Riemann Mapping Function Research Articles

On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set

We investigate Benford’s law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained as the limit of iterative sets, and the unique measures of their components follow a geometric distribution, which is Benford in most bases. Building on this intuition, we aim to study this distribution in more complicated fractals. We examine the Laurent coefficients of a Riemann mapping and the Taylor coefficients of its reciprocal function from the exterior of the Mandelbrot set to the complement of the unit disk. These coefficients are 2-adic rational numbers, and through statistical testing, we demonstrate that the numerators and denominators are a good fit for Benford’s law. We offer additional conjectures and observations about these coefficients. In particular, we highlight certain arithmetic subsequences related to the coefficients’ denominators, provide an estimate for their slope, and describe efficient methods to compute them.

Отпечатки, лемнискаты и квадратичные дифференциалы

We discuss some aspects of the theory of recognition of two-dimensional shapes by means of fingerprints of Jordan curves. An interesting approach to problems on shape recognition suggested by P.~Ebenfelt, D.~Khavinson, and H.~Shapiro and extended further by M.~Younsi reveals the fact that the fingerprints of polynomial lemniscates and, more generally, fingerprints of rational lemniscates can be obtained as solutions to certain functional equations involving Blaschke products. Our main goal here is to develop an approach which relates fingerprints of Jordan curves composed of arcs of trajectories and orthogonal trajectories of certain quadratic differentials with solutions of functional equations involving pullbacks of these quadratic differentials under appropriate Riemann mapping functions. In particular, we show that the previous results of P.~Ebenfelt, D.~Khavinson, and H.~Shapiro and the recent results of M.~Younsi follow from our more general theorems as special cases.

Asymptotic behaviour of the Riemann mapping function at analytic cusps

The well-known Riemann Mapping Theorem states the existence of a conformal map of a simply connected proper domain of the complex plane onto the upper half plane. One of the main topics in geometric function theory is to investigate the behaviour of the mapping functions at the boundary of such domains. In this work, we always assume that a piecewise analytic boundary is given. Hereby, we have to distinguish regular and singular boundary points. While the asymptotic behaviour for regular boundary points can be investigated by using the Schwarz Reflection at analytic arcs, the situation for singular boundary points is far more complicated. In the latter scenario two cases have to be differentiated: analytic corners and analytic cusps. The first part of the thesis deals with the asymptotic behaviour at analytic corners where the opening angle is greater than 0. The results of Lichtenstein and Warschawski on the asymptotic behaviour of the Riemann map and its derivatives at an analytic corner are presented as well as the much stronger result of Lehman that the mapping function can be developed in a certain generalised power series which in turn enables to examine the o-minimal content of the Riemann Mapping Theorem. To obtain a similar statement for domains with analytic cusps, it is necessary to investigate the asymptotic behaviour of a Riemann map at the cusp and based on this result to determine the asymptotic power series expansion. Therefore, the aim of the second part of this work is to investigate the asymptotic behaviour of a Riemann map at an analytic cusp. A simply connected domain has an analytic cusp if the boundary is locally given by two analytic arcs such that the interior angle vanishes. Besides the asymptotic behaviour of the mapping function, the behaviour of its derivatives, its inverse, and the derivatives of the inverse are analysed. Finally, we present a conjecture on the asymptotic power series expansion of the mapping function at an analytic cusp.

ORTHONORMAL BASIS FOR THE BERGMAN SPACE

We construct an orthonormal basis for the Bergman space associated to a simply connected domain. We use the or-thonormal basis for the Hardy space consisting of the Szegő kernel and the Riemann mapping function and rewrite their area integrals in terms of arc length integrals using the complex Green's identity. And we make a note about the matrix of a Toeplitz operator with respect to the orthonormal basis constructed in the paper.

A Question of Ahlfors

Abstract. In 1963, Ahlfors posed in [1] (and repeated in his book [2]) thefollowing question which gave rise to various investigations of quasiconformalextendibility of univalent functions. Question. Let � be a conformal map of the disk (or half-plane) ontoa domain with quasiconformal boundary (quasicircle). How can this map becharacterized? He conjectured that the characterization should be in analytic properties ofthe logarithmic derivative log� ′ = � ′′ /� ′ , and indeed, many results on quasicon-formal extensions of holomorphic maps have been established using � ′′ /� ′ andother invariants (see, e.g., the survey [9] and the references there).This question relates to another still not solved problem in geometric com-plex analysis: To what extent does the Riemann mapping function � of a Jordan domain �⊂ C determine the geometric and conformal invariants (characteristics) ofcomplementary domain � * = C ∖� ? The purpose of this paper is to provide a qualitative answer to these ques-tions, which discovers how the inner features of biholomorphy determine theadmissible bounds for quasiconformal dilatations and determine the Kobayashidistance for the corresponding points in the universal Teichmuller space.¨

The Hessenberg matrix and the Riemann mapping function

We consider a Jordan arc Γ in the complex plane \({\mathbb C}\) and a regular measure μ whose support is Γ. We denote by D the upper Hessenberg matrix of the multiplication by z operator with respect to the orthonormal polynomial basis associated with μ. We show in this work that, if the Hessenberg matrix D is uniformly asymptotically Toeplitz, then the symbol of the limit operator is the restriction to the unit circle of the Riemann mapping function ϕ(z) which maps conformally the exterior of the unit disk onto the exterior of the support of the measure μ. We use this result to show how to approximate the Riemann mapping function for the support of μ from the entries of the Hessenberg matrix D.

The Green's function and the Ahlfors map

The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting.

The C 2-convergence of circle packings with bounded degree to the Riemann mapping§

It is known that hexagonal circle packings can be used to approximate the Riemann mappings. Here, the convergence of the hexagonal circle packings is generalized to that of circle packings with bounded degree. We shall show that the first two derivatives of the bounded degree circle packings converge to the Riemann mapping function's first two derivatives, respectively; moreover we give the explicit estimates for the rates of convergence. §Dedicated to Professor Jian-Ke Lu on the ocassion of his 85th birthday.

The [formula omitted]-convergence of SG circle patterns to the Riemann mapping

Thurston conjectured that the Riemann mapping function from a simply connected region onto the unit disk can be approximated by regular hexagonal packings. Schramm introduced circle patterns with combinatorics of the square grid (SG) and showed that SG circle patterns converge to meromorphic functions. He and Schramm proved that hexagonal disk packings converge in C ∞ to the Riemann mapping. In this paper we show a similar C ∞ -convergence for SG circle patterns.

A Bound for Reflections Across Jordan Curves

Abstract The properties of logarithmic derivative of the Riemann mapping function of a quasidisk are applied to quantitative estimation of quasiconformal reflections across its boundary.

Uniqueness Theorems for Some Free Boundary Problems in Univalent Functions

Let Φ be a bounded positive continuous function in the complex plane C. We will consider the following problem. We can find a simply connected region Ω containing the origin such that the Riemann mapping function f Ω mapping the unit disk D onto Ω will satisfy <artwork name="GCOV31013eu1"> We show that there is a unique solution which is starlike with respect to the origin when Φ satisfies <artwork name="GCOV31013eu2"> We also show that if in addtion, ∂Φ/∂|w|>0 and −log Φ is subharmonic, then Ω will be a convex region.

$L^2$-Approximations of power and logarithmic functions with applications to numerical conformal mapping

For a bounded Jordan domain G with quasiconformal boundary L, two-sided estimates are obtained for the error in best \(L^2(G)\) polynomial approximation to functions of the form \((z-\tau)^{\beta}, {\beta}>-1\), and \((z-\tau)^m\log^l(z-\tau), m>-1, l\ne 0\), where \(\tau\in L\). Furthermore, Andrievskii's lemma that provides an upper bound for the \(L^\infty(G)\) norm of a polynomial \(p_n\) in terms of the \(L^2(G)\) norm of \(p_n^\prime\) is extended to the case when a finite linear combination (independent of n) of functions of the above form is added to \(p_n\). For the case when the boundary of G is piecewise analytic without cusps, the results are used to analyze the improvement in rate of convergence achieved by using augmented, rather than classical, Bieberbach polynomial approximants of the Riemann mapping function of G onto a disk. Finally, numerical results are presented that illustrate the theoretical results obtained.

Representations of analytic functions in terms of local values by means of the riemann mapping function

For any analytic function f on a Riemann surface S, we give a “good” formula representing f on a large simplyconnected domain containing any given point pεS in terms of the values of f in any neighborhood at any fixed point p p oεS, by using some Riemann mapping function.

Numerical conformal mapping via the Bergman kernel using the generalized minimum residual method

The Bergman kernel function is known to satisfy a certain boundary integral equation of the second kind. For boundaries that possess symmetrical qualities, the integral equation can be transformed into another integral equation that uses only a small part of the original boundary. This paper applies an iterative procedure known as the generalized minimum residual method for the computation of the Riemann mapping function via the Bergman kernel. The complexity of this procedure is O( n 2), where n is the number of collocation points on the boundary of the region. Numerical implementation on some test regions is also presented.

Numerical conformal mapping via the Bergman kernel

A new method to compute the Riemann mapping function via the Bergman kernel is presented. The method expresses the Bergman kernel as the solution of a second-kind integral equation involving the Neumann kernel. For symmetric regions, the integral equation can be transformed into a new one that uses only a small part of the original boundary. Numerical implementations on some test regions are also presented.

A Right Inverse of the Askey-Wilson Operator

We establish an integral representation of a right inverse of the Askey-Wilson finite difference operator on ${L^2}$ with weight ${(1 - {x^2})^{ - 1/2}}$. The kernel of this integral operator is $\vartheta _4’/\vartheta _4$ and is the Riemann mapping function that maps the interior of an ellipse conformally onto the open unit disc.

A right inverse of the Askey-Wilson operator

We establish an integral representation of a right inverse of the Askey-Wilson finite difference operator on L 2 {L^2} with weight ( 1 − x 2 ) − 1 / 2 {(1 - {x^2})^{ - 1/2}} . The kernel of this integral operator is ϑ 4 ′ / ϑ 4 \vartheta _4’/\vartheta _4 and is the Riemann mapping function that maps the interior of an ellipse conformally onto the open unit disc.

An Inverse Problem for Circle Packing and Conformal Mapping

Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius $\varepsilon$ in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then, after suitable normalization, the correspondence of circles converges to the Riemann mapping function as $\varepsilon \to 0$ (see [15]). In the present paper an inverse of this result is obtained as illustrated by Figure 1.2; namely, if the unit disk is almost packed with $\varepsilon$-circles there is an isomorphic circle packing almost filling the region such that, after suitable normalization, the circle correspondence converges to the conformal map of the disk onto the region as $\varepsilon \to 0$. Note that this set up yields an approximate triangulation of the region by joining the centers of triples of mutually tangent circles. Since this triangulation is intimately related to the Riemann mapping it may be useful for grid generation [18].

An inverse problem for circle packing and conformal mapping

Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius ε \varepsilon in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then, after suitable normalization, the correspondence of circles converges to the Riemann mapping function as ε → 0 \varepsilon \to 0 (see [15]). In the present paper an inverse of this result is obtained as illustrated by Figure 1.2; namely, if the unit disk is almost packed with ε \varepsilon -circles there is an isomorphic circle packing almost filling the region such that, after suitable normalization, the circle correspondence converges to the conformal map of the disk onto the region as ε → 0 \varepsilon \to 0 . Note that this set up yields an approximate triangulation of the region by joining the centers of triples of mutually tangent circles. Since this triangulation is intimately related to the Riemann mapping it may be useful for grid generation [18].

Exponential integrability of the quasi-hyperbolic metric on Hölder domains

A proper subdomain D of Rn is called a Holder domain if for a fixed y in D, the quasi-hyperbolic metric ko(r,y) is bounded by a constant plus a constant multiple of the logarithm of the Euclidean distance from c to the boundary of D. For simply connected planar domains D, it is known that these domains are characterized by the fact that the Riemann mapping function of the unit disk onto D satisfies a Holder condition with some positive exponent. For gr in D fixed, we prove that exp(r,bp(r,y)) is integrable over D for some r > 0. One corollary of this is that the boundaries of these domains have Hausdorff dimension less than n. Other applications pertain to Poincard domains and to averaging domains. Our method involves extending some recent results of Carleson-Jones and Jones-Makarov on simply connected planar domains to multiply connected domains in ,R' by using the quasi-hyperbolic metric'