Continuity of the Multifractal Spectrum of a Random Statistically Self-Similar Measure

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 μ be such a measure, for example on the boundary of a c-ary tree equipped with the standard ultrametric distance; for α≥0, denote by E _α the set of the points where μ possesses a local Hölder exponent equal to α, and dim E _α the Hausdorff dimension of E _α; then, there exists a deterministic open interval I⊂$\mathbb{R}$*₊ and a function f: I→$\mathbb{R}$*₊ such that for all α in I, with probability one, dim E _α=f(α). This statement is not completely satisfactory. Indeed, the main result in this paper is: with probability one, for all α∈I, dim E _α=f(α). 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 α=inf(I) or α=sup(I), one does not know whether E _α is empty or not. Under suitable assumptions, we show that E _α≠ø and calculate dim E _α.