Comptes Rendus
Dynamical Systems
Lipschitz equivalence of self-similar sets
[Equivalence Lipschitz d'ensembles autosimilaires]
Comptes Rendus. Mathématique, Volume 342 (2006) no. 3, pp. 191-196.

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)(M/5+2/5)(M/5+4/5) et M=(M/5)(M/5+3/5)(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)(M/5+2/5)(M/5+4/5) and M=(M/5)(M/5+3/5)(M/5+4/5). We answer this question positively.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.12.016

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
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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/

[1] D. Cooper; T. Pignataro On the shape of Cantor sets, J. Differential Geom., Volume 28 (1988), pp. 203-221

[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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