Quasisymmetric rigidity of square Sierpinski carpets

By M. Bonk, S. Merenkov. 2011

96 citations recorded by the source corpus.

We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpi\'nski carpet $S_3$ is a Euclidean isometry. For carpets in a more general family, the standard $1/p$-Sierpi\'nski 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/10044049b3a7d80b06cac0b52ca23b35c6c2cafc

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Uniformization of Sierpiński carpets in the plane (2010, background): “Applications of the carpet modulus to proving rigidity statements for Sierpiński carpets are studied in [BM].”