A metric space （X, d） is called bi-Lipschitz homogeneous if for any points x, y ∈X, there exists a self-homeomorphism h of X such that both h and h-1 are Lipschitz and h（x） = y. Let 2（x,d） denote the family of ...A metric space （X, d） is called bi-Lipschitz homogeneous if for any points x, y ∈X, there exists a self-homeomorphism h of X such that both h and h-1 are Lipschitz and h（x） = y. Let 2（x,d） denote the family of all non-empty compact subsets of metric space （X, d） with the Hausdorff metric. In 1985, Hohti proved that 2（[0,1],d） is not bi-Lipschitz homogeneous, where d is the standard metric on [0, 1]. We extend this result in two aspects. One is that 2（[0,1],e ） is not bi-Lipschitz homogeneous for an admissible metric Q satisfying some conditions. Another is that 2（X,d） is not bi-Lipschitz homogeneous if （X, d） has a nonempty open subspace which is isometric to an open subspace of m-dimensional Euclidean space R^m.展开更多
Suppose C r = (r C r ) ∪ (r C r + 1 ? r) is a self-similar set with r ∈ (0, 1/2), and Aut(C r ) is the set of all bi-Lipschitz automorphisms on C r . This paper proves that there exists f* ∈ Aut(C r ) such that $$ ...Suppose C r = (r C r ) ∪ (r C r + 1 ? r) is a self-similar set with r ∈ (0, 1/2), and Aut(C r ) is the set of all bi-Lipschitz automorphisms on C r . This paper proves that there exists f* ∈ Aut(C r ) such that $$ blip(f*) = inf\{ blip(f) > 1:f \in Aut(C_r )\} = min\left[ {\frac{1} {r},\frac{{1 - 2r^3 - r^4 }} {{(1 - 2r)(1 + r + r^2 )}}} \right], $$ where $ lip(g) = sup_{x,y \in C_r ,x \ne y} \frac{{\left| {g(x) - g(y)} \right|}} {{\left| {x - y} \right|}} $ and blip(g) = max(lip(g), lip(g ?1)).展开更多
For a given self-similar set ERd satisfying the strong separation condition,let Aut(E) be the set of all bi-Lipschitz automorphisms on E.The authors prove that {fAut(E):blip(f)=1} is a finite group,and the gap propert...For a given self-similar set ERd satisfying the strong separation condition,let Aut(E) be the set of all bi-Lipschitz automorphisms on E.The authors prove that {fAut(E):blip(f)=1} is a finite group,and the gap property of bi-Lipschitz constants holds,i.e.,inf{blip(f)=1:f∈Aut(E)}>1,where lip(g)=sup x,y∈E x≠y(|g(x)-g(y)|)/|x-y| and blip(g)=max(lip(g),lip(g-1)).展开更多
This paper is devoted to the study of some fundamental properties of the sewing home- omorphism induced by a Jordan domain. In particular, using conformal invariants such as harmonic measure, extremal distance, and re...This paper is devoted to the study of some fundamental properties of the sewing home- omorphism induced by a Jordan domain. In particular, using conformal invariants such as harmonic measure, extremal distance, and reduced extremal distance, we give several necessary and sumcient conditions for the sewing homeomorphism to be bi-Lipschitz or bi-Holder. Furthermore, equivalent conditions for a Jordan curve to be a quasicircle are also obtained.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.10971125)
文摘A metric space （X, d） is called bi-Lipschitz homogeneous if for any points x, y ∈X, there exists a self-homeomorphism h of X such that both h and h-1 are Lipschitz and h（x） = y. Let 2（x,d） denote the family of all non-empty compact subsets of metric space （X, d） with the Hausdorff metric. In 1985, Hohti proved that 2（[0,1],d） is not bi-Lipschitz homogeneous, where d is the standard metric on [0, 1]. We extend this result in two aspects. One is that 2（[0,1],e ） is not bi-Lipschitz homogeneous for an admissible metric Q satisfying some conditions. Another is that 2（X,d） is not bi-Lipschitz homogeneous if （X, d） has a nonempty open subspace which is isometric to an open subspace of m-dimensional Euclidean space R^m.
基金supported by National Natural Science Foundation of China (Grant Nos. 10671180, 10571140,10571063, 10631040, 11071164)Morningside Center of Mathematics
文摘Suppose C r = (r C r ) ∪ (r C r + 1 ? r) is a self-similar set with r ∈ (0, 1/2), and Aut(C r ) is the set of all bi-Lipschitz automorphisms on C r . This paper proves that there exists f* ∈ Aut(C r ) such that $$ blip(f*) = inf\{ blip(f) > 1:f \in Aut(C_r )\} = min\left[ {\frac{1} {r},\frac{{1 - 2r^3 - r^4 }} {{(1 - 2r)(1 + r + r^2 )}}} \right], $$ where $ lip(g) = sup_{x,y \in C_r ,x \ne y} \frac{{\left| {g(x) - g(y)} \right|}} {{\left| {x - y} \right|}} $ and blip(g) = max(lip(g), lip(g ?1)).
基金supported by the National Natural Science Foundation of China (Nos.10671180,10571140,10571063,10631040,11071164) and the Morningside Center of Mathematics
文摘For a given self-similar set ERd satisfying the strong separation condition,let Aut(E) be the set of all bi-Lipschitz automorphisms on E.The authors prove that {fAut(E):blip(f)=1} is a finite group,and the gap property of bi-Lipschitz constants holds,i.e.,inf{blip(f)=1:f∈Aut(E)}>1,where lip(g)=sup x,y∈E x≠y(|g(x)-g(y)|)/|x-y| and blip(g)=max(lip(g),lip(g-1)).
基金The first author is partially supported by National Natural Science Foundation of China(Grant Nos.11371268and 11471117)Science and Technology Commission of Shanghai Municipality(Grant No.13dz2260400)+1 种基金the third author is partially supported by National Natural Science Foundation of China(Grant No.11471117)by PERS of Emory
文摘This paper is devoted to the study of some fundamental properties of the sewing home- omorphism induced by a Jordan domain. In particular, using conformal invariants such as harmonic measure, extremal distance, and reduced extremal distance, we give several necessary and sumcient conditions for the sewing homeomorphism to be bi-Lipschitz or bi-Holder. Furthermore, equivalent conditions for a Jordan curve to be a quasicircle are also obtained.
