Alastair Fletcher, Daniel A. Nicks
In this article, we investigate the boundary of the escaping set I(f) for quasiregular mappings on R^n, both in the uniformly quasiregular case and in the polynomial type case. The aim is to show that the boundary of I(f) is the Julia set J(f) when the latter is defined, and shares properties with the Julia set when J(f) is not defined.
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Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps
本文证明单位球上一致拟正则自映射的Denjoy-Wolff定理，利用Schwarz-Pick引理和链度量，并应用于稳定Fatou分量的分类，构造了非单射例子。
Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia iteration theory of rational functions extend to quasiregular self-maps of the Riemann sphere for which the degree exceeds the dilatation. We show that in this context, in contrast to the case of rational functions, the Julia set may have Hausdorff dimension zero. On the other hand, we exhibit a gauge function depending on the degree and the dilatation such that the Hausdorff measure with respect to this gauge function is always positive, but may be finite.
Interpolating quasiregular power mappings
本文通过在 R³ 圆环内对不同次数的幂映射进行新型拟正则插值，构造出具有游荡、有界中空 quasi-Fatou 分支及球形 Julia 集分支的拟正则映射，并可任意控制其增长快慢。
We prove that every non-constant quasiregular selfmap of the $n$-sphere $\mathbb{S}^{n}$ admits a harmonic extension to the hyperbolic space $\mathbb{H}^{n+1}$ for $n\ge 2$.
Examples of Uniformly Quasiregular Mappings with Fatou Set Having Infinitely Many Components
构造了球面上Fatou集由无穷多分支组成的均匀拟正则映射，特别给出了多项式z→z²-1的高维类比。
We show that wandering domains can exist in the Fatou set of a polynomial type quasiregular mapping of the plane. We also give an example of a quasiregular mapping of the plane, with an essential singularity at infinity, which has a sequence of wandering domains contained in a bounded part of the plane. This contrasts with the situation in the analytic case, where wandering domains are impossible for polynomials and, for transcendental entire functions, the existence of wandering domains in a bounded part of the plane has been an open problem for many years.
Hausdorff dimension in quasiregular dynamics
本文研究三维空间中拟正则自映射，证明其快逃逸集的Hausdorff维数可取[1,3]内任意值，并在增长条件下估计Julia集的维数。
In complex dynamics, the bungee set is defined as the set points whose orbit is neither bounded nor tends to infinity. In this paper we study, for the first time, the bungee set of a quasiregular map of transcendental type. We show that this set is infinite, and shares many properties with the bungee set of a transcendental entire function. By way of contrast, we give examples of novel properties of this set in the quasiregular setting. In particular, we give an example of a quasiconformal map of the plane with a non-empty bungee set; this behaviour is impossible for an analytic homeomorphism.
We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure $μ_f$, which is balanced and invariant under $f$ and non-atomic, and whose support agrees with the Julia set of $f$. Furthermore we show that $f$ is strongly mixing with respect to the measure $μ_f$. We also characterize the measure $μ_f$ using an approximation property by iterated pullbacks of points under $f$ up to a set of exceptional initi…
Complex dynamics deals with the iteration of holomorphic functions. As is well- known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of Julia sets and the Mandelbrot set. In the same spirit, this article aims to study the dynamics of the simplest non-trivial quasiregular mappings. These are mappings in R^2 which are a composition of a quadratic polynomial and an affine stretch.
We construct a new type of locally homeomorphic quasiregular mappings in the 3-sphere and discuss their relation to the M.A.Lavrentiev problem, the Zorich map with an essential singularity at infinity, the Fatou's problem and a quasiregular analogue of domains of holomorphy in complex analysis. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms $M$ with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such locally homeomorphic quasiregular mappings are defined in the 3-sphere $S^…
This article studies the iterative behaviour of a quasiregular mapping S:\R^d\to\R^d that is an analogue of a sine function. We prove that the periodic points of S form a dense subset of \R^d. We also show that the Julia set of this map is \R^d in the sense that the forward orbit under S of any non-empty open set is the whole space \R^d. The map S was constructed by Bergweiler and Eremenko who proved that the escaping set I(S) is also dense in \R^d.
An analogue of Green's Functions for Quasiregular Maps
本文为二次次数且常数复伸缩的平面拟正则映射构造格林函数类比，其半共轭到实数平方映射，但正则性不同于多项式情形，用于研究逃逸集边界并展示新行为。
The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties…
We consider accumulation of periodic points in local uniformly quasiregular dynamics. Given a local uniformly quasiregular mapping $f$ with a countable and closed set of isolated essential singularities and their accumulation points on a closed Riemannian manifold, we show that points in the Julia set are accumulated by periodic points. If, in addition, the Fatou set is non-empty and connected, the accumulation is by periodic points in the Julia set itself. We also give sufficient conditions for the density of repelling periodic points.
Let h:C \to C be an R-linear map. In this article, we explore the dynamics of the quasiregular mapping H(z)=h(z)^2. Via the Böttcher type coordinate constructed in "On Böttcher coordinates and quasiregular maps" by Fletcher and Fryer, we are able to obtain results for any degree two mapping of the plane with constant complex dilatation. We show that any such mapping has either one, two or three fixed external rays, that all cases can occur, and exhibit how the dynamics changes in each case. We use results from complex dynamics to prove that these mappings are nowhere uniformly quasiregular in …
Carathéodory boundary extensions for generalized quasiregular mappings
研究了具有有界和有限扭曲的映射的边界行为，证明了在特定条件下，这些映射具有连续边界扩展，且结果适用于具有适当模条件的-homeomorphism和开放离散闭映射。
We introduce a higher dimensional quasiregular map analogous to the trigonometric functions and we use the dynamics of this map to define, for d>1, a partition of d-dimensional Euclidean space into curves tending to infinity such that two curves may intersect only in their endpoints and such that the union of the curves without their endpoints has Hausdorff dimension one.
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