A note on quasisymmetric homeomorphisms (original) (raw)

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.

On Uniformly Quasisymmetric Groups of Circle Diffeomorphisms

2006

This article deals with the conjugacy problem of uniformly quasisymmetric groups of circle homeomorphims to groups of Mobius transformations. We prove that if the involved maps have some degree of regularity and the uniform quasisymmetry can be detected by some natural L 1 -cocycle associated to the action, then the conjugacy is, in fact, smooth.

A note to "Radial limits of quasiregular Local Homeomorphisms

arXiv (Cornell University), 2023

In this short note, we consider quasiregular local homeomorphisms on uniform domains. We prove that such mappings always can be extended to some boundary points along John curves, which extends the corresponding result of Rajala [Amer. J. Math. 2008].

On the Regularity of the Displacement Sequence of an Orientation Preserving Circle Homeomorphism

2015

We investigate the regularity properties of the displacement sequence () () () (), 2 exp , 1 mod 1 ix z x x z n n n π = Φ − Φ = η − where R R → Φ : is a lift of an orientation preserving circle homeomorphism. If the rotation number () q p = ϕ is rational, then () z n η is asymptotically periodic with semi-period q. This WACŁAW MARZANTOWICZ and JUSTYNA SIGNERSKA 12 convergence to a periodic sequence is uniform in z if we admit that some points are iterated backward instead of taking only forward iterations for all z. This leads to the notion of an basins'-ε edge, which we illustrate by the numerical example. If () , Q ∈ / ϕ then some classical results in topological dynamics yield that the displacement sequence also exhibits some regularity properties, which we define and prove in the second part of the paper.

Real bounds and quasisymmetric rigidity of multicritical circle maps

Transactions of the American Mathematical Society, 2017

Let f, g : S 1 → S 1 be two C 3 critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if h : S 1 → S 1 is a topological conjugacy between f and g and h maps the critical points of f to the critical points of g, then h is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T. Clark and S. van Strien [5]. However, unlike the proof given in [5], which relies on heavy complex-analytic machinery, our proof uses purely realvariable methods, and is valid for non-integer critical exponents as well. We do not require h to preserve the power-law exponents at corresponding critical points.

Quasisymmetric Structure and Quasisymmetry on 1-manifolds

2007

The purpose of this paper is to study the group QS 2 of quasisymmetric homeomorphisms of a one-dimensional manifold J with respect to a given quasisymmetric structure on J. Under a natural neighborhood system of the identity, the group QS 2 is a partial topological group. Its characteristic topological subgroup is identified as the collection of all elements in QS 2 with vanishing ratio distortion. When J is a Jordan curve in the plane, denote the group of quasisymmetric homeomorphisms of J with respect to the Euclidean metric by QS 1. Sufficient conditions and necessary conditions are established for the two groups QS 1 and QS 2 to coincide with each other.

Displacement sequence of an orientation preserving circle homeomorphism

We give a complete description of the behaviour of the sequence of displacements etan(z)=Phin(x)−Phin−1(x)rmod1\eta_n(z)=\Phi^n(x) - \Phi^{n-1}(x) \ \rmod \ 1etan(z)=Phin(x)−Phin−1(x)rmod1, z=exp(2pirmix)z=\exp(2\pi \rmi x)z=exp(2pirmix), along a trajectory varphin(z)\{\varphi^{n}(z)\}varphin(z), where varphi\varphivarphi is an orientation preserving circle homeomorphism and Phi:mathbbRtomathbbR\Phi:\mathbb{R} \to \mathbb{R}Phi:mathbbRtomathbbR its lift. If the rotation number varrho(varphi)=fracpq\varrho(\varphi)=\frac{p}{q}varrho(varphi)=fracpq is rational then etan(z)\eta_n(z)etan(z) is asymptotically periodic with semi-period qqq. This convergence to a periodic sequence is uniform in zzz if we admit that some points are iterated backward instead of taking only forward iterations for all zzz. If varrho(varphi)notinmathbbQ\varrho(\varphi) \notin \mathbb{Q}varrho(varphi)notinmathbbQ then the values of etan(z)\eta_n(z)etan(z) are dense in a set which depends on the map gamma\gammagamma (semi-)conjugating varphi\varphivarphi with the rotation by varrho(varphi)\varrho(\varphi)varrho(varphi) and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if varphi\varphivarphi is C1C^1C1-diffeomorphism and show approximation of the displa...