The strong open set condition for self-conformal random fractals

Patzschke, Norbert

doi:10.1090/S0002-9939-03-07077-1

The strong open set condition for self-conformal random fractals
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by Norbert Patzschke

Proc. Amer. Math. Soc. 131 (2003), 2347-2358

DOI: https://doi.org/10.1090/S0002-9939-03-07077-1

Published electronically: March 18, 2003

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Abstract:

We prove that the open set condition and the strong open set condition are equivalent for self-conformal random fractals.

References

Matthias Arbeiter and Norbert Patzschke, Random self-similar multifractals, Math. Nachr. 181 (1996), 5–42. MR 1409071, DOI 10.1002/mana.3211810102
K. J. Falconer, Random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 559–582. MR 857731, DOI 10.1017/S0305004100066299
Siegfried Graf, Statistically self-similar fractals, Probab. Theory Related Fields 74 (1987), no. 3, 357–392. MR 873885, DOI 10.1007/BF00699096
Ka-Sing Lau, Hui Rao, and Yuan-Ling Ye, Corrigendum: “Iterated function system and Ruelle operator” [J. Math. Anal. Appl. 231 (1999), no. 2, 319–344; MR1669203 (2001a:37013)] by Lau and A. H. Fan, J. Math. Anal. Appl. 262 (2001), no. 1, 446–451. MR 1857239, DOI 10.1006/jmaa.2001.7297
R. Daniel Mauldin and S. C. Williams, Random recursive constructions: asymptotic geometric and topological properties, Trans. Amer. Math. Soc. 295 (1986), no. 1, 325–346. MR 831202, DOI 10.1090/S0002-9947-1986-0831202-5
Norbert Patzschke, The strong open set condition in the random case, Proc. Amer. Math. Soc. 125 (1997), no. 7, 2119–2125. MR 1377002, DOI 10.1090/S0002-9939-97-03816-1
Norbert Patzschke, Self-conformal multifractal measures, Adv. in Appl. Math. 19 (1997), no. 4, 486–513. MR 1479016, DOI 10.1006/aama.1997.0557
Yuval Peres, MichałRams, Károly Simon, and Boris Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc. 129 (2001), no. 9, 2689–2699. MR 1838793, DOI 10.1090/S0002-9939-01-05969-X
Andreas Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), no. 1, 111–115. MR 1191872, DOI 10.1090/S0002-9939-1994-1191872-1