Multifractal formalism for self-similar measures with weak separation condition |
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Authors: | De-Jun Feng Ka-Sing Lau |
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Affiliation: | aDepartment of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong |
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Abstract: | For any self-similar measure μ on satisfying the weak separation condition, we show that there exists an open ball U0 with μ(U0)>0 such that the distribution of μ, restricted on U0, is controlled by the products of a family of non-negative matrices, and hence μ|U0 satisfies a kind of quasi-product property. Furthermore, the multifractal formalism for μ|U0 is valid on the whole range of dimension spectrum, regardless of whether there are phase transitions. Moreover the dimension spectra of μ and μ|U0 coincide for q0. This result unifies and improves many of the recent works on the multifractal structure of self-similar measures with overlaps. |
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Keywords: | Self-similar measures Multifractal formalism Weak separation condition Moran structure |
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