方形Sierpinski地毯的拟对称刚度
Quasisymmetric rigidity of square Sierpinski carpets
摘要：
证明了标准1/3-Sierpienski地毯$S_3$的每一个拟对称自同胚都是欧氏等距.对于地毯在一个更一般的家庭，标准的$1/p$ -Sierpienski地毯$S_p$，$p\ge 3$奇数，我们表明，集团的拟对称自映射是有限的二面角。我们还建立了$S_p$和$S_q$是拟对称等价的，仅当$p=q$。在证明这些事实的主要工具是一个新的不变量-一定的离散模的路径家庭-这是保持下拟对称地图的地毯。
英文摘要：
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.
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