Equivalence of positive hausdorff measure and the open set condition for self-conformal sets

Yuval Peres, Michal Rams, Käroly Simon, Boris Solomyak

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Abstract

A compact set K is self-conformal if it is a finite union of its images by conformai contractions. It is well known that if the conformai contractions satisfy the "open set condition" (OSC), then K has positive sdimensional Hausdorff measure, where s is the solution of Bowen's pressure equation. \Ve prove that the OSC, the strong OSC, and positivity of the s-dimensional Hausdorff measure are equivalent for conformai 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.

Original languageEnglish
Pages (from-to)2689-2699
Number of pages11
JournalProceedings of the American Mathematical Society
Volume129
Issue number9
DOIs
StatePublished - 2001
Externally publishedYes

Keywords

  • Hausdorfl measure
  • Open set condition
  • Self-conformal set

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