Dimensions of random statistically self-affine Sierpinski sponges in [formula omitted

Abstract

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge K⊂Rk (k≥2) obtained by using some percolation process in [0,1]k. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on K. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing dimH⁡K as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on K, and show that the supremum is uniquely attained. The value of dimH⁡K is also expressed in terms of the weighted pressure function of some deterministic potential. As a by-product, when k=2, we give an alternative approach to the Hausdorff dimension of K, which was first obtained by Gatzouras and Lalley [27]. The value of the box counting dimension of K and its equality with dimH⁡K are also studied. We also obtain a variational formula for the Hausdorff dimensions of some orthogonal projections of K, and for statistically self-affine measures supported on K, we establish a dimension conservation property through these projections.