Dimensions of random statistically self-affine Sierpinski sponges in $\mathbb R^k$

Julien Barral (1) , De-Jun Feng (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Department of Mathematics CUHK

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge K ⊂ R k (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-similar measures supported on K. This formula presents a new feature with respect to the deterministic case or the random dynamical version. Then, we establish a variational principle expressing dim K as the supremum of the Hausdorff dimensions of statistically self-similar measures supported on K, which is shown to be uniquely reached. The value of dim 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 obtained by Gatzouras and Lalley. This alternative concerns both the sharp lower and upper bounds for the dimension. The value of the box counting dimension of K and its equality with dim K are also studied. We also obtain a variational formula for the Hausdorff dimensions of the natural orthogonal projections of K to the linear subspaces generated by the eigensubspaces of the diagonal endomorphism used to generate K (contrarily to what happens in the deterministic case, these projections are not of the same nature as K). Finally, we prove a dimension conservation formula associated to any Mandelbrot measure supported on K, that of its orthogonal projection to such subspace, and the dimension of almost every associated conditional measure.

Domaines: Géométrie métrique [math.MG]; Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Julien Barral, De-Jun Feng. Dimensions of random statistically self-affine Sierpinski sponges in $\mathbb R^k$. 2020. ⟨hal-02458498v1⟩