Multifractal Components of Multiplicative Set Functions

We analyze the multifractal spectrumof multiplicative set functions on a self-similar set with open set condition. We show that the multifractal components 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 constraints. Our analysis covers the case of multifractal components of self-similar measures, the case of Besicovitch normal sets of points, the multifractal spectrum of the relative logarithmic density of a pair of self-similar measures, the multifractal spectrum of the Liapunov exponent of the shift mapping and the intersections of all these sets. We show that the dimension of an arbitrary union of multifractal components is the supremum of the dimensions of the multifractal components in the union. The multidimensional Legendre transform is introduced to obtain the dimension of the intersection of finitely many multifractal components.