Quasiconformal Deformations Research Articles

Quasiconformal Deformations of Holomorphic Functions

This paper deals with holomorphic functions from Bergman spaces Bp in the disk and provides the existence of deformations (variations) which do not increase the norm of functions and preserve some other prescribed properties. Admissible variations are constructed (for even integer p ≥ 2) using special quasiconformal maps of the complex plane (in line with a new approach to variational problems for holomorphic functions in Banach spaces).

Unbounded components in parameter space of rational maps

Using pinching deformations of Riemann surfaces, we give several sufficient criteria for the space of quasiconformal deformations of rational map R R of degree d d to have non-compact closure in the space R a t d Rat_{d} of rational maps of degree d d modulo conjugation by Möbius transformations.

On the extremality of quasiconformal mappings and quasiconformal deformations

Given a family of quasiconformal deformations F (w, t) such that ∂F has a uniform bound M , the solution f(z, t)(f(z, 0) = z) of the Lownertype differential equation dw dt = F (w, t) is an e2Mt-quasiconformal mapping. An open question is to determine, for each fixed t > 0, whether the extremality of f(z, t) is equivalent to that of F (w, t). The note gives this a negative approach in both directions. §

No elliptic limits for quadratic maps

We establish bounds for the multipliers of those periodic orbits of $R_\mu(z) = z(z+\mu)/(1+\overline\mu z) $, which have a Poincaré rotation number $ p/q $. The bounds are given in terms of $ p/q $ and the (logarithmic) hororadius of $\mu$ to $e^{2\pi ip/q} $. The principal tool is a new construction denoted a ‘star’ of an immediate attracting basin. The bounds are used to prove properties of the space of Möbius conjugacy classes of quadratic rational maps. These properties are related to the mating and non-mating conjecture for quadratic polynomials lsqb;Ta]. Moreover they are also reminiscent of Chuckrows theorem on the non-existence of elliptic limits of loxodromic elements in quasiconformal deformations of Kleinian groups. We bear this analogy further by proving an analog of Chuckrows theorem for deformations of certain holomorphic maps.

Strict uniformization of real algebraic curves and global real analytic coordinates on real Teichmülhler spaces

We construct a global system of real analytic coordinates on the real Teichmuller space of a compact real algebraic curve X, using so-called strict uniformization of the real algebraic curve X. A global coordinate system is then obtained via real quasiconformal deformations of the Kleinian subgroup of PGL2(R) obtained as a group of covering transformations of a strict uniformization of X.

Equivariant extension of quasiconformal deformations into the complex unit ball

Equivariant extension of quasiconformal deformations into the complex unit ball

Uniformly quasiregular mappings of Lattès type

Using an analogy of the Lattès’ construction of chaotic rational functions, we show that there are uniformly quasiregular mappings of the n n -sphere R ¯ n \overline {{\Bbb R}}^n whose Julia set is the whole sphere. Moreover there are analogues of power mappings, uniformly quasiregular mappings whose Julia set is S n − 1 {\Bbb S}^{n-1} and its complement in S n {\Bbb S}^{n} consists of two superattracting basins. In the chaotic case we study the invariant conformal structures and show that Lattès type rational mappings are either rigid or form a 1-parameter family of quasiconformal deformations.

Semialgebraic Description of Teichmiiller Space

We give a concrete semialgebraic description of Teichmiiller space T_g of the closed surface group Γ_g of genus g(≥ 2) . Our result implies that for any \textit{SL}_2(\mathbf R) -representation of Γ_g , we can determine whether this representation is discrete and faithful or not by using 4g-6 explicit trace inequalities. We also show the connectivity and contractibility of T_g from the point of view of \textit{SL}_2(\mathbf R) -representations of Γ_g . Previously, these properties of _Tg _had been proved by using hyperbolic geometry and quasi-conformal deformations of Fuchsian groups. Our method is simple and only uses topological properties of the space of \textit{SL}_2(\mathbf R) -representations of Γ_g .

AN EXTENSION PRINCIPLE FOR QUASICONFORMAL DEFORMATIONS OF THE PLANE

An extension principle for quasiconformal deformations of the plane is described. The constructibility and diversity of extensions can be of interest in the solution of certain boundary value problems in mechanics. In the theory of quasiconformal mappings such extensions are interesting from the point of view of certain extremal problems in this theory. Bibliography: 16 titles.

Ergodic properties of discrete groups; inheritance to normal subgroups and invariance under quasiconformal deformations

Ergodic properties of discrete groups; inheritance to normal subgroups and invariance under quasiconformal deformations

Strong convergence of Kleinian groups and Carathéodory convergence of domains of discontinuity

In the theory of Kleinian groups, important examples of Kleinian groups are frequently constructed as algebraic limits of known sequences of Kleinian groups, for example quasi-conformal deformations of a Kleinian group. It is an important problem to determine the properties of the limit Kleinian group from information on the sequence converging to the limit. The topological properties of the domain of discontinuity and the limit set of a Kleinian group provide valuable pieces of information about the Kleinian group. It is reasonable to expect that in a good situation, the domain of discontinuity of the limit Kleinian group is the Carathéodory limit of the domains of discontinuity of the sequence, or, equivalently, the limit set of the limit Kleinian group is the Hausdorif limit of the limit sets of the sequence. Our main theorem (Theorem 5) shows that this is true when the sequence strongly converges to the limit, and the Kleinian groups of the sequence and the limit preserve the parabolicity in both directions and satisfy the condition (*) introduced by Bonahon.

Holomorphic families of geodesic discs in infinite dimensional Teichmüller spaces

The theory of quasiconformal mappings plays an important role in Teichmüller theory. The Teichmüller spaces of Riemann surfaces are defined as quotient spaces of the spaces of Beltrami differentials, and the Teichmüller distances are defined to measure quasiconformal deformations between the Riemann surfaces representing points in the Teichmüller spaces. The Teichmüller spaces are complex Banach manifolds equipped with natural complex structures such that the canonical projections are holomorphic. It is known (see Gardinar [4]) that the Teichmüller distance, defined independently of the complex structures, coincides with the Kobayashi distance.In spite of the naturality of the definition of a Teichmüller space as a quotient of Beltrami differentials, for given two Beltrami differentials it is hard to determine whether they are equivalent or not. For this reason, it is not trivial to describe geodesic lines with respect to the Teichmüller-Kobayashi metric.

Julia-Fatou-Sullivan theory for real one-dimensional dynamics

Our aim is to show that the Julia-Fatou--Sullivan structure theory for the dynamics of rational maps is also valid for smooth endomorphisms of the circle (and of the interval) under extremely mild smoothness and non-flatness conditions. In order to stress the similarity between real and complex one-dimensional dynamics let us recall the main results from the Fatou-Julia-Sullivan theory. If f is a rational map then there is a dynamical decomposition of the Riemann sphere into the disjoint union of two totally invariant (i.e., both forward and backward invariant) sets J(f), F(f). Here, F(f) is the domain of normality of the family of iterates of f , and is called the Fatou set. Its complement, which is called the Julia set off , is a compact set, which contains all the complications of the dynamics off . The connected components of the open set F(f) are mapped onto each other by Hence the orbit of a component of F(f) is the union of some components of F(f). Julia proved in the beginning of the century that if a component of F(f) is periodic and contains an attracting periodic point then the orbit of this component must contain a critical point. Sullivan, in the remarkable paper [Su], showed via quasi-conformal deformations that the components are eventually periodic and fall into finitely many orbits. Let N be either the circle S J or a compact interval of the real line and f. N---~N be a smooth endomorphism. A critical point of f is a point where the derivative vanishes. A critical point is non-flat if some (higher) derivative is non-zero. A critical point is an inflection point if it has a neighbourhood where f is monotone. Otherwise it is called a turning point. Assume f is not a homeomorphism ( i f f is a homeomorphism then these maps correspond to degree + 1 rational maps of the Riemann sphere and the situation is

Riemann surfaces and the geometrization of 3-manifolds

About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of hyperbolic geometry, as natural as Euclid’s regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it exists [Mos], so topology and geometry mesh harmoniously in dimension 3. This remarkable theorem applies to all 3-manifolds which can be built up in an inductive way from 3-balls, i.e. Haken manifolds. Thurston’s construction of a hyperbolic structure is also inductive. At the inductive step one must find the right geometry on an open 3-manifold so that its ends may be glued together. Using quasiconformal deformations, the gluing problem can be formulated as a fixed-point problem for a map of Teichmuller space to itself. Thurston proposes to find the fixed point by iterating this map. Here we outline Thurston’s construction, and sketch a new proof that the iteration converges. Our argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently

Ricci curvature and quasiconformal deformations of a Riemannian manifold

The theory of quasiconformal deformations of a Riemannian manifold (M, g) of dimensionn leads in a natural way to the AhlforsS operator being the symmetric and trace free part of the Levi-Civita connection Δ on 1-forms, and to the Ahlfors Laplacian \(L = S*S = 2\delta d + \frac{{4n - 4}}{n}d\delta- 4R\), whereR is the Ricci action. It is well known that there are no conformal deformations on compact Riemannian manifoldsM with negative Ricci curvature. The question arises, how close to being conformal a deformation on suchM can be, i.e. the question on the minimal constant of quasiconformality. Using spectral properties ofL, we derive several lower bounds for the constant of quasiconformality for the normalized deformations of compact manifolds with the positive definite, negative definite, or vanishing Ricci tensorR. As a result, we also obtain that there are no conformal deformations if the Ricci curvature is positive definite but small enough (Corollary 5.17).

Multiloop Calculation on the Partition Function of Polyakov's String Model

Explicit formulas are deduced for partition functions , with , of the closed string. The modular space , may be parametrized by length-twist parameters arising from the quasi-conformal deformations of Riemann surface .

A theorem on the limit sets of quasiconformal deformations of infinitely generated Fuchsian groups of the first kind

A theorem on the limit sets of quasiconformal deformations of infinitely generated Fuchsian groups of the first kind

Some differential operators connected with quasiconformal deformations on manifold

Some differential operators connected with quasiconformal deformations on manifold

A finiteness theorem for a dynamical class of entire functions

AbstractWe define a class Σ of entire functions whose covering properties are similar to those of rational maps. The set Σ is closed under composition of functions, and we show that when regarded as dynamical systems of the plane, the elements of Σ share many properties with rational maps. In particular, they have finite dimensional spaces of quasiconformal deformations, and they contain no wandering domains in their stable sets.

On the measurable dynamics of z → ez

AbstractWe study the measure theoretic properties of the complex exponential map E(z) = ez.An particular, we show that the equivalence relation generated by E is recurrent and that E has no quasi-conformal deformations. This enables us to give some information concerning Devaney's semi-conjugacy between E and the shift map on sequences of integers.