TY - JOUR

T1 - Equivalence of positive hausdorff measure and the open set condition for self-conformal sets

AU - Peres, Yuval

AU - Rams, Michal

AU - Simon, Käroly

AU - Solomyak, Boris

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

KW - Hausdorfl measure

KW - Open set condition

KW - Self-conformal set

UR - http://www.scopus.com/inward/record.url?scp=33646836094&partnerID=8YFLogxK

U2 - 10.1090/s0002-9939-01-05969-x

DO - 10.1090/s0002-9939-01-05969-x

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AN - SCOPUS:33646836094

SN - 0002-9939

VL - 129

SP - 2689

EP - 2699

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 9

ER -