Quasisymmetric conjugacies between unimodal maps (original) (raw)

A note on quasisymmetric homeomorphisms

Annales Academiae Scientiarum Fennicae Mathematica, 2020

By means of some integral operators and kernel functions, we characterize when a sense preserving homeomorphism h on the unit circle S 1 is quasisymmetric, symmetric or pintegrable asymptotic affine. As an application, we use these results to characterize the pull-back operator, induced by a quasisymmetric homeomorphism on S 1. 1 π¨∆ φ h (ζ, z)ψ(z) dx dy, ζ ∈ ∆.

Dilatations and exponents of quasisymmetric homeomorphisms

Annales Academiae Scientiarum Fennicae Mathematica, 2016

Given a quasisymmetric homeomorphism, we introduce the concept of quasisymmetric exponent and explore its relations to other conformal invariants. As a consequence, we establish a necessary and sufficient condition on the equivalence of the dilatation and the maximal dilatation of a quasisymmetric homeomorphism by using the quasisymmetric exponent. A classification on the elements of the universal Teichmüller space is obtained by using this necessary and sufficient condition.

Conjugacy of real diffeomorphisms. A survey

St. Petersburg Mathematical Journal, 2011

Given a group G, the conjugacy problem in G is the problem of giving an effective procedure for determining whether or not two given elements f, g ∈ G are conjugate, i. e., whether there exists h ∈ G with f h = hg. This paper is about the conjugacy problem in the group Diffeo(I) of all diffeomorphisms of an interval I ⊂ R. There is much classical work on the subject, solving the conjugacy problem for special classes of maps. Unfortunately, it is also true that many results and arguments known to the experts are difficult to find in the literature, or simply absent. We try to repair these lacunae, by giving a systematic review, and we also include new results about the conjugacy classification in the general case. §1. Informal introduction 1.1. Objective. We are going to work with diffeomorphisms defined just on various intervals (open, closed, or half-open, bounded or unbounded). Let Diffeo(I) denote the group of (infinitely differentiable) diffeomorphisms of the interval I ⊂ R, under the operation of composition. We denote the (normal) subgroup of orientation-preserving diffeomorphisms of the interval by Diffeo + (I). If an endpoint c belongs to I, then statements about derivatives at c should be interpreted as referring to one-sided derivatives. Our objective is to classify the conjugacy classes, i.e., to determine when two given maps f and g are conjugate in Diffeo(I). The reason this problem is important is that conjugate elements correspond to one another under a "change of variables". For most applications, a change of variables will not alter anything essential, so only the conjugacy class of an element is significant. From the viewpoint of group theorists, it is also usual to regard only the conjugacy classes as having "real" meaning in a group. Throughout the paper, we will use the term smooth to mean infinitely differentiable. There is a good deal of valuable and delicate work on conjugacy problems for functions that are merely C k , but we will not delve into this (apart from an occasional remark), in order to keep the discussion within bounds. Apart from its intrinsic interest, the conjugacy problem has applications to the holonomy theory of codimension-one foliations. Mather established a connection between the homotopy of Haefliger's classifying space for foliations and the cohomology of the group G of compactly supported diffeomorphisms of the line [15, 16]. Mather also used a conjugacy classification of a subset of the group G in order to establish that G is perfect.

On the quasisymmetry of quasiconformal mappings and its applications

2012

Abstract: Suppose that $ D $ is a proper domain in IRn\ IR^ n IRn and that $ f $ is a quasiconformal mapping from $ D $ onto a John domain $ D'$ in IRn\ IR^ n IRn. First, we show that if $ D $ and $ D'$ are bounded, and $ D $ is a broad domain, then for an arcwise connected subset $ A $ in $ D ,, , f (A) $ is $ LLC_2 $ with respect to deltaD′\ delta_ {D'} deltaD′ in $ D'$ if and only if the restriction $ f| _A: A\ to f (A) $ is quasisymmetric in the metrics deltaD\ delta_D deltaD and deltaD′\ delta_ {D'} deltaD′.

Criteria of convergence for quasiconformal mappings and their generalizations

Ukrainian Mathematical Journal, 1996

We establish necessary and sufficient conditions for the convergence of normalized homeomorphisms of Sobolev class in terms of the Fourier transforms of complex characteristics in the case where the upper bound of dilations is exponentially bounded in measure. This allows us to construct various metrics generating locally uniform convergence of mappings.

Quasiconformal Extensions of Quasisymmetric Mappings

2003

W hen loo k ing at mappings of the plane, the nicest mappings one can imagine are conformal mappings. That is mappings that preserve angles bet w een lines. These mappings are really the comple x analytic functions that have many nice properties. H ow ever they also have many restrictions. For ex ample we cannot conformally map a rectangle onto another rectangle w ith a diff erent aspect ratio w hile mapping corners to corners.