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Graph directed Markov systems on Hilbert spaces

Published online by Cambridge University Press:  01 September 2009

R. D. MAULDIN ,
T. SZAREK  and
M. URBAŃSKI
R. D. MAULDIN
Affiliation:
Department of Mathematics, University of North Texas, 1155 Union Circle 311430, Denton TX 76203-5017, U.S.A. e-mail: mauldin@unt.edu
T. SZAREK
Affiliation:
Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland and Institute of Mathematics Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland. e-mail: szarek@intertele.pl
M. URBAŃSKI
Affiliation:
Department of Mathematics, University of North Texas, 1155 Union Circle 311430, Denton TX 76203-5017, U.S.A. e-mail: urbanski@unt.edu
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Abstract

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 Bowen's 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 2 , September 2009 , pp. 455 - 488
Copyright
Copyright © Cambridge Philosophical Society 2009

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