Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets |
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| Authors: | Yuval Peres Michal Rams Ká roly Simon Boris Solomyak |
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| Institution: | Institute of Mathematics, Hebrew University, Jerusalem, Israel and Department of Statistics, University of California, Berkeley, California 94720 ; Polish Academy of Sciences, Warsaw, Poland ; Department of Stochastics, Institute of Mathematics, Technical University of Budapest, 1521 Budapest, P.O. Box 91, Hungary ; Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195 |
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| Abstract: | A compact set  is self-conformal if it is a finite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the ``open set condition' (OSC), then  has positive  -dimensional Hausdorff measure, where  is the solution of Bowen's pressure equation. We prove that the OSC, the strong OSC, and positivity of the  -dimensional Hausdorff measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting. |
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| Keywords: | Hausdorff measure self-conformal set open set condition |
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