Relations between topological and metrical properties of self-affine Sierpi$\acute{\text{n}}$ski sponges

TL;DR

This paper explores the relationship between topological and metric properties of self-affine Sierpiński sponges, introducing new invariants and characterizing their connectivity features through projections.

Contribution

It establishes conditions linking the maximal power law property and perfect disconnectedness of self-affine Sierpiński sponges to their projections' connectedness.

Findings

E satisfies the maximal power law iff E and its projections contain trivial connected components.

E is perfectly disconnected iff E and its projections are totally disconnected.

Introduces the new property of perfect disconnectedness for these fractals.

Abstract

We investigate two Lipschitz invariants of metric spaces defined by $\delta$-connected components, called the maximal power law property and the perfectly disconnectedness. The first property has been studied in literature for some self-similar sets and Bedford-McMullen carpets, while the second property seems to be new. For a self-affine Sierpi$\acute{\text{n}}$ski sponge $E$, we first show that $E$ satisfies the maximal power law if and only if $E$ and all its major projections contain trivial connected components; secondly, we show that $E$ is perfectly disconnected if and only if $E$ and all its major projections are totally disconnected.