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

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- by Yuval Peres, Michał Rams, Károly Simon and Boris Solomyak PDF
- Proc. Amer. Math. Soc.
**129**(2001), 2689-2699 Request permission

## Abstract:

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

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## Additional Information

**Yuval Peres**- Affiliation: Institute of Mathematics, Hebrew University, Jerusalem, Israel and Department of Statistics, University of California, Berkeley, California 94720
- MR Author ID: 137920
- Email: peres@math.huji.ac.il
**Michał Rams**- Affiliation: Polish Academy of Sciences, Warsaw, Poland
- MR Author ID: 656055
- Email: rams@snowman.impan.gov.pl
**Károly Simon**- Affiliation: Department of Stochastics, Institute of Mathematics, Technical University of Budapest, 1521 Budapest, P.O. Box 91, Hungary
- MR Author ID: 250279
- Email: simonk@math.bme.hu
**Boris Solomyak**- Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195
- MR Author ID: 209793
- Email: solomyak@math.washington.edu
- Received by editor(s): January 18, 2000
- Published electronically: February 9, 2001
- Additional Notes: 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. - Communicated by: David Preiss
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**129**(2001), 2689-2699 - MSC (2000): Primary 28A78; Secondary 28A80, 37C45, 37C70
- DOI: https://doi.org/10.1090/S0002-9939-01-05969-X
- MathSciNet review: 1838793