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Finer geometric rigidity of limit sets of conformal IFS

Authors: Volker Mayer and Mariusz Urbanski
Journal: Proc. Amer. Math. Soc. 131 (2003), 3695-3702
MSC (2000): Primary 37D45, 37D20, 28Exx
Published electronically: July 17, 2003
MathSciNet review: 1998176
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Abstract: We consider infinite conformal iterated function systems in the phase space $\mathbb{R}^d$ with $d\ge 3$. Let $J$ be the limit set of such a system. Under a mild technical assumption, which is always satisfied if the system is finite, we prove that either the Hausdorff dimension of $J$ exceeds the topological dimension $k$of the closure of $J$ or else the closure of $J$ is a proper compact subset of either a geometric sphere or an affine subspace of dimension $k$. A similar dichotomy holds for conformal expanding repellers.

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Additional Information

Volker Mayer
Affiliation: Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d’Ascq Cedex, France

Mariusz Urbanski
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430

Received by editor(s): November 18, 2001
Published electronically: July 17, 2003
Additional Notes: The second author was supported in part by the NSF Grant no. DMS 0100078
Communicated by: Michael Handel
Article copyright: © Copyright 2003 American Mathematical Society

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