Multifractal components of multiplicative set functions

We analyze the multifractal spectrum of multiplicative set functions on a s elf-similar set with open set condition. We show that the multifractal comp onents carry self-similar measures which maximize the dimension. This gives the dimension of a multifractal component as the solution of a problem of maximization of a quasiconcave function satisfying a set of linear constrai nts. Our analysis covers the case of multifractal components of self-simila r measures, the case of Besicovitch normal sets of points, the multifractal spectrum of the relative logarithmic density of a pair of selfsimilar meas ures, the multifractal spectrum of the Liapunov exponent of the shift mappi ng and the intersections of all these sets. We show that the dimension of a n arbitrary union of multifractal components is the supremum of the dimensi ons of the multifractal components in the union. The multidimensional Legen dre transform is introduced to obtain the dimension of the intersection of finitely many multifractal components.