The L^q spectrum of self-affine measures on sponges

In this paper a sponge in ℝ^d is the attractor of an iterated function system consisting of finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable separation condition is introduced under which a variational formula is proved for the L^q spectrum of any self-affine measure defined on a sponge for all q ∈ ℝ. We give sufficient conditions for the formula to have a closed form. In particular, this is always the case for the box dimension of the sponge. The Frostman and box dimension of these measures is also determined. The approach unifies several existing results and extends them to arbitrary dimensions. The result is derived from a more general variational principle for calculating box counting quantities on sponges which resembles the Ledrappier-Young formula for Hausdorff dimension and could be of interest to further study in its own right.