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Dimension spectrum and graph directed Markov systems.

In this dissertation we study graph directed Markov systems (GDMS) and limit sets associated with these systems. Given a GDMS S, by the Hausdorff dimension spectrum of S we mean the set of all positive real numbers which are the Hausdorff dimension of the limit set generated by a subsystem of S. We say that S has full Hausdorff dimension spectrum (full HD spectrum), if the dimension spectrum is the interval [0, h], where h is the Hausdorff dimension of the limit set of S. We give necessary conditions for a finitely primitive conformal GDMS to have full HD spectrum. A GDMS is said to be regular if the Hausdorff dimension of its limit set is also the zero of the topological pressure function. We show that every number in the Hausdorff dimension spectrum is the Hausdorff dimension of a regular subsystem. In the particular case of a conformal iterated function system we show that the Hausdorff dimension spectrum is compact. We introduce several new systems: the nearest integer GDMS, the Gauss-like continued fraction system, and the Renyi-like continued fraction system. We prove that these systems have full HD spectrum. A special attention is given to the backward continued fraction system that we introduce and we prove that it has full HD spectrum. This system turns out to be a parabolic iterated function system and this makes the analysis more involved. Several examples have been constructed in the past of systems not having full HD spectrum. We give an example of such a system whose limit set has positive Lebesgue measure.

Identiferoai:union.ndltd.org:unt.edu/info:ark/67531/metadc5226
Date05 1900
CreatorsGhenciu, Eugen Andrei
ContributorsMauldin, R. Daniel, UrbaƄski, Mariusz, Kallman, Robert
PublisherUniversity of North Texas
Source SetsUniversity of North Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis or Dissertation
FormatText
RightsUse restricted to UNT Community, Copyright, Ghenciu, Eugen Andrei, Copyright is held by the author, unless otherwise noted. All rights reserved.

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