Multifractal Structure of Convolution of the Cantor Measure

The multifractal structure of measures generated by iterated function systems (IFS) with overlaps is, to a large extend, unknown. In this paper we study the local dimension of the m-time convolution of the standard Cantor measure μ. By using some combinatoric techniques, we show that the set E of at...

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Veröffentlicht in:Advances in applied mathematics 2001-07, Vol.27 (1), p.1-16
Hauptverfasser: Hu, Tian-You, Lau, Ka-Sing
Format: Artikel
Sprache:eng
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Zusammenfassung:The multifractal structure of measures generated by iterated function systems (IFS) with overlaps is, to a large extend, unknown. In this paper we study the local dimension of the m-time convolution of the standard Cantor measure μ. By using some combinatoric techniques, we show that the set E of attainable local dimensions of μ contains an isolated point. This is rather surprising because when the IFS satisfies the open set condition, the set E is an interval. The result implies that the multifractal formalism fails without the open set condition.
ISSN:0196-8858
1090-2074
DOI:10.1006/aama.2000.0683