scholarly journals On the existence of cocycle-invariant Borel probability measures

2019 ◽  
Vol 40 (11) ◽  
pp. 3150-3168
Author(s):  
BENJAMIN D. MILLER
Keyword(s):  
Probability Measure ◽  
Borel Probability Measure ◽  
Natural Generalization ◽  
Probability Measures ◽  
Borel Probability

We show that a natural generalization of compressibility is the sole obstruction to the existence of a cocycle-invariant Borel probability measure.

scholarly journals Free minimal actions of countable groups with invariant probability measures

2020 ◽  
pp. 1-21
Author(s):  
GÁBOR ELEK
Keyword(s):  
Probability Measure ◽  
Borel Probability Measure ◽  
Countable Group ◽  
Cantor Set ◽  
Probability Measures ◽  
Continuous Action ◽  
Borel Probability

We prove that for any countable group $\unicode[STIX]{x1D6E4}$ , there exists a free minimal continuous action $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{C}}$ on the Cantor set admitting an invariant Borel probability measure.

A minimal distal map on the torus with sub-exponential measure complexity

2018 ◽  
Vol 40 (4) ◽  
pp. 953-974 ◽  
Author(s):  
WEN HUANG ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Borel Probability Measure ◽  
Skew Product ◽  
Topological Dynamical System ◽  
Borel Probability ◽  
Product Map

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on 𝕋

2009 ◽  
Vol 30 (1) ◽  
pp. 151-157 ◽  
Author(s):  
MANFRED EINSIEDLER ◽  
ALEXANDER FISH
Keyword(s):  
Probability Measure ◽  
Polynomial Growth ◽  
Borel Probability Measure ◽  
Finite Support ◽  
Multiplicative Semigroup ◽  
Logarithmic Density ◽  
Circle Group ◽  
Borel Probability

AbstractWe prove that if a Borel probability measure on the circle group is invariant under the action of a ‘large’ multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then the measure is either Lebesgue or has finite support.

Slices of Hewitt–Stromberg measures and co-dimensions formula

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Bilel Selmi
Keyword(s):  
Probability Measure ◽  
Borel Probability Measure ◽  
Compactly Supported ◽  
Borel Probability ◽  
The Relationship

Abstract This paper studies the behavior of the lower and upper multifractal Hewitt–Stromberg functions under slices onto ( n - m ) {(n-m)} -dimensional subspaces. More precisely, we discuss the relationship between the multifractal Hewitt–Stromberg functions of a compactly supported Borel probability measure and those of slices or sections of the measure. In addition, we prove that if μ has a finite m-energy and q lies in a certain somewhat restricted interval, then these functions satisfy the expected adding of co-dimensions formula.

scholarly journals Moments of the weighted Cantor measures

2019 ◽  
Vol 52 (1) ◽  
pp. 256-273
Author(s):  
Steven N. Harding ◽  
Alexander W. N. Riasanovsky
Keyword(s):  
Probability Measure ◽  
Generating Function ◽  
Exponential Decay ◽  
Legendre Polynomials ◽  
Borel Probability Measure ◽  
Moment Generating Function ◽  
Unit Interval ◽  
Uniform Error ◽  
Natural Case ◽  
Borel Probability

AbstractBased on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying {\mu ^\alpha }(E) = \sum\nolimits_{n = 0}^{N - 1} {{\alpha _n}{\mu ^\alpha }(\varphi _n^{ - 1}(E))} where ϕn : x ↦ (x + n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L_{{\mu ^\alpha }}^2 [0, 1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xm dμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first m moments within uniform error ε in O((log log(1/ε)) · m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2, 1/2].

Multifractal formalism for the inverse of random weak Gibbs measures

2019 ◽  
Vol 20 (04) ◽  
pp. 2050024
Author(s):  
Zhihui Yuan
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Gibbs Measures ◽  
Borel Probability Measure ◽  
Cantor Set ◽  
Multifractal Formalism ◽  
The Real ◽  
Random Dynamics ◽  
Borel Probability

Any Borel probability measure supported on a Cantor set included in [Formula: see text] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.

scholarly journals Dimension, entropy and Lyapunov exponents

1982 ◽  
Vol 2 (1) ◽  
pp. 109-124 ◽  
Author(s):  
Lai-Sang Young
Keyword(s):  
Hausdorff Dimension ◽  
Probability Measure ◽  
Lyapunov Exponents ◽  
Borel Probability Measure ◽  
Fractional Dimension ◽  
Present Context ◽  
Rényi Dimension ◽  
Borel Probability

AbstractWe consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.

scholarly journals A Note on Variational Principle of Subsets for Nonautonomous Dynamical Systems

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.

scholarly journals Haar Null Sets and the Consistent Reflection of Non-meagreness

2014 ◽  
Vol 66 (2) ◽  
pp. 303-322 ◽  
Author(s):  
Márton Elekes ◽  
Juris Steprāns
Keyword(s):  
Probability Measure ◽  
Continuum Hypothesis ◽  
Borel Probability Measure ◽  
Baire Category ◽  
Borel Set ◽  
Polish Group ◽  
Borel Probability ◽  
Haar Null Sets ◽  
Definition Of ◽  
Null Set

AbstractA subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

Study on Strong Sensitivity of Systems Satisfying the Large Deviations Theorem

2021 ◽  
Vol 31 (10) ◽  
pp. 2150151
Author(s):  
Risong Li ◽  
Tianxiu Lu ◽  
Xiaofang Yang ◽  
Yongxi Jiang
Keyword(s):  
Probability Measure ◽  
Characteristic Function ◽  
Large Deviations ◽  
Open Subset ◽  
Metric Space ◽  
Borel Probability Measure ◽  
Compact Metric Space ◽  
Full Support ◽  
Upper Density ◽  
Borel Probability

Let [Formula: see text] be a nontrivial compact metric space with metric [Formula: see text] and [Formula: see text] be a continuous self-map, [Formula: see text] be the sigma-algebra of Borel subsets of [Formula: see text], and [Formula: see text] be a Borel probability measure on [Formula: see text] with [Formula: see text] for any open subset [Formula: see text] of [Formula: see text]. This paper proves the following results : (1) If the pair [Formula: see text] has the property that for any [Formula: see text], there is [Formula: see text] such that [Formula: see text] for any open subset [Formula: see text] of [Formula: see text] and all [Formula: see text] sufficiently large (where [Formula: see text] is the characteristic function of the set [Formula: see text]), then the following hold : (a) The map [Formula: see text] is topologically ergodic. (b) The upper density [Formula: see text] of [Formula: see text] is positive for any open subset [Formula: see text] of [Formula: see text], where [Formula: see text]. (c) There is a [Formula: see text]-invariant Borel probability measure [Formula: see text] having full support (i.e. [Formula: see text]). (d) Sensitivity of the map [Formula: see text] implies positive lower density sensitivity, hence ergodical sensitivity. (2) If [Formula: see text] for any two nonempty open subsets [Formula: see text], then there exists [Formula: see text] satisfying [Formula: see text] for any nonempty open subset [Formula: see text], where [Formula: see text] there exist [Formula: see text] with [Formula: see text].

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