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2009/2010 Perturbed Iterated Function Systems and the Exact Hausdorff Measure of their Attractors
Nicholas Freeman
Real Anal. Exchange 35(1): 91-120 (2009/2010).


We define a perturbed iterated function system (pIFS) in $\R^d$ as, loosely speaking, a sequence of iterated function systems (IFSs) whose constituent transformations converge towards some limiting IFS. We define the attractor of such a system in a similar style to that of an IFS, and prove that such a set exists uniquely. We define a partially perturbed IFS (ppIFS) to be a perturbed IFS with a constant tail. In a setup with similitudes and the strong separation condition we show that a pIFS attractor can be approximated by a sequence of ppIFS attractors in such a way that the Hausdorff measure is preserved in the limit. We use this result to calculate the exact Hausdorff measure of the pIFS attractor from that of the limiting IFS.


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Nicholas Freeman. "Perturbed Iterated Function Systems and the Exact Hausdorff Measure of their Attractors." Real Anal. Exchange 35 (1) 91 - 120, 2009/2010.


Published: 2009/2010
First available in Project Euclid: 27 April 2010

MathSciNet: MR2657290

Primary: 28A78
Secondary: 28A80

Keywords: Cantor set , deranged Cantor set , Fractal , Hausdorff dimension , Hausdorff measure , IFS , iterated function system , perturbed iterated function system , pIFS

Rights: Copyright © 2009 Michigan State University Press

Vol.35 • No. 1 • 2009/2010
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