From Lipschitz embedding to Lipschitz equivalence between dust-like self-similar sets

TL;DR

This paper investigates conditions under which Lipschitz embeddings between dust-like self-similar sets imply Lipschitz equivalence, revealing the roles of measure intersection, homogeneity, and algebraic dependence of scaling ratios.

Contribution

It establishes new criteria linking Lipschitz embeddings to Lipschitz equivalence for self-similar sets, especially highlighting the importance of measure intersection and algebraic dependence.

Findings

Lipschitz image intersection with positive Hausdorff measure implies Lipschitz surjection.

Homogeneity of the set leads to algebraic dependence of scaling ratios and Lipschitz equivalence.

Lipschitz equivalence can fail without the homogeneity assumption.

Abstract

Let $K,F\subset\mathbb{R}^d$ be two dust-like self-similar sets sharing the same Hausdorff dimension. We consider when the mere existence of a Lipschitz embedding from $K$ to $F$ already implies their Lipschitz equivalence. Our main result is threefold: (1) if the Lipschitz image of $K$ intersects $F$ in a set of positive Hausdorff measure, then $K$ admits a Lipschitz surjection onto $F$; (2) if $F$ is in addition homogeneous, then the generating iterated function systems of $K, F$ should have algebraically dependent ratios and consequently, $K$ and $F$ are Lipschitz equivalent; (3) the Lipschitz equivalence can fail without the homogeneity assumption. This answers two questions in Balka and Keleti [Adv. Math. 446 (2024), 109669].