Dimensions of random statistically self-affine Sierpinski sponges in $\mathbb R^k$
Résumé
We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset \R^k$ ($k\ge 2$) 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 $\dim_H K$ 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 $\dim_H K$ 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 \cite{GL94}. The value of the box counting dimension of $K$ and its equality with $\dim_H K$ 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.
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-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. RÉSUMÉ. Nous calculons les dimensions de Hausdorff des tapis de Sierpinski aléatoires auto-affines en loi dans R k (k ≥ 2) obtenus grâceà certains processus de percolation dans [0, 1] k. Etant donné un tel ensemble K, en premier lieu nous exhibons une formule de type Ledrappier-Young pour les dimensions de Hausdorff des mesures statistiquement auto-affines portées par K; cette formule présente une nouvelle caractéristique comparée aux cas où K est déterministe ou associéà une dynamique aléatoire. Puis nousétablissons un principe variationnel qui donne dimH K comme le supremum des dimensions de ces mesures, et montrons que ce supremum est atteint de façon unique. La valeur de dimH K est aussi donnée comme la pression pondérée d'un potentiel déterministe. Cette approche fournit une alternativeà celle développée par Gatzouras et Lalley [27] dans le cas où k = 2. Nousétudionségalement la valeur de la dimension de boîte de K et sonégalité avec dimH K. Nous obtenons aussi un principe variationnel pour les dimensions de Hausdorff de certaines projections orthogonales de K, et pour les mesures auto-similaires en loi portées par K, nousétablissons une propriété de conservation de dimension au travers de ces projections.
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