LIPSCHITZ EQUIVALENCE OF CANTOR SETS AND ALGEBRAIC PROPERTIES OF CONTRACTION RATIOS

作者：Rao, Hui; Ruan, Huo-Jun^*; Wang, Yang

来源：Transactions of the American Mathematical Society, 2012, 364(3): 1109-1126.

DOI：10.1090/S0002-9947-2011-05327-4

• Lipschitz equivalence
• dust-like self-similar sets
• matchable condition
• algebraic rank
• uniform contraction ratio

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.

• 出版日期2012-3
• 单位华中师范大学; 浙江大学