Lipschitz equivalence of cantor sets and algebraic properties of contraction ratios

Abstract

In this paper we investigate the Lipschitz equivalence of dust-like self-similar sets in ℝ^d. One of the fundamental results by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223-233] establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. In this paper we extend the study by examining deeper such connections. A key ingredient of our study is the introduction of a new equivalent relation between two dust-like self-similar sets called a matchable condition. Thanks to a certain measure-preserving property of bi-Lipschitz maps between dust-like self-similar sets, we show that the matchable condition is a necessary condition for Lipschitz equivalence. Using the matchable condition we 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. We also completely characterize the Lipschitz equivalence of dust-like self-similar sets with two branches (i.e., they are generated by IFS with two contractive similarities). Some other results are also presented, including a complete characterization of Lipschitz equivalence when one of the self-similar sets has uniform contraction ratio. © 2011 American Mathematical Society.