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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Lipschitz equivalence of Cantor sets and algebraic properties of contraction ratios


Authors: Hui Rao, Huo-Jun Ruan and Yang Wang
Journal: Trans. Amer. Math. Soc. 364 (2012), 1109-1126
MSC (2010): Primary 28A80
Published electronically: October 13, 2011
MathSciNet review: 2869169
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Abstract: In this paper we investigate the Lipschitz equivalence of dust-like self-similar sets in $ \mathbb{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.


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Additional Information

Hui Rao
Affiliation: Department of Mathematics, Hua Zhong Normal University, Wuhan 430079, People’s Republic of China
Email: hrao@mail.ccnu.edu.cn

Huo-Jun Ruan
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China – and – Department of Mathematics, Cornell University, Ithaca, New York 14853
Email: ruanhj@zju.edu.cn

Yang Wang
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: ywang@math.msu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05327-4
PII: S 0002-9947(2011)05327-4
Keywords: Lipschitz equivalence, dust-like self-similar sets, matchable condition, algebraic rank, uniform contraction ratio
Received by editor(s): December 8, 2009
Published electronically: October 13, 2011
Additional Notes: The research of the first author was supported by the NSFC grants 10971013 and 11171128.
The research of the second author, who is the corresponding author, was supported in part by the NSFC grant 10601049 and by the Future Academic Star project of Zhejiang University.
The research of the third author was supported in part by NSF Grant DMS-0813750.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.