Constructing quasisymmetric functions via graph-directed iterated function systems

Using Tukia’s method for representing a quasisymmetric function as a quasisymmetric sieve, we generalize his modification to the Salem scheme and find a sufficient condition for the collection of functions that realize a structure parametrization of a graph-directed function system of a particular form (a one-dimensional multizipper) to consist of quasisymmetric functions. We give an asymptotically sharp estimate for the quasisymmetry coefficient of these functions in terms of the dilation coefficients of the mappings constituting a given multizipper, which yields a substantially more general method for constructing quasisymmetric functions than Tukia’s construction.     

间隔序列与拟对称映射

本文研究了欧式空间上拟对称映射的不变量问题,利用定义集合间隔序列的方法,获得了一个新的d-维拟对称映射的不变量,深化了Lipschitz不变量研究中类似的结果.     

度量空间中拟对称映射与拟双曲一致域的研究

本文主要考虑度量空间中拟双曲一致域与拟对称映射之间的关系，并证明了度量空间中拟双曲一致域在拟对称映射下仍然是保持不变的.     

拟共形映射与调和函数

本文利用调和函数的Ｄｉｒｉｃｈｌｅｔ积分来估计具有给定边界值的拟共形映射的极值伸缩商,回答Ｄ．Ｐａｒｔｙｋａ关于拟对称函数谱值和特征值的一个问题。     

Sobolev classes of Banach space-valued functions and quasiconformal mappings

We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality. J. H. supported by NSF grant DMS9970427. P. K. supported by the Academy of Finland, project 39788. N. S. supported in part by Enterprise Ireland. J. T. T. supported by an NSF Postdoctoral Research Fellowship.     

A new matrix characterization of fredholm eigenvalues of quasicircles

We give a new characterization of the Fredholm eigenvalues of a quasicircle or of a quasisymmetric transformation. This leads to a matrix eigenvalue problem for a suitable Hermitian matrix. There are connections to extremal quasiconformal mappings and reflections.     

On generalized variational inequalities

We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained. Dedicated to the memory of T. Rapcsák.     

Hamilton sequences for extremal quasiconformal mappings in the unit disk

By studying the mapping by heights for quadratic differentials introduced by Strebel, some relations have been established between the maximal norm sequence for quasisymmetric functions and the Hamilton sequence for extremal quasiconformal mappings in the unit disk. Consequently it is proved that a Hamilton sequence is only determined by e quasisymmetric function. Project supported by the National Natural Science Foundation of China (Grant No. 19871002).     

Peak quasisymmetric functions and Eulerian enumeration

Via duality of Hopf algebras, there is a direct association between peak quasisymmetric functions and enumeration of chains in Eulerian posets. We study this association explicitly, showing that the notion of cd-index, long studied in the context of convex polytopes and Eulerian posets, arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by Stembridge. Thus Eulerian posets having a nonnegative cd-index (for example, face lattices of convex polytopes) correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis. We diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions, and study the g-polynomial for Eulerian posets as an algebra homomorphism.     

Mappings of Finite Distortion: Generalized Hausdorff Dimension Distortion

We study how planar Sobolev homeomorphisms distort sets of Hausdorff dimension strictly less than two. We measure the image size by means of a generalized Hausdorff measure. As an application, we obtain a sharp generalized dimension distortion estimate for mappings of exponentially integrable distortion.     

Approximation of Quasisymmetries Using Circle Packings

The Universal Teichmüller Space has recently gained the attention of physicists as a setting for developing string theory. Two common models for this space are the set of quasicircles and the set of quasisymmetries. The link between these models lies in the fact that the Riemann maps from D and D ^* to the complementary domains of quasicircles induce quasisymmetric automorphisms of . We develop a means of approximating these quasisymmetries given their associated quasicircles. This provides a concrete method for switching from the quasicircle to the quasisymmetry model of the Universal Teichmüller Space. Our approach uses discrete analytic functions induced by circle packings to approximate the Riemann maps. Received May 19, 1999, and in revised form April 6, 2000. Online publication September 22, 2000.     

Metric Subregularity of Composition Set-Valued Mappings with Applications to Fixed Point Theory

In this paper we underline the importance of the parametric subregularity property of set-valued mappings, defined with respect to fixed sets. We show that this property appears naturally for some very simple mappings which play an important role in the theory of metric regularity. We prove a result concerning the preservation of metric subregularity at generalized compositions. Then we obtain, in purely metric setting, several fixed point assertions for set-valued mappings in local and global frameworks.     

Generalized differentiation and fixed points sets behaviors with respect to Fisher convergence

We study the stability of a new concept of generalized differentiation for set-valued mappings involving positively homogeneous maps with respect to a variational convergence deriving from the proximal topology. This notion of convergence, well-suited for set-valued mappings, allows us as well to establish the continuous dependence of fixed points sets of set-valued contractions.     

Multiplicity Free Schur, Skew Schur, and Quasisymmetric Schur Functions

In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape have distinct descent sets. We conclude by providing such a classification for quasisymmetric Schur function families, giving a classification of Schur functions that are in some sense almost F-multiplicity free.     

Boundary Dilatation and Degenerating Hamilton Sequences

In this paper, we study the boundary dilatation of quasiconformal mappings in the unit disc. By using Strebel mapping by heights theory we show that a degenerating Hamilton sequence is determined by a quasisymmetric function.     

齐次集及其拟Lipschitz等价

本文定义并研究一类齐次分形,该类分形包含所有的(拟)Ahlfors-David正则集和许多非正则的Moran集,这里如果一个分形的Hausdorff维数与packing维数不相等,则称它是非正则的.对于这类齐次分形,本文得出它们的分形维数,并且给出在适当分离条件下两个齐次分形拟Lipschitz等价的充要条件.随后,本文将这些结果应用到非正则的Moran集上.     

On a conjecture of S. Reich

Simeon Reich (1974) proved that the fixed point theorem for single-valued mappings proved by Boyd and Wong can be generalized to multivalued mappings which map points into compact sets. He then asked (1983) whether his theorem can be extended to multivalued mappings whose range consists of bounded closed sets. In this note, we provide an affirmative answer for a certain subclass of Boyd-Wong contractive mappings.

     

Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings

We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of Rn affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.     

Analytic families of semihyperbolic generalized polynomial-like mappings

We show that the Hausdorff dimension of Julia sets in any analytic family of semihyperbolic generalized polynomial-like mappings (GPL) depends in a real-analytic manner on the parameter. For the proof we introduce abstract weakly regular analytic families of conformal graph directed Markov systems. We show that the Hausdorff dimension of limit sets in such families is real-analytic, and we associate to each analytic family of semihyperbolic GPLs a weakly regular analytic family of conformal graph directed Markov systems with the Hausdorff dimension of the limit sets equal to the Hausdorff dimension of the Julia sets of the corresponding semihyperbolic GPLs.