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

Authors:
Yuval Peres, Michal Rams, Károly Simon and Boris Solomyak

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2689-2699

MSC (2000):
Primary 28A78; Secondary 28A80, 37C45, 37C70

Published electronically:
February 9, 2001

MathSciNet review:
1838793

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

**[1]**Christoph Bandt and Siegfried Graf,*Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure*, Proc. Amer. Math. Soc.**114**(1992), no. 4, 995–1001. MR**1100644**, 10.1090/S0002-9939-1992-1100644-3**[2]**T. Bedford, Applications of dynamical systems theory to fractals -- a study of cookie-cutter Cantor sets, in*Fractal Geometry and Analysis, J. Bélair and S. Dubuc (eds.)*, 1-44, Kluwer, (1991). CMP**92:05****[3]**Rufus Bowen,*Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR**0442989****[4]**Kenneth Falconer,*Techniques in fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1997. MR**1449135****[5]**A. H. Fan, K.-S. Lau, Iterated function systems and Ruelle operator.*J. Math. Analysis Applic.***231**(1999), 319-344. CMP**99:09****[6]**John E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, 10.1512/iumj.1981.30.30055**[7]**R. Daniel Mauldin,*Infinite iterated function systems: theory and applications*, Fractal geometry and stochastics (Finsterbergen, 1994) Progr. Probab., vol. 37, Birkhäuser, Basel, 1995, pp. 91–110. MR**1391972**, 10.1007/978-3-0348-7755-8_5**[8]**Norbert Patzschke,*Self-conformal multifractal measures*, Adv. in Appl. Math.**19**(1997), no. 4, 486–513. MR**1479016**, 10.1006/aama.1997.0557**[9]**C. A. Rogers and S. J. Taylor,*Functions continuous and singular with respect to a Hausdorff measure.*, Mathematika**8**(1961), 1–31. MR**0130336****[10]**David Ruelle,*Repellers for real analytic maps*, Ergodic Theory Dynamical Systems**2**(1982), no. 1, 99–107. MR**684247****[11]**Andreas Schief,*Separation properties for self-similar sets*, Proc. Amer. Math. Soc.**122**(1994), no. 1, 111–115. MR**1191872**, 10.1090/S0002-9939-1994-1191872-1

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
28A78,
28A80,
37C45,
37C70

Retrieve articles in all journals with MSC (2000): 28A78, 28A80, 37C45, 37C70

Additional Information

**Yuval Peres**

Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem, Israel and Department of Statistics, University of California, Berkeley, California 94720

Email:
peres@math.huji.ac.il

**Michal Rams**

Affiliation:
Polish Academy of Sciences, Warsaw, Poland

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

Email:
simonk@math.bme.hu

**Boris Solomyak**

Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195

Email:
solomyak@math.washington.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-05969-X

Keywords:
Hausdorff measure,
self-conformal set,
open set condition

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

Article copyright:
© Copyright 2001
American Mathematical Society