Dynamical Systems
Lipschitz equivalence of self-similar sets
Comptes Rendus. Mathématique, Volume 342 (2006) no. 3, pp. 191-196.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.12.016
Rao, Hui 1; Ruan, Huo-Jun 2; Xi, Li-Feng 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
@article{CRMATH_2006__342_3_191_0,
     author = {Rao, Hui and Ruan, Huo-Jun and Xi, Li-Feng},
     title = {Lipschitz equivalence of self-similar sets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {191--196},
     publisher = {Elsevier},
     volume = {342},
     number = {3},
     year = {2006},
     doi = {10.1016/j.crma.2005.12.016},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2005.12.016/}
}
TY  - JOUR
AU  - Rao, Hui
AU  - Ruan, Huo-Jun
AU  - Xi, Li-Feng
TI  - Lipschitz equivalence of self-similar sets
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 191
EP  - 196
VL  - 342
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2005.12.016/
DO  - 10.1016/j.crma.2005.12.016
LA  - en
ID  - CRMATH_2006__342_3_191_0
ER  - 
%0 Journal Article
%A Rao, Hui
%A Ruan, Huo-Jun
%A Xi, Li-Feng
%T Lipschitz equivalence of self-similar sets
%J Comptes Rendus. Mathématique
%D 2006
%P 191-196
%V 342
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2005.12.016/
%R 10.1016/j.crma.2005.12.016
%G en
%F CRMATH_2006__342_3_191_0
Rao, Hui; Ruan, Huo-Jun; Xi, Li-Feng. 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. http://www.numdam.org/articles/10.1016/j.crma.2005.12.016/

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

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

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

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

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

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

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

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

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

Cited by Sources: