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On the open set condition for self-similar fractals
Authors:
Christoph Bandt, Nguyen Viet Hung and Hui Rao
Journal:
Proc. Amer. Math. Soc. 134 (2006), 1369-1374
MSC (2000):
Primary 28A80; Secondary 28A75
Posted:
October 6, 2005
MathSciNet review:
2199182
Full-text PDF Free Access
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Additional Information
Abstract: For self-similar sets, the existence of a feasible open set is a natural separation condition which expresses geometric as well as measure-theoretic properties. We give a constructive approach by defining a central open set and characterizing those points which do not belong to feasible open sets.
References
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C. Bandt and S. Graf, Self-similar sets VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc. 114 (1992), 995-1001. MR 1100644 (93d:28014)
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K.J. Falconer, Fractal Geometry, Wiley, 1990. MR 1102677 (92j:28008)
- 3.
J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 0625600 (82h:49026)
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M. Moran, Dynamical boundary of a self-similar set. Fund. Math. 160 (1999), 1-14. MR 1694399 (2000f:28009)
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P.A.P. Moran, Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42 (1946), 15-23. MR 0014397 (7:278f)
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Additional Information
Christoph Bandt
Affiliation:
Institute for Mathematics, Arndt University, 17487 Greifswald, Germany
Email:
bandt@uni-greifswald.de
Nguyen Viet Hung
Affiliation:
Institute for Mathematics, Arndt University, 17487 Greifswald, Germany
Email:
nvh0@yahoo.com
Hui Rao
Affiliation:
Department of Mathematics, Tsinghua University, P.O. Box 100084, Beijing, People's Republic of China
Email:
HRao@math.tsinghua.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002-9939-05-08300-0
PII:
S 0002-9939(05)08300-0
Keywords:
Self-similar,
fractals,
iterated function system
Received by editor(s):
October 26, 2004,
Received by editor(s) in revised form:
December 2, 2004
Posted:
October 6, 2005
Additional Notes:
The third author was supported by the German Research Foundation (DFG), and the second author was supported by the Vietnamese Government and the German Academic Exchange Service (DAAD)
Communicated by:
David Preiss
Article copyright:
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
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