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REPTILES WITH HOLES

Published online by Cambridge University Press:  15 September 2005

Francis Jordan  and
Sze-Man Ngai
Francis Jordan
Affiliation:
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA (fjordan@georgiasouthern.edu; ngai@gsu.mat.georgiasouthern.edu)
Sze-Man Ngai
Affiliation:
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA (fjordan@georgiasouthern.edu; ngai@gsu.mat.georgiasouthern.edu)
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Abstract

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Croft, Falconer and Guy asked: what is the smallest integer $n$ such that an $n$-reptile in the plane has a hole? Motivated by this question, we describe a geometric method of constructing reptiles in $\mathbb{R}^d$, especially reptiles with holes. In particular, we construct, for each even integer $n\ge4$, an $n$-reptile in $\mathbb{R}^2$ with holes. We also answer some questions concerning the topological properties of a reptile whose interior consists of infinitely many components.

Keywords

Primary 52C20Secondary 28A80reptileself-similar setiterated function system
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 48 , Issue 3 , October 2005 , pp. 651 - 671
Copyright
Copyright © Edinburgh Mathematical Society 2005