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

Abstract

Let $(E,\rho )$ be a metric space and let $h_E\left(\delta \right)$ be the cardinality of the set of $\delta$-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 $h_E(\delta )\asymp {\delta }^{-{\dim }_{B}E},\text{ when }\delta \to 0.$ 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 $\mu$ such that for any cylinder $R$, it holds that $h_R(\delta )\asymp \mu (R){\delta }^{-{\dim }_{B}E},\text{ when }\delta \to 0.$