Distribution of δ-connected components of self-affine sponges of Lalley-Gatzouras type

Abstract

Let (E,ρ) be a metric space and let hE(δ) 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 relationhE(δ)≍δ−dimB⁡E, when δ→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 W, it holds thathW(δ)≍μ(W)δ−dimB⁡E, when δ→0.