2020/02/03 by Julien Barral, De‐Jun Feng, Barral, Julien +1
Computer Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Probability (math.PR) #Theoretical and Computational Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2002.00642
openalex publication_date 2020/02/03 · openalex created_date 2021/03/01 · openalex updated_date 2026/07/28
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 \citeGL94. 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.