Quasisymmetric rigidity of square Sierpinski carpets

Details Quasisymmetric rigidity of square Sierpinski carpets Quasisymmetric rigidity of square Sierpinski carpets

2011-02-16

arxiv.org/abs/1102.3224v1

Mathematics

Complex Variables

56 pages, 4 figures

Scientific paper

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.

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