### On typical Markov operators acting on Borel measures.

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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.

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.

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.

We prove a new sufficient condition for the asymptotic stability of Markov operators acting on measures. This criterion is applied to iterated function systems.

A sufficient condition for the asymptotic stability of Markov operators acting on measures defined on Polish spaces is presented.

It is shown that Markov operators with equicontinuous dual operators which overlap supports have at most one invariant measure. In this way we extend the well known result proved for Markov operators with the strong Feller property by R. Z. Khas'minski.

A new sufficient condition for the existence of an invariant measure for Markov operators defined on Polish spaces is presented. This criterion is applied to iterated function systems.

We formulate some criteria on the existence of an invariant measure for Markov chains and Markov processes. We also show their utylities in the theory of function systems and stochastic differential equations

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.

Let $K$ be a compact quasi self-similar set in a complete metric space $X$ and let ${\mathfrak{M}}_{1}\left(K\right)$ 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 ${\mathfrak{M}}_{1}\left(K\right)$ the lower concentration dimension is equal to $0$, while the upper concentration dimension is equal to the Hausdorff dimension of $K$.

Let X be a locally compact, separable metric space. We prove that $di{m}_{T}X=infdi{m}_{L}{X}^{\text{'}}:{X}^{\text{'}}ishomeomorphictoX$, where $di{m}_{L}X$ and $di{m}_{T}X$ stand for the concentration dimension and the topological dimension of X, respectively.

We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems $({S}_{i},{p}_{i}):i=1,...,N$ where ${S}_{i}$ are bi-lipschitzean transformations.

It is shown that every Polish space X with $di{m}_{T}X\ge d$ admits a compact subspace Y such that $di{m}_{H}Y\ge d$ where $di{m}_{T}$ and $di{m}_{H}$ denote the topological and Hausdorff dimensions, respectively.

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