scholarly journals Pure strictly uniform models of non-ergodic measure automorphisms

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tomasz Downarowicz ◽  
Benjamin Weiss
Keyword(s):  
Compact Space ◽  
Ergodic Measure ◽  
Classical Theorem ◽  
Upper Semicontinuous ◽  
System A ◽  
Ergodic Measures ◽  
Uniform Model

<p style='text-indent:20px;'>The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model.</p>

scholarly journals Measure-theoretic chaos

2012 ◽  
Vol 34 (1) ◽  
pp. 110-131 ◽  
Author(s):  
TOMASZ DOWNAROWICZ ◽  
YVES LACROIX
Keyword(s):  
Ergodic Measure ◽  
Uniform Measure ◽  
Ergodic System ◽  
Topological System ◽  
Ergodic Measures ◽  
Topological Chaos ◽  
Probability Spaces ◽  
Entropy Zero ◽  
Standard Probability

AbstractWe define new isomorphism invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic$^+$ chaos. These notions are analogs of the topological chaos DC2 and its slightly stronger version (which we denote by $\text {DC}1\frac 12$). We prove that: (1) if a topological system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is topologically DC2 $(\text {DC}1\frac 12)$ chaotic; (2) every ergodic system with positive Kolmogorov–Sinai entropy is measure-theoretically$^+$ chaotic (even in a slightly stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, that is, of a system of entropy zero with uniform measure-theoretic$^+$chaos.

scholarly journals Extensions of best approximation and coincidence theorems

1997 ◽  
Vol 20 (4) ◽  
pp. 689-698 ◽  
Author(s):  
Sehie Park
Keyword(s):  
Continuous Function ◽  
Vector Space ◽  
Topological Vector Space ◽  
Compact Space ◽  
Best Approximation ◽  
Upper Semicontinuous ◽  
Coincidence Theorems ◽  
Continuous Seminorm ◽  
Upper Semicontinuous Multifunction

LetXbe a Hausdorff compact space,Ea topological vector space on whichE*separates points,F:X→2Ean upper semicontinuous multifunction with compact acyclic values, andg:X→Ea continuous function such thatg(X)is convex andg−1(y)is acyclic for eachy∈g(X). Then either (1) there exists anx0∈Xsuch thatgx0∈Fx0or (2) there exist an(x0,z0)on the graph ofFand a continuous seminormponEsuch that0<p(gx0−z0)≤p(y−z0)         for all         y∈g(X). A generalization of this result and its application to coincidence theorems are obtained. Our aim in this paper is to unify and improve almost fifty known theorems of others.

scholarly journals Relative equilibrium states and class degree

2017 ◽  
Vol 39 (4) ◽  
pp. 865-888
Author(s):  
MAHSA ALLAHBAKHSHI ◽  
JOHN ANTONIOLI ◽  
JISANG YOO
Keyword(s):  
Finite Type ◽  
Upper Bound ◽  
Relative Equilibrium ◽  
Ergodic Measure ◽  
Equilibrium States ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Ergodic Measures ◽  
Factor Code ◽  
Special Case

Given a factor code $\unicode[STIX]{x1D70B}$ from a shift of finite type $X$ onto a sofic shift $Y$, an ergodic measure $\unicode[STIX]{x1D708}$ on $Y$, and a function $V$ on $X$ with sufficient regularity, we prove an invariant upper bound on the number of ergodic measures on $X$ which project to $\unicode[STIX]{x1D708}$ and maximize the measure pressure $h(\unicode[STIX]{x1D707})+\int V\,d\unicode[STIX]{x1D707}$ among all measures in the fiber $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D708})$. If $\unicode[STIX]{x1D708}$ is fully supported, this bound is the class degree of $\unicode[STIX]{x1D70B}$. This generalizes a previous result for the special case of $V=0$ and thus settles a conjecture raised by Allahbakhshi and Quas.

An extension of the ergodic closing lemma

2009 ◽  
Vol 30 (3) ◽  
pp. 773-808 ◽  
Author(s):  
SHUHEI HAYASHI
Keyword(s):  
Lyapunov Exponents ◽  
Small Perturbation ◽  
Approximation Theorem ◽  
Ergodic Measure ◽  
Homoclinic Tangency ◽  
Hyperbolic Structure ◽  
Extended Version ◽  
Closing Lemma ◽  
One Dimensional ◽  
Ergodic Measures

AbstractAn extended version of the ergodic closing lemma of Mañé is proved. As an application, we show that, C1 densely in the complement of the closure of Morse–Smale diffeomorphisms and those with a homoclinic tangency, there exists a weakly hyperbolic structure (dominated splittings with average hyperbolicity at almost every point on hyperbolic parts, and one-dimensional center direction when zero Lyapunov exponents are involved) over the supports of all non-atomic ergodic measures. As another application, we prove an approximation theorem, which claims that approximating the Lyapunov exponents of any non-atomic ergodic measure by those of an atomic ergodic measure by a C1 small perturbation is possible.

scholarly journals NON-HYPERBOLIC ERGODIC MEASURES AND HORSESHOES IN PARTIALLY HYPERBOLIC HOMOCLINIC CLASSES

2019 ◽  
Vol 19 (5) ◽  
pp. 1765-1792 ◽  
Author(s):  
Dawei Yang ◽  
Jinhua Zhang
Keyword(s):  
Topological Entropy ◽  
Classical Result ◽  
Ergodic Measure ◽  
Rich Family ◽  
Ergodic Measures ◽  
Partially Hyperbolic ◽  
Hyperbolic Sets ◽  
Homoclinic Classes

We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak$\ast$-topology and in entropy. For hyperbolic ergodic measures, it is a classical result of A. Katok. The novelty here is to deal with non-hyperbolic ergodic measures. As a consequence, we obtain the continuity of topological entropy.

Generalisations of the Euler adic

2010 ◽  
Vol 150 (2) ◽  
pp. 241-256 ◽  
Author(s):  
GABRIEL STRASSER
Keyword(s):  
Closed Form ◽  
Ergodic Measure ◽  
Eulerian Numbers ◽  
Ergodic Measures ◽  
Dynamical Properties

AbstractWe consider generalisations of the so-called Euler adic and investigate dynamical properties like ergodicity and total ergodicity. We prove the existence of a unique fully-supported ergodic measure for these generalisations. We also investigate the structure of non-fully-supported ergodic measures and in addition show that each of these measures (fully- and non-fully-supported) is also totally ergodic. In order to determine these dynamical properties we find closed-form expressions for the generalised Eulerian numbers. Additionally we extend a result given by Frick and Petersen to a wider class of adic transformations.

Equivalence singularity dichotomies for a class of ergodic measures

1977 ◽  
Vol 81 (2) ◽  
pp. 249-252 ◽  
Author(s):  
Marek Kanter
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Borel Subset ◽  
Ergodic Measure ◽  
Unit Mass ◽  
Probability Measures ◽  
Borel Sets ◽  
Ergodic Measures ◽  
Countable Subgroup ◽  
Two Measures

Let µ be a probability measure on the Borel subsets of R∞. If D is a countable subgroup of R∞ we say that µ is D-ergodic if (1) for any D invariant Borel subset A of R we have µ(A) = 0 or 1 and (2) if µ*δx ≈ µ for all x ∈ D (where δx stands for unit mass at x while the equivalence relation ≈ signifies that the two measures have the same null sets.) We say that x is an admissible translate for µ if µ*δx ≈ µ. We say that µ is D-smooth if sx is an admissible translate for µ for all x ∈ D and all s ∈ R. We say that µ is a smooth ergodic measure if µ is D-ergodic and D-smooth for some countable subgroup D as above. In this paper we show that any two smooth ergodic probability measures µl, µ2 are either equivalent or singular (where the latter means that there exist disjoint Borel sets Al, A2 ⊂ R∞ such that µi(Ai) = 1 and is signified by µ1 ┴ µ2). It is important to note that the countable subgroup D1 associated with µl need not be the same as the subgroup D2 associated with µ2.

scholarly journals On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

2018 ◽  
Vol 39 (11) ◽  
pp. 2932-2967 ◽  
Author(s):  
CHRISTIAN BONATTI ◽  
JINHUA ZHANG
Keyword(s):  
Lyapunov Exponent ◽  
Periodic Orbits ◽  
Dense Subset ◽  
Ergodic Measure ◽  
Compact Set ◽  
List Type ◽  
Hyperbolic Diffeomorphisms ◽  
Shadowing Lemma ◽  
Single Orbit ◽  
Ergodic Measures

Gorodetski et al. [Nonremovability of zero Lyapunov exponents. Funktsional. Anal. i Prilozhen. 39(1) (2005), 27–38 (in Russian); Engl. Transl. Funct. Anal. Appl. 39(1) (2005), 21–30] and Bochi et al. [Robust criterion for the existence of nonhyperbolic ergodic measures. Comm. Math. Phys. 344(3) (2016), 751–795] propose two very different ways for building non-hyperbolic measures, Gorodetski et al. (2005) building such a measure as the limit of periodic measures and Bochi et al. (2016) as the $\unicode[STIX]{x1D714}$ -limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in Gorodetski et al. (2005) has been used in a generic setting in Bonatti et al. [Non-hyperbolic ergodic measures with large support. Nonlinearity 23(3) (2010), 687–705] and Díaz and Gorodetski [Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes. Ergod. Th. &amp; Dynam. Sys. 29(5) (2009), 1479–1513], as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in Bochi et al. (2016) are accumulated by periodic measures. In this paper we use a shadowing lemma from Gan [A generalized shadowing lemma. Discrete Contin. Dyn. Syst. 8(3) (2002), 527–632]: ∙for getting the periodic orbits in Gorodetski et al. (2005) without perturbing the dynamics;∙for recovering the compact set in Bochi et al. (2016) with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits. As a consequence, we prove that there exists an open and dense subset ${\mathcal{U}}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in {\mathcal{U}}$ , there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset ${\mathcal{V}}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in {\mathcal{V}}$ , there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures.

scholarly journals Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

2017 ◽  
Vol 39 (7) ◽  
pp. 1805-1823 ◽  
Author(s):  
CHENG CHENG ◽  
SYLVAIN CROVISIER ◽  
SHAOBO GAN ◽  
XIAODONG WANG ◽  
DAWEI YANG
Keyword(s):  
Ergodic Measure ◽  
Homoclinic Class ◽  
Ergodic Measures ◽  
Homoclinic Classes

We prove that, for $C^{1}$-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-trivial non-hyperbolic ergodic measure supported on it. This proves a conjecture by Díaz and Gorodetski.

scholarly journals Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

2012 ◽  
Vol 33 (3) ◽  
pp. 831-850 ◽  
Author(s):  
YONGLUO CAO ◽  
HUYI HU ◽  
YUN ZHAO
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Topological Pressure ◽  
Compact Sets ◽  
Equivalent Definition ◽  
Ergodic Measures ◽  
Nonadditive Measure ◽  
Definition Of ◽  
Conformal Repeller

AbstractWithout any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact sets, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the superadditive measure-theoretic pressure which has similar formalism to the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Hausdorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

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