Continuity of the multifractal spectrum of a random statistically self-similar measure

Until now [see Kahane;((19)) Holley and Waymire;((46)) Falconer;((14)) Olse n;((29)) Molchan;((28)) Arbeiter and Patzschke;((1)) and Barral((3))] one d etermines the multifractal spectrum of a statistically self-similar positiv e measure of the type introduced. in particular by Mandelbrot,((26, 27)) on ly in the following way: let mu be such a measure, for example on the bound ary of a c-ary tree equipped with the standard ultrametric distance; for al pha greater than or equal to0, denote by E-alpha the set of the prints wher e 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 int erval l subset of R-+(*) and a function f : l --> R-+(*) such that for all alpha in l, with probability one, dim E-alpha = f(x). This statement is not completely satisfactory. Indeed, the main result in this paper is: with pr obability one , for all alpha is an element of l, dim E-alpha = f(x). This holds also for a new type of statistically self-similar measures deduced fr om a result recently obtained by Liu.((22)) We also study another problem l eft open in the previous works on the subject: if alpha = inf(I) or alpha = sup(l), one does not know whether E-alpha is empty or not. Under suitable assumptions, we show that E-alpha not equal phi and calculate dim E-alpha.