Multifractal analysis for projections of Gibbs and related measures

Let n > m >= 1 be two integers. At first we obtain general results for the multifractal analysis of the orthogonal projections on m-dimensional linear subspaces of singular measures mu on R-n satisfying the multifractal formalism. The results hold for gamma(n,m)-almost every such subspace, where gamma(n,m) is the uniform measure on the Grassmannian manifold G(n,m). Let mu be such a measure and suppose that its upper Hausdorff dimension is less than or equal to m. Let I stand for the interval over which the singularity spectrum of mu is increasing. We prove that there exists a non-trivial subinterval (I) over tilde of I such that for every alpha is an element of (I) over tilde, for gamma(n,m)-almost every m-dimensional subspace V, the multifractal formalism holds at alpha for mu(V), the orthogonal projection of mu on V. Moreover, in some cases the result is optimal in the sense that the interval (I) over tilde is maximal in I. Also, we determine the L-q-spectrum tau(mu V) (q) on the minimal interval J necessary to recover the singularity spectrum of mu(V) over (I) over tilde as the Legendre transform of tau(mu V). The interval J and the function tau(mu V) (q) do not depend on V, and tau(mu V) (q) can differ from tau(mu) on a non-trivial interval. For Gibbs measures and some of their discrete counterparts, we show the stronger uniform result: for gamma(n,m)-almost every m-dimensional subspace V, the multifractal formalism holds for mu(V) over the whole interval (I) over tilde. As an application, we obtain a part of the singularity spectrum of some self-similar measures on attractors of iterated function systems which do not satisfy the weak separation condition.