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.

LIAO STYLE NUMBERS OF DIFFERENTIAL SYSTEMS

2004 ◽  
Vol 06 (02) ◽  
pp. 279-299 ◽  
Author(s):  
XIONGPING DAI
Keyword(s):  
Differential System ◽  
Compact Riemannian Manifold ◽  
Ergodic Measure ◽  
Ergodic System ◽  
View Point ◽  
Differential Systems ◽  
Linearly Independent ◽  
Characterization Theorems ◽  
Positive Integers ◽  
Style Number

For any C1 differential system S on a compact Riemannian manifold M of dimension d with d≥2, this paper studies the Liao style numbers, κ(S) (or respectively, κ*(S)) of S from the view-point of ergodic theory. Here κ(S) (κ*(S)) is the largest number of moving vectors (or respectively, conjugate-) of the differential system S that are mean linearly independent. For any ergodic measure ν of S, two positive integers κ*(ν) and κ(ν), called the reduced and non-reduced style number of ν respectively, are introduced. The connection between the style numbers of the system (M,S) and ones of the ergodic system (M,S; ν) are discovered by the variational principle of style number proved in the paper. Several characterization theorems with respect to the style numbers κ*(S), κ*(ν), κ(S) and κ(ν) are presented respectively.

scholarly journals Entropy of systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1118-1126
Author(s):  
RADU B. MUNTEANU
Keyword(s):  
Positive Integer ◽  
Probability Space ◽  
Bernoulli Shift ◽  
Ergodic Measure ◽  
Measure Preserving Transformation ◽  
Zero Entropy ◽  
Standard Probability

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is $\text{AT}(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not $\text{AT}(n)$. We also give an example of a transformation which has zero entropy but does not have property $\text{AT}(n)$ for any integer $n\geq 1$.

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.

scholarly journals Weak containment of measure-preserving group actions

2019 ◽  
Vol 40 (10) ◽  
pp. 2681-2733 ◽  
Author(s):  
PETER J. BURTON ◽  
ALEXANDER S. KECHRIS
Keyword(s):  
Global Structure ◽  
Equivalence Classes ◽  
The Other ◽  
Unitary Representations ◽  
Weak Equivalence ◽  
Interesting Fact ◽  
Weak Containment ◽  
Probability Spaces ◽  
Measure Preserving Actions ◽  
Standard Probability

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

scholarly journals Groups with infinite FC-center have the Schmidt property

2021 ◽  
pp. 1-46
Author(s):  
YOSHIKATA KIDA ◽  
ROBIN TUCKER-DROB
Keyword(s):  
Probability Space ◽  
Amenable Group ◽  
Ergodic Measure ◽  
Countable Group ◽  
Full Group ◽  
Orbit Equivalence ◽  
Central Sequence ◽  
Property T ◽  
Finite Conjugacy ◽  
Standard Probability

Abstract We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property.

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.

Probability in quantum computation and quantum computational logics: a survey

2014 ◽  
Vol 24 (3) ◽  
Author(s):  
MARIA LUISA DALLA CHIARA ◽  
ROBERTO GIUNTINI ◽  
GIUSEPPE SERGIOLI
Keyword(s):  
Quantum Computation ◽  
Logic Gates ◽  
Point Of View ◽  
Quantum Processes ◽  
Classical Probability ◽  
Quantum Logic Gates ◽  
Quantum Computational Logics ◽  
Probability Spaces ◽  
Formal Point ◽  
Standard Probability

Quantum computation and quantum computational logics give rise to some non-standard probability spaces that are interesting from a formal point of view. In this framework, events represent quantum pieces of information (qubits, quregisters, mixtures of quregisters), while operations on events are identified with quantum logic gates (which correspond to dynamic reversible quantum processes). We investigate the notion of Shi–Aharonov quantum computational algebra. This structure plays the role for quantum computation that is played by σ-complete Boolean algebras in classical probability theory.

Sign in / Sign up

Export Citation Format

Share Document