论文标题
在$ \ mathbb r^k $中随机统计上的自动sierpinski海绵的尺寸
Dimensions of random statistically self-affine Sierpinski sponges in $\mathbb R^k$
论文作者
论文摘要
我们计算了任何随机统计上自我的sierpinski Sponge $ k \ subset \ mathbb {r}^k $（$ k \ ge 2 $）的Hausdorff尺寸，该尺寸是通过在$ [0,1]^k $中使用的某些渗透过程获得的。为此，我们首先为$ k $支持的统计上自我采用的措施的Hausdorff尺寸展示了Ledrappier-Young类型的公式。 该公式与其确定性或随机动力学版本相比提出了一项新功能。然后，我们建立了一个表达$ \ dim_h k $的变异原则，是$ k $支持的统计上自我措施的Hausdorff尺寸的至高无上的，并表明了超级巨人是独特的。 $ \ dim_h k $的值也根据某些确定性潜力的加权压力函数表示。 作为副产品，当$ k = 2 $时，我们为$ k $的Hausdorff尺寸提供了另一种方法，这是Gatzouras和Lalley \ cite \ cite {gl94}首先获得的。还研究了$ k $的盒子计数尺寸及其与$ \ dim_h k $的平等的值。我们还获得了一些$ K $的正交预测的Hausdorff尺寸的变异公式，并且对于〜$ K $支持的统计上自我措施的措施，我们通过这些预测建立了维度保护属性。
We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset \mathbb{R}^k$ ($k\ge 2$) 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.
