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Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems

In the context of fractal geometry, the natural extension of volume in Euclidean space is given by Hausdorff and packing measures. These measures arise naturally in the context of iterated function systems (IFS). For example, if the IFS is finite and conformal, then the Hausdorff and packing dimensions of the limit sets agree and the corresponding Hausdorff and packing measures are positive and finite. Moreover, the map which takes the IFS to its dimension is continuous. Developing on previous work, we show that the map which takes a finite conformal IFS to the numerical value of its packing measure is continuous. In the context of self-similar sets, we introduce the super separation condition. We then combine this condition with known density theorems to get a better handle on finding balls of maximum density. This allows us to extend the work of others and give exact formulas for the numerical value of packing measure for classes of Cantor sets, Sierpinski N-gons, and Sierpinski simplexes.

Identiferoai:union.ndltd.org:unt.edu/info:ark/67531/metadc1011825
Date08 1900
CreatorsReid, James Edward
ContributorsUrbaƄski, Mariusz, Fishman, Lior, 1964-, Jackson, Steve, 1957-
PublisherUniversity of North Texas
Source SetsUniversity of North Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis or Dissertation
Formatv, 107 pages, Text
RightsPublic, Reid, James Edward, Copyright, Copyright is held by the author, unless otherwise noted. All rights Reserved.

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