When is the union of two unit intervals a self-similar set satisfying the open set condition? |
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Authors: | De-Jun Feng Su Hua Yuan Ji |
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Institution: | (1) The Chinese University of Hong Kong, Shatin, Hong Kong;(2) Tsinghua University, Beijing, P.R. China |
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Abstract: | Let U
λ be the union of two unit intervals with gap λ. We show that U
λ is a self-similar set satisfying the open set condition if and only if U
λ can tile an interval by finitely many of its affine copies (admitting different dilations). Furthermore, each such λ can
be characterized as the spectrum of an irreducible double word which represents a tiling pattern. Some further considerations
of the set of all such λ’s, as well as the corresponding tiling patterns, are given.
The first author was partially supported by the RGC grant and the direct grant in CUHK, Fok Ying Tong Education Foundation
and NSFC (10571100). The second author was partially supported by NSFC (70371074) and NFSC (10571104). |
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Keywords: | 2000 Mathematics Subject Classification: 28A80 28A75 |
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