Multifractal Analysis Of Inhomogeneous Multinomial Measures

Let (?)1={0,1…,c1-1}and (?)2={0,1,…c2-1} be two alphabets. Let{Tk} be a time sequence which is increasing rapidly. Let (ai) and (bj) be two probability vectors such that the numbers of ai and bj are compatible with the alphabets respectively. We first define the mixed symbolic space with respect to the triplet{(?)1,(?)2,{Tk}} and then construct a Borel probability measure on the mixed symbolic space (?)2. By analysing the multifractal properties of this measure, we compute its Olsen multifractal dimensional functions B and b, and find that they are respectively the maximum and minimum of two convex analytic functions. The alphabets need not be the same and the same to the parameters (ai) and (bj). We call this measure an inhomogeneous multinomial measure. Our purpose is twofold.Firstly we look for exact dimensional measures whose support is the whole inter-val [0,1] and whose Olsen’s multifractal functions b and B are real analytic and agree at two points only. These measures v satisfy an extended multifractal formalism in the sense that, for α in some interval, the Hausdorff dimension of the level sets X(α) of the local Holder exponent of v is the value of the Legendre transform of b at α whereas their packing dimension is the value of the Legendre transform of B at α. To this end, we first work on the symbolic spaces. By counting the numbers and orders of the zeros of the corresponding generalized Dirichlet polynomials, we extract a family of inhomogeneous multinomial measures μ such that their b and B functions are analytic. Also, it is shown that μ satisfies an extended multifractal formalism for all α in an open interval. Then one obtains measures on [0,1] having the same properties by using the natural projection after having used a Gray code.Secondly we show that inhomogeneous multinomial measure μ shares the same mul-tifractal properties with the image measure v of μ, under the projection of the natural code map. This means that μ and v have the same Olsen’s b and B functions. Moreover, the weak doubling property of the measure v is obtained, by which one shows that μ and v satisfy the same extended multifractal formalism for the same range of α. So to the first purpose, the use of the Gray code is not necessary to get the results, although we are dealing with non-doubling measures.