A class of quasisymmetric mappings with substantial points

It is proved that Kq(h)=K0(h) for every h in some class of quasisymmetric mappings of the unit circle with substantial points, where Kq(h):=sup{M(h(Q))/M(Q); Qis a quadrilateral with the domain unit disk} and K0(h) is the extremal maximum dilatation of h.

Quasisymmetric boundary correspondence and Grtzsch problem

Two necessary and sufficient conditions for the validity of the conjecture K 0(h)=K 1(h) are given, which are independent of the complex dilatations of extremal quasiconformal mappings, where K 0(h) is the maximal conformal modulus dilatation of the boundary homeomorphism h, K 1(h) is the maximal dilatation of extremal quasiconformal mappings that agree with h on the boundary. In addition, when the complex dilatation of an extremal quasiconformal mapping is known, the proof of the result simplifies Reich and Chen Jixiu-Chen Zhiguo’s result.

Quasisymmetric Mappings on Moran Sets

This paper studies the quasisymmetric mappings on Moran sets. We introduce a gener- alized form of weak quasisymmetry and prove that, on Moran set satisfying the small gap condition, a generalized weakly quasisymmetric mapping is quasisymmetric. We further give a criterion for the quasisymmetry of mappings between Moran sets with some regular structure.

Hausdorff Dimensions of Quasilines and Differentiability of Quasisymmetric Homeomorphisms

In this paper,we try to describe the relationship between the differentiability of a quasisymmetric homeomorphism and the local Hausdorff dimension of the quasiline at a point.

Quasisymmetric geometry of the Julia sets of McMullen maps Dedicated to the Memory of Professor Lei Tan

We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z^m+ λ/z~?,where λ∈ C \ {0} and ? and m are positive integers satisfying 1/? + 1/m < 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet(which is also standard in some sense).If the free critical points are not escaped, we give a sufficient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.

Quadrilaterals, extremal quasiconformal extensions and Hamilton sequences

The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrigues many mathematicians. It had been a conjecture for some time that the dilatations Ko（h） and K1（h） of h are equal before Anderson and Hinkkanen disproved this by constructing concrete counterexamples. The independent work of Wu and of Yang completely characterizes the condition for Ko（h） = K1 （h） when h has no substantial boundary point. In this paper, we give a necessary and sufficient condition to determine the equality for h admitting a substantial boundary point.

Estimation for the Dilatation of the Generalized Beurling-Ahlfors' Extension

The Beurling-Ahlfors’ extension is studied under relatively general conditions and its dilatation fonction is estimated. Particularly, the classic Deurling -Ahlfors’ theorem can be obtained under the M-condition.

A remark on "an approximation condition and extremal quasiconformal extension

LET D be the unit disk in the complex plane C and f be a sense preserving quasisymmetrichomeomorphism of D onto itself. Denote by Q a quadrilateral D （z1, z2, z3, z4 ） with do-main D and vertices z1, z2, z3, z4 ∈D, and by M（Q） its conformal modulus. We are in-

A counterexample theorem in quasiconformal mapping theory

Given a quasisymmetric homeomorphismh of the unit circle onto itself, denote byK n * ,H h andK h the extremal maximal dilatation, boundary dilatation and maximal dilatation ofh, respectively. It is proved that there exists a family of quasisymmetric homeomorphismsh such thatK h <H h =K h * This gives a negative answer to a problem asked independently by Wu and Yang. Furthermore, some related topics are also discussed.

A problem in extremal quasiconformal extensions

A constantK 0 (m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK 0(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established forK 0 (m) (h) =K 1(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK 0 (m) (h)<K 1(h), wherek 1(h) is the maximal dilatation ofh.

Kobayashi’s and Teichmuller’s Metrics and Bers Complex Manifold Structure on Circle Diffeomorphisms

Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm(u| ")ller space TC^1+H as the union of over all0 <α≤1,which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC^1+H ,Kobayashi’s metric and Teichmuller’s metric coincide.

Generalized Beurling-Ahlfors'Theorem

A generalized Beurling-Ahlfors’ Theorem for the self homeomorphism f of the upper half plane with the sphere dilatation H(z,f) L (H) is established and the property of weighted quasi-isometry for the generalized Beurling-Ahlfors’ extension is studied.

Non-Strebel points and variability sets

This paper studies the subset of the non-Strebel points in the universal Teichmüller spaceT. Let z0 ∈ Δ be a fixed point. Then we prove that for every non-Strebel pointh, there is a holomorphic curve γ: [0, 1] →T withh as its initial point satisfying the following conditions. (1) The curve γ is on a sphere centered at the base-point ofT, i.e.d T (id, γ(t))=d T (id, h), (t∈[0, 1]). (2) For everyt ∈ (0,1], the variability set Vγ(t)[z0] of γ(t) has non-empty interior, i.e. .

Relative Quasisymmetry and Quasim?bius Mappings in Quasi-metric Spaces

The aim of this paper is to investigate the relationship between relative quasisymmetry and quasimöbius in quasi-metric spaces,and show that a homeomorphism f isη-quasisymmetric relative to A if and only if it isθ-quasimöbius relative to A between two both bounded quasi-metric spaces,where A⊆X and X is a quasi-metric space.

The cross-ratio distortion of integrably asymptotic affine homeomorphism of unit circle

Let QS* (S 1) be the space of quasisymmetric homeomorphisms of the unit circle such that the corresponding subspace of the universal Teichmu¨ller space has Weil-Petersson metric.In this paper we give a necessary condition for a quasisymmetric homeomorphism to belong to QS *(S 1) from the aspect of cross-ratio distortion.