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

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. - See more at: http://www.ams.org/journals/proc/2001-129-09/S0002-9939-01-05969-X/#sthash.Pd60169C.dpuf