Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets
Yuval Peres, Michal Rams, Károly Simon and Boris Solomyak
Proc. Amer. Math. Soc. 129 (2001), 2689-2699
Primary 28A78; Secondary 28A80, 37C45, 37C70
February 9, 2001
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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.
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
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
Bowen, Equilibrium states and the ergodic theory of Anosov
diffeomorphisms, Lecture Notes in Mathematics, Vol. 470,
Springer-Verlag, Berlin, 1975. MR 0442989
Falconer, Techniques in fractal geometry, John Wiley &
Sons Ltd., Chichester, 1997. MR 1449135
A. H. Fan, K.-S. Lau, Iterated function systems and Ruelle operator. J. Math. Analysis Applic. 231 (1999), 319-344. CMP 99:09
E. Hutchinson, Fractals and self-similarity, Indiana Univ.
Math. J. 30 (1981), no. 5, 713–747. MR 625600
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
Patzschke, Self-conformal multifractal measures, Adv. in Appl.
Math. 19 (1997), no. 4, 486–513. MR 1479016
A. Rogers and S.
J. Taylor, Functions continuous and singular with respect to a
Hausdorff measure., Mathematika 8 (1961), 1–31.
0130336 (24 #A200)
Ruelle, Repellers for real analytic maps, Ergodic Theory
Dynamical Systems 2 (1982), no. 1, 99–107. MR 684247
Schief, Separation properties for self-similar
sets, Proc. Amer. Math. Soc.
122 (1994), no. 1,
111–115. MR 1191872
- 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
- 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
- R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Mathematics 470, Springer, (1975). MR 56:1364
- K. J. Falconer, Techniques in fractal geometry. Wiley (1997). MR 99f:28013
- A. H. Fan, K.-S. Lau, Iterated function systems and Ruelle operator. J. Math. Analysis Applic. 231 (1999), 319-344. CMP 99:09
- J. E. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713-747. MR 82h:49026
- 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
- N. Patzschke, Self-conformal multifractal measures. Adv. Appl. Math. 19 (1997), 486-513. MR 99c:28020
- C. A. Rogers, S. J. Taylor, Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1962), 1-31. MR 24:A200
- D. Ruelle, Repellers for real analytic maps, Ergod. Th. and Dynam. Sys. 2 (1982), 99-107. MR 84f:58095
- A. Schief, Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122 (1994), 111-115. MR 94k:28012
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Institute of Mathematics, Hebrew University, Jerusalem, Israel and Department of Statistics, University of California, Berkeley, California 94720
Polish Academy of Sciences, Warsaw, Poland
Department of Stochastics, Institute of Mathematics, Technical University of Budapest, 1521 Budapest, P.O. Box 91, Hungary
Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195
open set condition
Received by editor(s):
January 18, 2000
February 9, 2001
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.
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