Iterated function systems with non-distinct fixed points

The dimension theory of self-similar sets is quite well understood in the cases when some separation conditions (open set condition or weak separation condition) or the so-called transversality condition hold. Otherwise the study of the Hausdorff dimension is far from well understood. We investigate...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2011-11, Vol.383 (1), p.244-258
Main Author: BARANY, Balázs
Format: Article
Language:English
Subjects:
Box dimension
Exact sciences and technology
Global analysis, analysis on manifolds
Hausdorff dimension
Logic and foundations
Mathematical analysis
Mathematical logic, foundations, set theory
Mathematics
Measure and integration
Ordinary differential equations
Sciences and techniques of general use
Self-similar sets
Set theory
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Transversality
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Summary:The dimension theory of self-similar sets is quite well understood in the cases when some separation conditions (open set condition or weak separation condition) or the so-called transversality condition hold. Otherwise the study of the Hausdorff dimension is far from well understood. We investigate the properties of the Hausdorff dimension of self-similar sets such that some functions in the corresponding iterated function system share the same fixed point. Then it is not possible to apply directly known techniques. In this paper we are going to calculate the Hausdorff dimension for almost all contracting parameters and calculate the proper dimensional Hausdorff measure of the attractor.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2011.05.023