The Convergence of Borel Probability Measures and Its Applications to Topological Dynamics

Existence of expansivity for group action [Formula: see text] depends on algebraic properties of [Formula: see text] and the topology of [Formula: see text]. We give an expansive action of a solvable group on [Formula: see text] while there is no expansive action of a solvable group on a dendrite [Formula: see text]. We prove that a continuous action [Formula: see text] on a compact metric space [Formula: see text] is expansive if and only if there exists an open cover [Formula: see text] such that for any other open cover [Formula: see text], [Formula: see text] for some finite set [Formula: see text]. In this paper, we introduce the notion of topological expansivity of a group action [Formula: see text] on a [Formula: see text]-paracompact space [Formula: see text]. If a [Formula: see text]-paracompact space [Formula: see text] admits topologically expansive action, then [Formula: see text] is Hausdorff space. We also show that a continuous action [Formula: see text] of a finitely generated group [Formula: see text] on a compact Hausdorff uniform space [Formula: see text] is expansive with an expansive neighborhood [Formula: see text] if and only if for every [Formula: see text] there is an entourage [Formula: see text] such that for every two [Formula: see text]-pseudo orbit [Formula: see text] if [Formula: see text] for all [Formula: see text], then [Formula: see text] for all [Formula: see text]. Finally, we introduce measure [Formula: see text]-expansive actions on a uniform space. The set of all [Formula: see text]-expansive measures with common expansive neighborhood for a group action [Formula: see text] is a convex, closed and [Formula: see text]-invariant subset of the set of all Borel probability measures on [Formula: see text]. Also, we show that a group action [Formula: see text] is expansive if all Borel probability measures are [Formula: see text]-expansive or all Dirac measures [Formula: see text], [Formula: see text], have a common expansive neighborhood.

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On the set of expansive measures

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500869 ◽

2018 ◽

Vol 20 (07) ◽

pp. 1750086 ◽

Cited By ~ 1

Author(s):

Keonhee Lee ◽

C. A. Morales ◽

Bomi Shin

Keyword(s):

Metric Space ◽

Weak Topology ◽

Hausdorff Metric ◽

Probability Measures ◽

Compact Metric Space ◽

Whole Space ◽

Isolated Points ◽

Borel Probability ◽

Heteroclinic Points

We prove that the set of expansive measures of a homeomorphism of a compact metric space is a [Formula: see text] subset of the space of Borel probability measures equipped with the weak* topology. Next that every expansive measure of a homeomorphism of a compact metric space can be weak* approximated by expansive measures with invariant support. In addition, if the expansive measures of a homeomorphism of a compact metric space are dense in the space of Borel probability measures, then there is an expansive measure whose support is both invariant and close to the whole space with respect to the Hausdorff metric. Henceforth, if the expansive measures are dense in the space of Borel probability measures, the set of heteroclinic points has no interior and the space has no isolated points.

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A Note on Variational Principle of Subsets for Nonautonomous Dynamical Systems

10.47260/tma/1131 ◽

2021 ◽

pp. 1-16

Author(s):

Jiao Yang

Keyword(s):

Dynamical Systems ◽

Variational Principle ◽

Variational Principles ◽

Borel Probability Measure ◽

Jel Classification ◽

Probability Measures ◽

Nonautonomous Dynamical Systems ◽

Nonautonomous Case ◽

Borel Probability ◽

Discrete Type

Abstract In this paper, we introduce measure-theoretic for Borel probability measures to characterize upper and lower Katok measure-theoretic entropies in discrete type and the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we establish new variational principles for Bowen topological entropy for nonautonomous dynamical systems. JEL classification numbers: 37A35. Keywords: Nonautonomous, Measure-theoretical entropies, Variational principles.

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Connecting ergodicity and dimension in dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000568x ◽

1990 ◽

Vol 10 (3) ◽

pp. 451-462 ◽

Cited By ~ 23

Author(s):

C. D. Cutler

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Metric Spaces ◽

Sufficient Conditions ◽

Probability Measures ◽

Pointwise Dimension ◽

Continuous Maps ◽

Borel Probability ◽

The Relationship

AbstractIn this paper we make precise the relationship between local or pointwise dimension and the dimension structure of Borel probability measures on metric spaces. Sufficient conditions for exact-dimensionality of the stationary ergodic distributions associated with a dynamical system are obtained. A counterexample is provided to show that ergodicity alone is not sufficient to guarantee exactdimensionality even in the case of continuous maps or flows.

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THE FREQUENCY SPACE OF A FREE GROUP

International Journal of Algebra and Computation ◽

10.1142/s0218196705002700 ◽

2005 ◽

Vol 15 (05n06) ◽

pp. 939-969 ◽

Cited By ~ 18

Author(s):

ILYA KAPOVICH

Keyword(s):

Conjugacy Class ◽

Free Group ◽

Outer Automorphism ◽

Probability Measures ◽

Natural Action ◽

Frequency Space ◽

Borel Probability

We analyze the structure of the frequency spaceQ(F) of a nonabelian free group F = F(a1,…,ak) consisting of all shift-invariant Borel probability measures on ∂F and construct a natural action of Out(F) on Q(F). In particular we prove that for any outer automorphism ϕ of F the conjugacy distortion spectrum of ϕ, consisting of all numbers ‖ϕ(w)‖/‖w‖, where w is a nontrivial conjugacy class, is the intersection of ℚ and a closed subinterval of ℝ with rational endpoints.

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Time-preserving conjugacies of geodesic flows

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700006581 ◽

1992 ◽

Vol 12 (1) ◽

pp. 67-74 ◽

Cited By ~ 16

Author(s):

Ursula Hamenstädt

Keyword(s):

Sufficient Conditions ◽

Universal Covering ◽

Probability Measures ◽

Geodesic Flows ◽

Ideal Boundary ◽

Negatively Curved ◽

Unit Tangent Bundle ◽

Borel Probability ◽

Necessary And Sufficient ◽

The Ideal

AbstractIn this note we study Borel-probability measures on the unit tangent bundle ofa compact negatively curved manifold M that are invariant under the geodesic flow. We interpret the entropy of such a measure as a Hausdorff dimension with respect to a natural family of distances on the ideal boundary of the universal covering of M. This in term yields necessary and sufficient conditions for the existence of time preserving conjugacies of geodesic flows.

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Invariant Borel probability measures for discrete long-wave-short-wave resonance equations

Applied Mathematics and Computation ◽

10.1016/j.amc.2018.06.059 ◽

2018 ◽

Vol 339 ◽

pp. 853-865

Author(s):

Chengzhi Wang ◽

Gang Xue ◽

Caidi Zhao

Keyword(s):

Short Wave ◽

Probability Measures ◽

Long Wave ◽

Wave Resonance ◽

Borel Probability

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UNIVERSALLY RATIONAL BELIEF HIERARCHIES

International Game Theory Review ◽

10.1142/s0219198914400039 ◽

2014 ◽

Vol 16 (01) ◽

pp. 1440003

Author(s):

ELIAS TSAKAS

Keyword(s):

Probability Measure ◽

Rational Belief ◽

Probability Measures ◽

Rational Beliefs ◽

Natural Restriction ◽

Mathematical Economics ◽

Type Space ◽

Space Model ◽

Underlying Space ◽

Borel Probability

In a recent paper, Tsakas [2013 Rational belief hierarchies, Journal of Mathematical Economics, Maastricht University] introduced the notion of rational beliefs. These are Borel probability measures that assign a rational probability to every Borel event. Then, he constructed the corresponding Harsanyi type space model that represents the rational belief hierarchies. As he showed, there are rational types that are associated with a non-rational probability measure over the product of the underlying space of uncertainty and the opponent's types. In this paper, we define the universally rational belief hierarchies, as those that do not exhibit this property. Then, we characterize them in terms of a natural restriction imposed directly on the belief hierarchies.

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A MATRIX METHOD FOR APPROXIMATING FRACTAL MEASURES

International Journal of Bifurcation and Chaos ◽

10.1142/s021812749200015x ◽

1992 ◽

Vol 02 (01) ◽

pp. 167-175 ◽

Cited By ~ 2

Author(s):

A. BOYARSKY ◽

Y.S. LOU

Keyword(s):

Iterated Function System ◽

Markov Operator ◽

Measurable Subset ◽

Function System ◽

Probability Measures ◽

Bounded Subset ◽

Step Functions ◽

Iterated Function ◽

Fractal Measures ◽

Borel Probability

Let X be a bounded subset of Rn and let A be the Lebesgue measure on X. Let {X:τ1,…, τN} be an iterated function system (IFS) with attractor S. We associate probabilities p1,…, pN with τ1,…, τN, respectively. Let M(X) be the space of Borel probability measures on X, and let M: M(X)→M(X) be the Markov operator associated with the IFS and its probabilities given by: [Formula: see text] where A is a measurable subset of X. Then there exists a unique µ∈M (A) such that Mµ=µ; µ is referred to as the measure invariant under the iterated function system with the associated probabilities. The support of μ is the attractor S. We prove the existence of a sequence of step functions {fi}, which are the eigenvectors of matrices {Mi}, such that the measures {fidλ} converge weakly to µ. An algorithm is presented for the construction of Mi and an example is given.

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