Lipschitz equivalence of self-similar sets

Hui Rao; Huo-Jun Ruan; Li-Feng Xi

doi:10.1016/j.crma.2005.12.016

Dynamical Systems

Lipschitz equivalence of self-similar sets
[Equivalence Lipschitz d'ensembles autosimilaires]

Présenté par : Jean-Pierre Kahane

Hui Rao ¹ ; Huo-Jun Ruan ² ; Li-Feng Xi ³

¹ Department of Mathematics, Tsinghua University, Beijing, 100084, China
² Department of Mathematics, Zhejiang University, Hangzhou, 310027, China
³ Institute of Mathematics, Zhejiang Wanli University, Ningbo, 315100, China

Comptes Rendus. Mathématique, Volume 342 (2006) no. 3, pp. 191-196.

Résumé
Abstract

En 1997, David et Semmes ont posé la question de savoir s'il existe une application bi-lipschitzienne entre les deux compacts linéaires M et $M^{'}$ définis par les relations $M = (M / 5) \cup (M / 5 + 2 / 5) \cup (M / 5 + 4 / 5)$ et $M^{'} = (M^{'} / 5) \cup (M^{'} / 5 + 3 / 5) \cup (M^{'} / 5 + 4 / 5)$ . Nous répondons affirmativement à cette question.

In 1997 David and Semmes asked whether there exists a bilipschitz map between the two compact self-similar subset M and $M^{'}$ of the real line defined by the relations $M = (M / 5) \cup (M / 5 + 2 / 5) \cup (M / 5 + 4 / 5)$ and $M^{'} = (M^{'} / 5) \cup (M^{'} / 5 + 3 / 5) \cup (M^{'} / 5 + 4 / 5)$ . We answer this question positively.

Reçu le : 2005-11-23
Accepté le : 2005-12-07
Publié le : 2006-01-05

DOI : 10.1016/j.crma.2005.12.016

Affiliations des auteurs :

Hui Rao ¹ ; Huo-Jun Ruan ² ; Li-Feng Xi ³

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     title = {Lipschitz equivalence of self-similar sets},
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Hui Rao; Huo-Jun Ruan; Li-Feng Xi. Lipschitz equivalence of self-similar sets. Comptes Rendus. Mathématique, Volume 342 (2006) no. 3, pp. 191-196. doi : 10.1016/j.crma.2005.12.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.12.016/

Bibliographie
Cité par

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[2] G. David; S. Semmes Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure, Oxford Univ. Press, 1997

[3] K.J. Falconer; D.T. Marsh Classification of quasi-circles by Hausdorff dimension, Nonlinearity, Volume 2 (1989), pp. 489-493

[4] K.J. Falconer; D.T. Marsh On the Lipschitz equivalence of Cantor sets, Mathematika, Volume 39 (1992), pp. 223-233

[5] J.E. Hutchinson Fractals and self similarity, Indiana Univ. Math. J., Volume 30 (1981), pp. 713-747

[6] R.D. Mauldin; S.C. Williams Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., Volume 309 (1988), pp. 811-829

[7] H. Rao; Z.-Y. Wen A class of self-similar fractals with overlap structure, Adv. Appl. Math., Volume 20 (1998), pp. 50-72

[8] Z.-Y. Wen; L.-F. Xi Relations among Whitney sets, self-similar arcs and quasi-arcs, Israel J. Math., Volume 136 (2003), pp. 251-267

[9] L.-F. Xi Lipschitz equivalence of self-conformal sets, J. London Math. Soc., Volume 70 (2004) no. 2, pp. 369-382

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