Finer geometric rigidity of limit sets of conformal IFS

Mayer, Volker; Urbański, Mariusz

doi:10.1090/S0002-9939-03-07216-2

Finer geometric rigidity of limit sets of conformal IFS
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by Volker Mayer and Mariusz Urbański PDF

Proc. Amer. Math. Soc. 131 (2003), 3695-3702 Request permission

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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