Perturbed iterated function systems and the exact Hausdorff measure of their attractors

Nic P Freeman

Research output: Contribution to journalArticle (Academic Journal)peer-review

Abstract

We define a perturbed iterated function system (pIFS) in Rd 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.
Original languageEnglish
Pages (from-to)91-120
Number of pages30
JournalReal Analysis Exchange
Volume35
Issue number1
Publication statusPublished - 2009

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