Continuity of the Multifractal Spectrum of a Random Statistically Self-Similar Measure |
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Authors: | Julien Barral |
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Affiliation: | (1) Equipe d'Analyse Harmonique (URA 757 du CNRS), Mathématiques, Université Paris-Sud, Bât. 425, 91405 Orsay Cedex, France |
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Abstract: | Until now [see Kahane;(19) Holley and Waymire;(16) Falconer;(14) Olsen;(29) Molchan;(28) Arbeiter and Patzschke;(1) and Barral(3)] one determines the multifractal spectrum of a statistically self-similar positive measure of the type introduced, in particular by Mandelbrot,(26, 27) only in the following way: let be such a measure, for example on the boundary of a c-ary tree equipped with the standard ultrametric distance; for 0, denote by E the set of the points where possesses a local Hölder exponent equal to , and dim E the Hausdorff dimension of E; then, there exists a deterministic open interval I*+ and a function f: I*+ such that for all in I, with probability one, dim E=f(). This statement is not completely satisfactory. Indeed, the main result in this paper is: with probability one, for all I, dim E=f(). This holds also for a new type of statistically self-similar measures deduced from a result recently obtained by Liu.(22) We also study another problem left open in the previous works on the subject: if =inf(I) or =sup(I), one does not know whether E is empty or not. Under suitable assumptions, we show that Eø and calculate dim E. |
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Keywords: | multifractal analysis statistically self-similar measures Mandelbrot's martingales multiplicative cascades |
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