Distribution of $δ$-connected components of self-affine sponge of Lalley-Gatzouras type

Yanfang Zhang; Yongqiang Yang

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arXiv · 2302.12934

Distribution of $δ$-connected components of self-affine sponge of Lalley-Gatzouras type

Yanfang Zhang·Yongqiang Yang

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Abstract

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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BibTeX RIS

Yanfang Zhang, Yongqiang Yang. 2023-02-24. Distribution of $δ$-connected components of self-affine sponge of Lalley-Gatzouras type. https://arxiv.org/abs/2302.12934

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