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| Gap Property of Bi-Lipschitz Constants of Bi-Lipschitz Automorphisms on Self-similar Sets |
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Citation： |
Lifeng XI,Ying XIONG.Gap Property of Bi-Lipschitz Constants of Bi-Lipschitz Automorphisms on Self-similar Sets[J].Chinese Annals of Mathematics B,2010,31(2):211~218 |
| Page view： 1832
Net amount： 1770 |
Authors： |
Lifeng XI; Ying XIONG; |
Foundation： |
the National Natural Science Foundation of China (No. 10671180, 10571140,
10571063, 10631040, 11071164) and the Morningside Center of Mathematics. |
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| Abstract： |
For a given self-similar set $E\subset \mathbb{R}^{d}$ satisfying
the strong separation condition, let $\Aut(E)$ be the set of all
bi-Lipschitz automorphisms on $E.$ The authors prove that $\{f\in
\Aut(E):{\rm blip}(f)=1\}$ is a finite group, and the gap property
of bi-Lipschitz constants holds, i.e., $ \inf \{{\rm blip}(f)\neq
1\!:\!f\in \Aut(E)\}>1,$ where ${\rm lip}(g)=\sup\limits_{x,y\in E
\atop x\neq y}\frac{|g(x)-g(y)|}{|x-y|}$ and $ {\rm blip}(g)=\max
({\rm lip}(g),{\rm lip}(g^{-1})).$ |
Keywords： |
Fractal, Bi-Lipschitz automorphism, Self-similar set |
Classification： |
28A80 |
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