of these interesting and well-studied examples.

scholarly journals The Measure-Theoretic Entropy and Topological Entropy of Actions overℤm

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Chih-Hung Chang ◽  
Yu-Wen Chen
Keyword(s):  
Invariant Measure ◽  
Cellular Automata ◽  
Topological Entropy ◽  
Finite Ring ◽  
One Dimensional ◽  
Maximal Measure ◽  
Bernoulli Measure ◽  
Markov Measures ◽  
Infinite Sequences

This paper studies the quantitative behavior of a class of one-dimensional cellular automata, named weakly permutive cellular automata, acting on the space of all doubly infinite sequences with values in a finite ringℤm,m≥2. We calculate the measure-theoretic entropy and the topological entropy of weakly permutive cellular automata with respect to any invariant measure on the spaceℤmℤ. As an application, it is shown that the uniform Bernoulli measure is the unique maximal measure for linear cellular automata among the Markov measures.

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