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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
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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.

References [Enhancements On Off] (What's this?)

  • [1] C. Bandt, S. Graf, Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc. 114 (1992), 995-1001. MR 93d:28014
  • [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] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Mathematics 470, Springer, (1975). MR 56:1364
  • [4] K. J. Falconer, Techniques in fractal geometry. Wiley (1997). MR 99f:28013
  • [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] J. E. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713-747. MR 82h:49026
  • [7] R. D. Mauldin, Infinite iterated function systems: theory and applications, Fractals and Stochastics I, Proceedings of the Finstenbergen 1994 Conference, C. Bandt, S. Graf and M. Zähle (Editors), 91-110, Birkhäuser, (1995). MR 97c:28013
  • [8] N. Patzschke, Self-conformal multifractal measures. Adv. Appl. Math. 19 (1997), 486-513. MR 99c:28020
  • [9] C. A. Rogers, S. J. Taylor, Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1962), 1-31. MR 24:A200
  • [10] D. Ruelle, Repellers for real analytic maps, Ergod. Th. and Dynam. Sys. 2 (1982), 99-107. MR 84f:58095
  • [11] A. Schief, Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122 (1994), 111-115. MR 94k:28012

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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

Michal Rams
Affiliation: Polish Academy of Sciences, Warsaw, Poland

Károly Simon
Affiliation: Department of Stochastics, Institute of Mathematics, Technical University of Budapest, 1521 Budapest, P.O. Box 91, Hungary

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

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

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