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Spaces of σ-finite linear measure

Ihor Stasyuk, Edward D. Tymchatyn (2013)

Colloquium Mathematicae

Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear measure then...

Spectral properties of ergodic dynamical systems conjugate to their composition squares

Geoffrey R. Goodson (2007)

Colloquium Mathematicae

Let S and T be automorphisms of a standard Borel probability space. Some ergodic and spectral consequences of the equation ST = T²S are given for T ergodic and also when Tⁿ = I for some n>2. These ideas are used to construct examples of ergodic automorphisms S with oscillating maximal spectral multiplicity function. Other examples illustrating the theory are given, including Gaussian automorphisms having simple spectra and conjugate to their squares.

Spectrum of multidimensional dynamical systems with positive entropy

B. Kamiński, P. Liardet (1994)

Studia Mathematica

Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov ℤ d -action on a Lebesgue space has a countable Lebesgue spectrum. At the same time we extend this theorem to ℤ ∞ -actions. Next, using its relative version, we extend to ℤ ∞ -actions some other general results connecting spectrum and entropy.

Spontaneous clustering in theoretical and some empirical stationary processes*

T. Downarowicz, Y. Lacroix, D. Léandri (2010)

ESAIM: Probability and Statistics

In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper...

SRB-like Measures for C⁰ Dynamics

Eleonora Catsigeras, Heber Enrich (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

For any continuous map f: M → M on a compact manifold M, we define SRB-like (or observable) probabilities as a generalization of Sinai-Ruelle-Bowen (i.e. physical) measures. We prove that f always has observable measures, even if SRB measures do not exist. We prove that the definition of observability is optimal, provided that the purpose of the researcher is to describe the asymptotic statistics for Lebesgue almost all initial states. Precisely, the never empty set of all observable measures is...

Standardness of sequences of σ-fields given by certain endomorphisms

Jacob Feldman, Daniel Rudolph (1998)

Fundamenta Mathematicae

 Let E be an ergodic endomorphism of the Lebesgue probability space X, ℱ, μ. It gives rise to a decreasing sequence of σ-fields ℱ , E - 1 ℱ , E - 2 ℱ , . . . A central example is the one-sided shift σ on X = 0 , 1 ℕ with 1 2 , 1 2 product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphismis defined on (X× Y, μ× ν) by ( x , y ) → ( σ ( x ) , T x ( 1 ) ( y ) ) . Here ℱ is the σ-field of μ× ν-measurable sets. Each field is a two-point extension of the one beneath it. Vershik has defined as “standard” any decreasing sequence of σ-fields isomorphic...

Statistical convergence of a sequence of random variables and limit theorems

Sanjoy Ghosal (2013)

Applications of Mathematics

In this paper the ideas of three types of statistical convergence of a sequence of random variables, namely, statistical convergence in probability, statistical convergence in mean of order r and statistical convergence in distribution are introduced and the interrelation among them is investigated. Also their certain basic properties are studied.

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