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

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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 ... continued below

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Reid, James Edward August 2017.

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  • Reid, James Edward

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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.

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  • August 2017

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  • Oct. 9, 2017, 11:44 a.m.

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Reid, James Edward. Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems, dissertation, August 2017; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc1011825/: accessed September 25, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .