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Alastair Fletcher

Publications and source records attributed to Alastair Fletcher.

At least 19 recordsLinked to original sources
arXiv2025

Geometric Function Theory on Uniformly Quasiconformally Homogeneous Domains

Uniformly quasiconformally homogeneous domains in $\mathbb{R}^n$ carry a transitive collection of $K$-quasiconformal maps for a fixed $K\geq 1.$ In this paper, we study two questions in this setting. The first is to show that quasiconformality and quasisymmetry with respect to the quasihyperbolic metric are equivalent. The second is to study normal quasiregular maps from such a domain into $S^n$ or $\mathbb{R}^n$ and show they enjoy geometric properties such as a uniform Hölder condition.

math.CV↗
arXiv2024

An analogue of Green's Functions for Quasiregular Maps

Green's functions are highly useful in analyzing the dynamical behavior of polynomials in their escaping set. The aim of this paper is to construct an analogue of Green's functions for planar quasiregular mappings of degree two and constant complex dilatation. These Green's functions are dynamically natural, in that they semi-conjugate our quasiregular mappings to the real squaring map. However, they do not share the same regularity properties as Green's functions of polynomials. We use these Green's functions to investigate properties of the boundary of the escaping set and give several examples to illustrate behavior that does not occur for the dynamics of quadratic polynomials.

math.DS↗
arXiv2022

On the mean radius of quasiconformal mappings

We study the mean radius growth function for quasiconformal mappings. We give a new sub-class of quasiconformal mappings in $\mathbb{R}^n$, for $n\geq 2$, called bounded integrable parameterization mappings, or BIP maps for short. These have the property that the restriction of the Zorich transform to each slice has uniformly bounded derivative in $L^{n/(n-1)}$. For BIP maps, the logarithmic transform of the mean radius function is bi-Lipschitz. We then apply our result to BIP maps with simple infinitesimal spaces to show that the asymptotic representation is indeed quasiconformal by showing that its Zorich transform is a bi-Lipschitz map.

math.CV↗
arXiv2020

On Julia limiting directions in higher dimensions

In this paper we study, for the first time, Julia limiting directions of quasiregular mappings in $\mathbb{R}^n$ of transcendental-type. First, we give conditions under which every direction is a Julia limiting direction. Along the way, our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a contribution to the inverse problem in $\mathbb{R}^3$ of determining which compact subsets of $S^2$ can give rise to Julia limiting directions. The methods here will require showing that certain sectorial domains in $\mathbb{R}^3$ are ambient quasiballs, which is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball $\mathbb{B}^3$ under an ambient quasiconformal map of $\mathbb{R}^3$ to itself.

math.DS↗
arXiv2020

A Complete Realization of the orbits of generalized derivatives of Quasiregular Mappings

Quasiregular maps are differentiable almost everywhere maps which are analogous to holomorphic maps in the plane for higher real dimensions. Introduced by Gutlyanskii et al, the infinitesimal space is a generalization of the notion of derivatives for quasiregular maps. Evaluation of all elements in the infinitesimal space at a particular point is called the orbit space. We prove that any compact connected subset of $\R^n\setminus\{0\}$ can be realized as an orbit space of a quasiconformal map. To that end, we construct analogues of logarithmic spiral maps and interpolation between radial stretch maps in higher dimensions. For the construction of such maps, we need to implement a new tool called the Zorich transform, which is a direct analogue of the logarithmic transform. The Zorich transform could have further applications in quasiregular dynamics.

math.CV↗
arXiv2019

On Locally Quasiconformal Teichmuller Spaces

We define a universal Teichmüller space for locally quasiconformal mappings whose dilatation grows not faster than a certain rate. Paralleling the classical Teichmüller theory, we prove results of existence and uniqueness for extremal mappings in the generalized Teichmüller class. Further, we analyze the circle maps that arise.

math.CV↗
arXiv2018

Strongly automorphic mappings and Julia sets of uniformly quasiregular mappings

A theorem of Ritt states the a linearizer of a holomorphic function at a repelling fixed point is periodic only if the holomorphic map is conjugate to a power of $z$, a Chebyshev polynomial or a Lattès map. The converse, except for some exceptions, is also true. In this paper, we prove the analogous statement in the setting of strongly automorphic quasiregular mappings and uniformly quasiregular mappings in $\mathbb{R}^n$. Along the way, we characterize the possible automorphy groups that can arise via crystallographic orbifolds and a use of the Poincaré conjecture. We further give a classification of the behaviour of uniformly quasiregular mappings on their Julia set when the Julia set is a quasisphere, quasidisk or all of $\mathbb{R}^n$ and the Julia set coincides with the set of conical points. Finally, we prove an analogue of the Denjoy-Wolff Theorem for uniformly quasiregular mappings in $\mathbb{B}^3$, the first such generalization of the Denjoy-Wolff Theorem where there is no guarantee of non-expansiveness with respect to a metric.

math.DS↗
arXiv2018

On Simultaneous Linearization

Given a uniformly quasiregular mapping, there is typically no reason to assume any relationship between linearizers at different repelling periodic points. However, in the current paper we prove that in the case where the uqr map arises as a solution of a Schröder equation then, with some further natural assumptions, if $L$ is a linearizer at one repelling periodic point, then $L\circ T$ is a linearizer at another repelling periodic point, where $T$ is a translation. In this sense we say $L$ simultaneously linearizes $f$. In the plane, an example would be that $e^z$ simultaneously linearizes $z^2$. Our methods utilize generalized derivatives for quasiregular mappings, including a chain rule and inverse derivative formula, which may be of independent interest.

math.DS↗
arXiv2017

The Orbits of Generalized Derivatives

The infinitesimal space of a quasiregular mapping was introduced by Gutlyanskii et al and generalized the idea of a derivative for this class of mappings which is only differentiable almost everywhere. In this paper, we show that the infinitesimal space is either simple, that is, it consists of only one mapping, or it contains uncountable many. To achieve this, we define the orbit of a given point as its image under all elements of the infinitesimal space. We prove that this orbit is a compact and connected subset of $\mathbb{R}^n \setminus \{ 0 \}$ and moreover, every such set in dimension two can be realized as an orbit space. We conclude with some examples exhibiting features of orbits.

math.CV↗
arXiv2017

On Infinitesimal Strebel Points

In this paper, we prove that if $X$ is a Riemann surface of infinite analytic type and $[μ]_T$ is any element of Teichmüller space, then there exists $μ_1 \in [μ]_T$ so that $μ_1$ is an infinitesimal Strebel point.

math.CV↗
arXiv2016

On the infinitesimal space of uqr mappings

Generalized derivatives and infinitesimal spaces generalize the idea of derivatives to mappings which need not be differentiable. It is particularly powerful in the context of quasiregular mappings, where normal family arguments imply generalized derivatives always exist. The main result of this paper is to show that if $f$ is any uniformly quasiregular mapping with $x_0$ a topologically attracting or repelling fixed point, at which $f$ is locally injective, then $f$ may be conjugated to a uniformly quasiregular mapping $g$ with fixed point $0$ and so that the infinitesimal space of $g$ at $0$ contains uncountably many elements. This should be contrasted with the fact that $f$ (and also $g$) is conjugate to $x\mapsto x/2$ or $x\mapsto 2x$ in the attracting or repelling cases respectively.

math.DS↗
arXiv2015

Epicycloids and Blaschke products

It is well known that the bounding curve of the central hyperbolic component of the Multibrot set in the parameter space of unicritical degree $d$ polynomials is an epicycloid with $d-1$ cusps. The interior of the epicycloid gives the polynomials of the form $z^d+c$ which have an attracting fixed point. We prove an analogous result for unicritical Blaschke products: in the parameter space of degree $d$ unicritical Blaschke products, the parabolic functions are parameterized by an epicycloid with $d-1$ cusps and inside this epicycloid are the parameters which give rise to elliptic functions having an attracting fixed point in the unit disk. We further study in more detail the case when $d=2$ in which every Blaschke product is unicritical in the unit disk.

math.DS↗
arXiv2015

Fixed curves near fixed points

Let $H$ be a composition of an $\mathbb{R}$-linear planar mapping and $z\mapsto z^n$. We classify the dynamics of $H$ in terms of the parameters of the $\mathbb{R}$-linear mapping and the degree by associating a certain finite Blaschke product. We apply this classification to this situation where $z_0$ is a fixed point of a planar quasiregular mapping with constant complex dilatation in a neighbourhood of $z_0$. In particular we find how many curves there are that are fixed by $f$ and that land at $z_0$.

math.DS↗
arXiv2014

Unicritical Blaschke products and domains of ellipticity

Elliptic Möbius transformations of the unit disk are those for which there is a fixed point in $\mathbb{D}$. It is not hard to classify which Möbius transformations are elliptic in terms of the parameters. The set of parameters can be identified with the solid torus $S^1 \times \mathbb{D}$, and the set of elliptic parameters is called the domain of ellipticity. In this paper, we study the domain of ellipticity for non-trivial unicritical Blaschke products. We will also study the set corresponding to the Mandelbrot set for this family, and show how it can be obtained from the domain of ellipticity by adding one point.

math.CV↗
arXiv2014

On quasiregular linearizers

Linearization is a well-known concept in complex dynamics. If $p$ is a polynomial and $z_0$ is a repelling fixed point, then there is an entire function $L$ which conjugates $p$ to the linear map $z\mapsto p'(z_0)z$. This notion of linearization carries over into the quasiregular setting, in the context of repelling fixed points of uniformly quasiregular mappings. In this article, we investigate how linearizers arising from the same uqr mapping and the same repelling fixed point are related. In particular, any linearizer arising from a uqr solution to a Schröder equation is shown to be automorphic with respect to some quasiconformal group.

math.CV↗
arXiv2014

Superattracting fixed points of quasiregular mappings

We investigate the rate of convergence of the iterates of an n-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity.

math.DS↗