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

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 (mu) be such a measure, for example on the boundary of a c-ary tree equipped with the standard ultrametric distance; for (alpha) => 0, denote by E (alpha)the set of the points where (mu) possesses a local Holder exponent equal to (alpha), and dim E (alpha)the Hausdorff dimension of E (alpha); then, there exists a deterministic open interval I (sebset of) R *+ and a function f: I – R *+ such that for all (alpha)in I, with probability one, dim E (alpha)=f(alpha). This statement is not completely satisfactory. Indeed, the main result in this paper is: with probability one, for all (alpha) (element of) I, dim E (alpha)=f(alpha). This holds also for a new type of statistically selfsimilar 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 (alpha)=inf(I) or (alpha)=sup(I), one does not know whether E (alpha)is empty or not. Under suitable assumptions, we show that E (alpha)(not equal to) (empty set) and calculate dim E (alpha).