Lipschitz equivalence of Cantor sets and algebraic properties of contraction ratios (Q2880674)

The authors investigate the Lipschitz equivalence of dust-like self-similar sets in \({\mathbb{R}}^d\). One of the fundamental results by Falconer and Marsh establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. The authors extend the study by examining deeper such connections. A key ingredient of their study is the introduction of a new equivalent relation between two dust-like self-similar sets, called a \textit{matchable} condition. They show that the matchable condition is a necessary condition for Lipschitz equivalence. Using the matchable condition, they prove several conditions on the Lipschitz equivalence of dust-like self-similar sets based on the algebraic properties of the contraction ratios, which include a complete characterization of Lipschitz equivalence when the multiplication groups generated by the contraction ratios have full rank. They also completely characterize the Lipschitz equivalence of dust-like self-similar sets with two branches (i.e., the sets generated by IFS with two contractive similarities). Some other results were also presented including a complete characterization of Lipschitz equivalence when one of the self-similar sets has uniform contraction ratio.