Self-similar sets with initial cubic patterns

Xi, Li-Feng; Xiong, Ying

doi:10.1016/j.crma.2009.12.006

Mathematical Analysis

Self-similar sets with initial cubic patterns
[Ensembles auto-similaires avec motifs initiaux cubiques]

Présenté par : Jean-Pierre Kahane

Xi, Li-Feng ¹ ; Xiong, Ying ²

¹ Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, PR China
² Department of Mathematics, South China University of Technology, Guangzhou 510641, PR China

Comptes Rendus. Mathématique, Tome 348 (2010) no. 1-2, pp. 15-20.

Résumé
Abstract

Si $A \subset {0, \dots, n - 1}^{m}$ , soit $E_{A}$ l'unique compact non vide de $R^{m}$ tel que $E_{A} = ⋃_{a \in A} (\frac{1}{n} E_{A} + \frac{a}{n})$ . Nous montrons que deux tels ensembles auto-similaires totalement discontinus $E_{A}$ et $E_{B}$ (avec $A, B \subset {0, \dots, n - 1}^{m}$ ) sont lipschitziennement équivalents si et seulement si $# A = # B$ .

For $A \subset {0, \dots, n - 1}^{m}$ , let $E_{A}$ be the unique nonempty compact subset of $R^{m}$ such that $E_{A} = ⋃_{a \in A} (\frac{1}{n} E_{A} + \frac{a}{n})$ . We show that two such self-similar sets $E_{A}$ and $E_{B}$ (for $A, B \subset {0, \dots, n - 1}^{m}$ ), supposed to be totally disconnected, are Lipschitz equivalent if and only if $# A = # B$ .

Reçu le : 2009-08-22
Accepté le : 2009-11-30
Publié le : 2009-12-24

DOI : 10.1016/j.crma.2009.12.006

Affiliations des auteurs :

Xi, Li-Feng ¹ ; Xiong, Ying ²

¹ Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, PR China
² Department of Mathematics, South China University of Technology, Guangzhou 510641, PR China

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     title = {Self-similar sets with initial cubic patterns},
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Xi, Li-Feng; Xiong, Ying. Self-similar sets with initial cubic patterns. Comptes Rendus. Mathématique, Tome 348 (2010) no. 1-2, pp. 15-20. doi : 10.1016/j.crma.2009.12.006. http://www.numdam.org/articles/10.1016/j.crma.2009.12.006/

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