Geodesic
Quasisymmetric rigidity of square Sierpiński carpets
Mario Bonk1 ; Sergei Merenkov2
1 University of California, Los Angeles, Los Angeles, CA
2 University of Illinois at Urbana-Champaign, Urbana, IL
Annals of mathematics, Tome 177 (2013) no. 2, pp. 591-643.

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We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpiński carpet $S_3$ is a Euclidean isometry. For carpets in a more general family, the standard $1/p$-Sierpiński carpets $S_p$, $p\ge 3$ odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that $S_p$ and $S_q$ are quasisymmetrically equivalent only if $p=q$. The main tool in the proof for these facts is a new invariant—a certain discrete modulus of a path family—that is preserved under quasisymmetric maps of carpets.
DOI : 10.4007/annals.2013.177.2.5

Mario Bonk 1 ; Sergei Merenkov 2

1 University of California, Los Angeles, Los Angeles, CA
2 University of Illinois at Urbana-Champaign, Urbana, IL 
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Mario Bonk; Sergei Merenkov. Quasisymmetric rigidity of  square Sierpiński carpets. Annals of mathematics, Tome 177 (2013) no. 2, pp. 591-643. doi : 10.4007/annals.2013.177.2.5. http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.177.2.5/

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