Uniformly quasiregular maps of $\mathbb{R}^n$ with Cantor Julia sets
For any $n\ge 3$, we construct uniformly quasiregular self-maps of $\mathbb{R}^n$ of polynomial type for which the Julia set is a Cantor set.
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arXiv subjects
Daniel A. Nicks
Publications and source records attributed to Daniel A. Nicks.
Interpolating quasiregular power mappings
We construct a quasiregular mapping in $\mathbb{R}^3$ that is the first to illustrate several important dynamical properties: the quasi-Fatou set contains wandering components; these quasi-Fatou components are bounded and hollow; and the Julia set has components that are genuine round spheres. The key tool in this construction is a new quasiregular interpolation in round rings in $\mathbb{R}^3$ between power mappings of differing degrees on the boundary components. We also exhibit the flexibility of constructions based on these interpolations by showing that we may obtain quasiregular mappings which grow as quickly, or as slowly, as desired.
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Normal families and quasiregular mappings
Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others, and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.
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Iterating the minimum modulus: functions of order half, minimal type
For a transcendental entire function $f$, the property that there exists $r>0$ such that $m^n(r)\to\infty$ as $n\to\infty$, where $m(r)=\min \{|f(z)|:|z|=r\}$, is related to conjectures of Eremenko and of Baker, for both of which order $1/2$ minimal type is a significant rate of growth. We show that this property holds for functions of order $1/2$ minimal type if the maximum modulus of $f$ has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg's method of constructing entire functions of small growth, which allows rather precise control of $m(r)$.
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Which sequences are orbits?
In the study of discrete dynamical systems, we typically start with a function from a space into itself, and ask questions about the properties of sequences of iterates of the function. In this paper we reverse the direction of this study. In particular, restricting to the complex plane, we start with a sequence of complex numbers and study the functions (if any) for which this sequence is an orbit under iteration. This gives rise to questions of existence and of uniqueness. We resolve some questions, and show that these issues can be quite delicate.
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The bungee set in quasiregular dynamics
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.
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Eremenko's conjecture for functions with real zeros: the role of the minimum modulus
We show that for many families of transcendental entire functions $f$ the property that $m^n(r)\to\infty$ as $n\to \infty$, for some $r>0$, where $m(r)=\min\{|f(z)|:|z|=r\}$, implies that the escaping set $I(f)$ of $f$ has the structure of a spider's web. In particular, in this situation $I(f)$ is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.
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The dynamics of quasiregular maps of punctured space
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 are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Marti-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.
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Slow escaping points of quasiregular mappings
This article concerns the iteration of quasiregular mappings on $\mathbb{R}^d$ and entire functions on $\mathbb{C}$. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let $f:\mathbb{R}^d\to\mathbb{R}^d$ be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of $f$ contains points at which the iterates $f^n$ tend to infinity arbitrarily slowly. We also prove that, for any large $R$, there is a point $x$ with modulus approximately $R$ such that the growth of $|f^n(x)|$ is asymptotic to the iterated maximum modulus $M^n(R,f)$.
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Periodic domains of quasiregular maps
We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^d$ to $\mathbb{R}^d$. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from $\mathbb{R}^3$ to $\mathbb{R}^3$ with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of biLipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^3$ to $\mathbb{R}^3$ which is equal to the identity map in a half-space.
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Hollow quasi-Fatou components of quasiregular maps
We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in $\mathbb{R}^d$ is called hollow if it has a bounded complementary component. We show that for each $d \geq 2$ there exists a quasiregular map of transcendental type $f: \mathbb{R}^d \to \mathbb{R}^d$ with a quasi-Fatou component which is hollow. Suppose that $U$ is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if $U$ is bounded, then $U$ has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if $U$ is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if $J(f)$ has an isolated point, or if $J(f)$ is not equal to the boundary of the fast escaping set. Finally, we deduce that if $J(f)$ has a bounded component, then all components of $J(f)$ are bounded.
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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.
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The Julia set and the fast escaping set of a quasiregular mapping
It is shown that for quasiregular maps of positive lower order the Julia set coincides with the boundary of the fast escaping set.
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Foundations for an iteration theory of entire quasiregular maps
The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire functions. Here the Julia set is defined as the set of all points such that complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.
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Chaotic dynamics of a quasiregular sine mapping
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.
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Iteration of quasiregular tangent functions in three dimensions
We define a new quasiregular mapping T in three dimensions that generalizes the tangent function on the complex plane and shares a number of its geometric properties. We investigate the dynamics of the family λT for λ>0, establishing results analogous to those of Devaney and Keen for the meromorphic family λtan z, λ>0, although the methods used are necessarily original.
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Wandering domains in quasiregular dynamics
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.
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Julia sets of uniformly quasiregular mappings are uniformly perfect
It is well-known that the Julia set J(f) of a rational map is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this article we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f in R^n is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.
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