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

Abstract

A compact set K K 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 K K has positive s s -dimensional Hausdorff measure, where s s is the solution of Bowen’s pressure equation. We prove that the OSC, the strong OSC, and positivity of the s s -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.