Self-similar sets with an open set condition and great variety of overlaps

For a very simple family of self-similar sets with two pieces, we prove, using a technique of Solomyak, that the intersection of the pieces can be a Cantor set with any dimension in [0,0.2] as well as a finite set of any cardinality 2^m. The main point is that the open set condition is fulfilled for...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2008-11, Vol.136 (11), p.3895-3903
Main Authors: Bandt, Christoph, Hung, Nguyen Viet
Format: Article
Language:English
Subjects:
Cantor set
Coefficients
Combinatorics
Exact sciences and technology
Fractals
General mathematics
General, history and biography
Geometric planes
Hausdorff dimensions
Mathematical theorems
Mathematics
Power series
Power series coefficients
Sciences and techniques of general use
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Summary:For a very simple family of self-similar sets with two pieces, we prove, using a technique of Solomyak, that the intersection of the pieces can be a Cantor set with any dimension in [0,0.2] as well as a finite set of any cardinality 2^m. The main point is that the open set condition is fulfilled for all these examples.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-08-09349-0