Bounded Geometry in the Supports of Ergodic Invariant Probability Measures

1998 ◽  
Vol 08 (10) ◽  
pp. 1957-1973 ◽  
Author(s):  
Jun Hu
Keyword(s):  
Hausdorff Dimension ◽  
Periodic Orbit ◽  
Probability Measure ◽  
Lebesgue Measure ◽  
Turning Points ◽  
Invariant Probability Measure ◽  
Cantor Set ◽  
Probability Measures ◽  
Bounded Geometry

In this paper we put some techniques and methods of [Misurewicz, 1979; Hu & Sullivan, 1997; Hu & Tresser, 1998; Blokh & Lyubich, 1990; Martens et al., 1992] together to show that if a map f, from an interval I into itself with finitely many turning points, satisfies a new smooth regularity in [Hu & Sullivan, 1997] and is on the boundary of chaos, and if μ is an ergodic f-invariant probability measure on I which is not concentrated on a periodic orbit of f, then the support K of μ is a Cantor set of bounded geometry, and hence has Lebesgue measure 0 and Hausdorff dimension strictly between 0 and 1. We also include some natural examples which satisfy this new smooth regularity rather than the traditional ones.

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 Closability of quadratic forms associated to invariant probability measures of SPDEs

2017 ◽  
Vol 20 (04) ◽  
pp. 1750023
Author(s):  
Michael Röckner ◽  
Feng-Yu Wang
Keyword(s):  
Probability Measure ◽  
Hamiltonian Systems ◽  
General Result ◽  
Quadratic Forms ◽  
Invariant Probability Measure ◽  
Markov Operator ◽  
Probability Measures ◽  
Infinite Dimensional ◽  
Stochastic Hamiltonian Systems ◽  
Integration By Parts Formula

By using the integration by parts formula of a Markov operator, the closability of quadratic forms associated to the corresponding invariant probability measure is proved. The general result is applied to the study of semilinear SPDEs, infinite-dimensional stochastic Hamiltonian systems, and semilinear SPDEs with delay.

On the Uniqueness Property for Products of Symmetric Invariant Probability Measures

2002 ◽  
Vol 9 (1) ◽  
pp. 75-82
Author(s):  
A. Kharazishvili
Keyword(s):  
Probability Measure ◽  
Invariant Probability Measure ◽  
Probability Measures ◽  
Strong Uniqueness ◽  
Uniqueness Property

Abstract Two symmetric invariant probability measures μ 1 and μ 2 are constructed such that each of them possesses the strong uniqueness property but their product μ 1 × μ 2 turns out to be a symmetric invariant probability measure without the uniqueness property.

scholarly journals Multifractality and intermittency in the solar wind

2007 ◽  
Vol 14 (6) ◽  
pp. 695-700 ◽  
Author(s):  
W. M. Macek
Keyword(s):  
Solar Wind ◽  
Probability Measure ◽  
Complex Dynamics ◽  
Invariant Probability Measure ◽  
Multifractal Spectrum ◽  
Cantor Set ◽  
Intermittent Turbulence ◽  
Plasma Parameters ◽  
Uniform Compression ◽  
Two Parameters

Abstract. Within the complex dynamics of the solar wind's fluctuating plasma parameters, there is a detectable, hidden order described by a chaotic strange attractor which has a multifractal structure. The multifractal spectrum has been investigated using Voyager (magnetic field) data in the outer heliosphere and using Helios (plasma) data in the inner heliosphere. We have also analyzed the spectrum for the solar wind attractor. The spectrum is found to be consistent with that for the multifractal measure of the self-similar one-scale weighted Cantor set with two parameters describing uniform compression and natural invariant probability measure of the attractor of the system. In order to further quantify the multifractality, we also consider a generalized weighted Cantor set with two different scales describing nonuniform compression. We investigate the resulting multifractal spectrum depending on two scaling parameters and one probability measure parameter, especially for asymmetric scaling. We hope that this generalized model will also be a useful tool for analysis of intermittent turbulence in space plasmas.

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.

Hausdorff dimension of sets of non-differentiability points of Cantor functions

1995 ◽  
Vol 117 (1) ◽  
pp. 185-191 ◽  
Author(s):  
Richard Darst
Keyword(s):  
Distribution Function ◽  
Hausdorff Dimension ◽  
Probability Measure ◽  
Relative Length ◽  
Cantor Set ◽  
Uniform Probability ◽  
Cantor Function ◽  
Lower Derivative

AbstractEach number a in the segment (0, ½) produces a Cantor set, Ca, by putting b = 1 − 2a and recursively removing segments of relative length b from the centres of the interval [0, 1] and the intervals which are subsequently generated. The distribution function of the uniform probability measure on Ca is a Cantor function, fa. When a = 1/3 = b, Ca is the standard Cantor set, C, and fa is the standard Cantor function, f. The upper derivative of f is infinite at each point of C and the lower derivative of f is infinite at most points of C in the following sense: the Hausdorff dimension of C is ln(2)/ln(3) and the Hausdorff dimension of S = {x ∈ C: the lower derivative of f is finite at x} is [ln(2)/ln(3)]2. The derivative of f is zero off C, the derivative of f is infinite on C — S, and S is the set of non-differentiability points of f. Similar results are established in this paper for all Ca: the Hausdorff dimension of Ca is ln (2)/ln (1/a) and the Hausdorff dimension of Sa is [ln (2)/ln (1/a)]2. Removing k segments of relative length b and leaving k + 1 intervals of relative length a produces a Cantor set of dimension ln(k + l)/ln(1/a); the dimension of the set of non-differentiability points of the corresponding Cantor function is [ln (k + l)/ln (1/a)]2.

scholarly journals Ergodic properties and ergodic decompositions of continuous-time Markov processes

2006 ◽  
Vol 43 (3) ◽  
pp. 767-781 ◽  
Author(s):  
O. L. V. Costa ◽  
F. Dufour
Keyword(s):  
Markov Process ◽  
Probability Measure ◽  
Continuous Time ◽  
Invariant Probability Measure ◽  
Probability Measures ◽  
Transition Kernel ◽  
Occupation Measure ◽  
Ergodic Properties ◽  
Necessary And Sufficient ◽  
Separable Metric Space

In this paper we obtain some ergodic properties and ergodic decompositions of a continuous-time, Borel right Markov process taking values in a locally compact and separable metric space. Initially, we assume that an invariant probability measure (IPM) μ exists for the process and, without making any further assumptions on the transition kernel, obtain some characterization results for the convergence of the expected occupation measure to a limit kernel. Under the same assumption, we present the so-called Yosida decomposition. Next, instead of assuming the existence of an IPM, we assume that the Markov process satisfies a certain condition, named the T'-condition. Under this condition it is shown that the Foster-Lyapunov criterion is necessary and sufficient for the existence of an IPM and that the process admits a Doeblin decomposition. Furthermore, it is shown that in this case the set of ergodic probability measures is countable and that every probability measure for the Markov process is nonsingular with respect to the transition kernel.

scholarly journals Ergodic properties and ergodic decompositions of continuous-time Markov processes

2006 ◽  
Vol 43 (03) ◽  
pp. 767-781
Author(s):  
O. L. V. Costa ◽  
F. Dufour
Keyword(s):  
Markov Process ◽  
Probability Measure ◽  
Continuous Time ◽  
Invariant Probability Measure ◽  
Probability Measures ◽  
Transition Kernel ◽  
Occupation Measure ◽  
Ergodic Properties ◽  
Necessary And Sufficient ◽  
Separable Metric Space

In this paper we obtain some ergodic properties and ergodic decompositions of a continuous-time, Borel right Markov process taking values in a locally compact and separable metric space. Initially, we assume that an invariant probability measure (IPM) μ exists for the process and, without making any further assumptions on the transition kernel, obtain some characterization results for the convergence of the expected occupation measure to a limit kernel. Under the same assumption, we present the so-called Yosida decomposition. Next, instead of assuming the existence of an IPM, we assume that the Markov process satisfies a certain condition, named theT'-condition. Under this condition it is shown that the Foster-Lyapunov criterion is necessary and sufficient for the existence of an IPM and that the process admits a Doeblin decomposition. Furthermore, it is shown that in this case the set of ergodic probability measures is countable and that every probability measure for the Markov process is nonsingular with respect to the transition kernel.

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

Sign in / Sign up

Export Citation Format

Share Document