Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets
Yuval Peres, Michal Rams, Károly Simon and Boris Solomyak
Proc. Amer. Math. Soc. 129 (2001), 2689-2699
Primary 28A78; Secondary 28A80, 37C45, 37C70
February 9, 2001
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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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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
open set condition
Received by editor(s):
January 18, 2000
February 9, 2001
The first author’s research was partially supported by NSF grant #DMS-9803597.
The second author’s research was supported in part by KBN grant No. 2 P03A 009 17, Foundation for Polish Science and the Hebrew University of Jerusalem.
The third author’s research was supported in part by the OTKA foundation grant F019099.
The fourth author’s research was supported in part by NSF grant #DMS 9800786, the Fulbright foundation, and the Institute of Mathematics at the Hebrew University, Jerusalem.
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