Tomasz Szarek

Graph directed Markov systems on Hilbert spaces

We deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowens formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by�product of the mainstream of our investigations we prove a 4r�covering theorem for all metric spaces. It enables us to establish appropriate co�Frostman type theorems.

Markov operators with a unique invariant measure

Let M be the set of all finite Borel measures on a Polish space X. Let P be a Markov operator on M and π the transition function corresponding to P. Set Γ(x)=suppπ(x,·), x∈X. It is proved that, if P admits a unique invariant measure μ∗, then μ∗(D)=0 or μ∗(⋂n=0∞Γn(D))=1 for every Borel set D such that Γ(D)⊂D. Moreover, if P is nonexpansive, then a trajectory of every Markov chain corresponding to P and starting from suppμ∗ is dense in suppμ∗. The last statement fails if we drop nonexpansivity condition.

The OSC does not imply the SOSC for infinite iterated function systems

It is shown that every class of contracting similitudes {f 1 ,...,f N } on R s satisfying the OSC and such that dimes K 0 < s, where K 0 denotes the corresponding fractal, can be extended to an infinite family of contracting similitudes which still satisfies the OSC but the SOSC does not hold.

RANDOMLY CONNECTED DYNAMICAL SYSTEMS ON BANACH SPACES

We give sufficient conditions for asymptotic stability of Markov operators governing the evolution of measures due to the action of randomly chosen dynamical systems on Banach spaces. We show that the existence of an invariant measure for the transition operator implies the existence of an invariant measure for a semigroup generated by the considered systems.

On harmonic analysis operators in Laguerre-Dunkl and Laguerre-symmetrized settings

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to

A Lower Estimation of the Hausdorff Dimension for Attractors with Overlaps

mathbb{Z}_2^d

Dimension of measures invariant with respect to the Ważewska partial differential equation

. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators,

Invariant measures for Markov operators with application to function systems

g

Markov operators acting on Polish spaces

-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calderon-Zygmund theory we prove that these operators are bounded on weighted

Analysis Related to All Admissible Type Parameters in the Jacobi Setting

L^p