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

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.