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

## TL;DR

This paper extends the understanding of the distribution of $oldsymbol{ ext{delta}}$-connected components in self-affine Lalley-Gatzouras sponges, establishing a measure-theoretic relation similar to known results for other fractals.

## Contribution

It generalizes the relation between $oldsymbol{h_E(oldsymbol{ ext{delta}})}$ and $oldsymbol{ ext{delta}^{- ext{dim}_B E}}$ to Lalley-Gatzouras type self-affine sponges, introducing a Bernoulli measure framework.

## Key findings

- Existence of a Bernoulli measure $oldsymbol{ extmu}$ for Lalley-Gatzouras sponges.
- Asymptotic relation $oldsymbol{h_R(oldsymbol{ ext{delta}}) oldsymbol{ extasymp} oldsymbol{ extmu}(R) oldsymbol{ ext{delta}^{- ext{dim}_B E}}$ as $oldsymbol{ ext{delta} o 0}$.
- Generalization of known fractal component distribution results to a broader class of self-affine sets.

## 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 Sierpi\'nski 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. $