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

Yuan Zhang; Liang-Yi Huang

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arXiv · 2011.10929

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

Yuan Zhang·Liang-Yi Huang

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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.

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BibTeX RIS

Yuan Zhang, Liang-Yi Huang. 2020-11-22. Relations between topological and metrical properties of self-affine Sierpi$\acute{\text{n}}$ski sponges. https://arxiv.org/abs/2011.10929

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