Lalley-Gatzouras型自仿射海绵的$δ$连通分支的分布
Distribution of $δ$-connected components of self-affine sponge of Lalley-Gatzouras type
摘要：
设$(E, ρ)$是一个度量空间，$h_E\left( δ\right)$是$E$的$δ$ -连通分支集的基数。在文献中，当$E$是满足开集条件的自共形集或$E$是自仿射Sierpienski海绵时，给出了关系式$ h_E(δ)\asymp δ^{-\dim_B E}, \text{ when }δ\to 0. $成立的充要条件。本文将上述结果推广到Lalley-Gatzouras型自仿射海绵;实际上，在这种情况下，我们证明了存在一个伯努利测度$μ$，使得对于任何圆柱$R$，它保持$ h_R(δ)\asymp μ(R) δ^{-\dim_B E}, \text{ when }δ\to 0. $
英文摘要：
Let $(E, ρ)$ be a metric space and let $h_E\left( δ\right)$ be the cardinality of the set of $δ$-connected components of $E$. In literature, in case of that $E$ is a self-conformal set satisfying the open set condition or $E$ is a self-affine Sierpiński sponge, necessary and sufficient condition is given for the validity of the relation $ h_E(δ)\asymp δ^{-\dim_B E}, \text{ when }δ\to 0. $ In this paper, we generalize the above result to self-affine sponges of Lalley-Gatzouras type; actually in this case, we show that there exists a Bernoulli measure $μ$ such that for any cylinder $R$, it holds that $ h_R(δ)\asymp μ(R) δ^{-\dim_B E}, \text{ when }δ\to 0. $
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