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Sharp Lipschitz constant of bi-Lipschitz automorphism on Cantor set

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【Author】 XIONG Ying1, WANG LiSha2 & XI LiFeng3 1 Department of Mathematics, South China University of Technology, Guangzhou 510641, China 2 Department of Mathematics, Hubei University, Wuhan 430062, China 3 Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China

【摘要】 Suppose Cr = (rCr) ∪ (rCr + 1 - r) is a self-similar set with r ∈ (0, 1/2), and Aut(Cr) is the set of all bi-Lipschitz automorphisms on Cr. This paper proves that there exists f* ∈ Aut(Cr) such that blip(f*) = inf{blip(f) > 1 : f ∈ Aut(Cr)} = min 1r , (1 -1 2 -r) 2(r13 + - r r +4 r2) , where lip(g) = supx,y∈Cr, x=y |g(x|x)--yg(| y)|and blip(g) = max(lip(g), lip(g-1)).

【Abstract】 Suppose Cr = (rCr) ∪ (rCr + 1 - r) is a self-similar set with r ∈ (0, 1/2), and Aut(Cr) is the set of all bi-Lipschitz automorphisms on Cr. This paper proves that there exists f* ∈ Aut(Cr) such that blip(f*) = inf{blip(f) > 1 : f ∈ Aut(Cr)} = min 1r , (1 -1 2 -r) 2(r13 + - r r +4 r2) , where lip(g) = supx,y∈Cr, x=y |g(x|x)--yg(| y)|and blip(g) = max(lip(g), lip(g-1)).

【关键词】 fractal； bi-Lipschitz automorphism； Cantor set；
【Key words】 fractal； bi-Lipschitz automorphism； Cantor set；
【基金】 supported by National Natural Science Foundation of China (Grant Nos. 10671180, 10571140,10571063, 10631040, 11071164);Morningside Center of Mathematics
  • 【文献出处】 Science in China(Series A:Mathematics) ,中国科学(A辑:数学)(英文版) , 编辑部邮箱 ,2009年04期
  • 【分类号】O415.5
  • 【被引频次】1
  • 【下载频次】31
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