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Invariant measures for nonexpansive Markov operators on Polish spaces

Tomasz Szarek — 2003

New sufficient conditions for the existence of an invariant measure for nonexpansive Markov operators defined on Polish spaces are presented. These criteria are applied to iterated function systems, stochastically perturbed dynamical systems and Poisson stochastic differential equations. We also estimate the Ledrappier version of capacity for invariant measures.

Invariant measures for iterated function systems

Tomasz Szarek — 2000

Annales Polonici Mathematici

A new criterion for the existence of an invariant distribution for Markov operators is presented. Moreover, it is also shown that the unique invariant distribution of an iterated function system is singular with respect to the Hausdorff measure.

Generic properties of learning systems

Tomasz Szarek — 2000

Annales Polonici Mathematici

It is shown that the set of learning systems having a singular stationary distribution is generic in the family of all systems satisfying the average contractivity condition.

Irreducible Markov systems on Polish spaces

Katarzyna HorbaczTomasz Szarek — 2006

Studia Mathematica

Contractive Markov systems on Polish spaces which arise from graph directed constructions of iterated function systems with place dependent probabilities are considered. It is shown that their stability may be studied using the concentrating methods developed by the second author [Dissert. Math. 415 (2003)]. In this way Werner's results obtained in a locally compact case [J. London Math. Soc. 71 (2005)] are extended to a noncompact setting.

Some generic properties of concentration dimension of measure

Józef MyjakTomasz Szarek — 2003

Bollettino dell'Unione Matematica Italiana

Let K be a compact quasi self-similar set in a complete metric space X and let M 1 K denote the space of all probability measures on K , endowed with the Fortet-Mourier metric. We will show that for a typical (in the sense of Baire category) measure in M 1 K the lower concentration dimension is equal to 0 , while the upper concentration dimension is equal to the Hausdorff dimension of K .

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