Quasiregular Mappings Research Articles

Entire functions whose Julia sets include any finitely many copies of quadratic Julia sets

We prove that for any finite collection of quadratic Julia sets, a polynomial and a transcendental entire function exist whose Julia sets include copies of the given quadratic Julia sets. In order to prove the result, we construct quasiregular maps with required dynamics and employ the quasiconformal surgery to obtain the desired functions.

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 the 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 bi-Lipschitz 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.

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.

Absolute continuity on paths of spatial open discrete mappings

We prove that open discrete mappings of Sobolev classes \(W_\mathrm{loc}^{1, p},\)\(p>n-1,\) with locally integrable inner dilatations admit \(ACP_p^{\,-1}\)-property, which means that these mappings are absolutely continuous on almost all preimage paths with respect to p-module. In particular, our results extend the well-known Poletskiĭ lemma for quasiregular mappings. We also establish the upper bounds for p-module of such mappings in terms of integrals depending on the inner dilatations and arbitrary admissible functions.

Estimates for Jacobian and Dilatation Coefficients of Open Discrete Mappings with Controlled p-Module

We study open discrete mappings whose p-module of curve families is integrally restricted and establish various estimates for the Jacobian and dilatation coefficients. We also show that such mappings are close to Lipschitz mappings, quasiregular mappings and mappings of finite distortion. The sharpness of these estimates is illustrated by several examples.

Quasiregular mapping that preserves plane

Quasiregular mappings are a natural generalization of analytic functions to higher dimensions. Quasiregular mappings have many properties. Our work in this paper is to prove the following theorem: If f  a b is a quasiregular mapping which maps the plane onto the plane, then f is a bijection. We do this by finding the connection between quasiregular and quasiconformal mappings.

Hollow quasi-Fatou components of quasiregular maps

AbstractWe define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd 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.

The size and topology of quasi-Fatou components of quasiregular maps

We consider the iteration of quasiregular maps of transcendental type from $\mathbb {R}^d$ to $\mathbb {R}^d$. In particular we study quasi-Fatou components, which are defined as the connected components of the complement of the Julia set. Many authors have studied the components of the Fatou set of a transcendental entire function, and our goal in this paper is to generalise some of these results to quasi-Fatou components. First, we study the number of complementary components of quasi-Fatou components, generalising, and slightly strengthening, a result of Kisaka and Shishikura. Second, we study the size of quasi-Fatou components that are bounded and have a bounded complementary component. We obtain results analogous to those of Zheng, and of Bergweiler, Rippon and Stallard. These are obtained using techniques which may be of interest even in the case of transcendental entire functions.

The visual angle metric and quasiregular maps

The distortion of distances between points under maps is studied. We first prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric. Then we investigate conversely the quasiconformality of a bilipschitz map with respect to the visual angle metric on convex domains. For the unit ball or half space, we prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. We also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric.

Rigidity of extremal quasiregularly elliptic manifolds

We show that for a closed n -manifold N admitting a quasiregular mapping from Euclidean n -space the following are equivalent: (1) order of growth of \pi_1(N) is n , (2) N is aspherical, and (3) \pi_1(N) is virtually \Z^n and torsion free.

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.

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on Rd and entire functions on 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:Rd→Rd 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 fn 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 |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f).

Integration by parts on generalized manifolds and applications on quasiregular maps

Integration by parts on generalized manifolds and applications on quasiregular maps

The branch set of a quasiregular mapping between metric manifolds

In this note, we announce some new results on quantitative countable porosity of the branch set of a quasiregular mapping in very general metric spaces. As applications, we solve a recent conjecture of Fässler et al., an open problem of Heinonen–Rickman, and an open question of Heinonen–Semmes.

Exponential integrability of mappings of finite distortion

We consider mappings with exponentially integrable distortion whose Jacobian determinants are integrable over the n -ball. We show that the boundary extensions of such mappings are exponentially integrable with bounds, and give examples to illustrate that there is not too much room for improvement. This extends the results of Beurling [2], and Chang and Marshall [3], [10] on analytic functions, and Poggi-Corradini and Rajala [14] on quasiregular mappings.

Some properties of open, discrete, generalized ring mappings

We study properties of open, discrete ring mappings satisfying generalized modular inequalities. Such mappings generalize the known class of quasiregular mappings and their extensions known as mappings of finite distortion. We apply our results to open discrete ring mappings satisfying condition (N) and having local inverses, and we focus especially on the case . We show that such mappings cannot have essential singularities and also that Zoric’s theorem can hold in this case under some conditions even if . This is in contrast even with the known case of quasiregular mappings.

Quasiregular Mappings on Sub-Riemannian Manifolds

We study mappings on sub-Riemannian manifolds which are quasiregular with respect to the Carnot–Caratheodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit is not available in the setting of general sub-Riemannian manifolds. In the present paper we adopt therefore a metric rather than analytic viewpoint. As a first main result, we prove that the sub-Riemannian lens space admits nontrivial uniformly quasiregular (UQR) mappings, that is, quasiregular mappings with a uniform bound on the distortion of all the iterates. In doing so, we also obtain new examples of UQR maps on the standard sub-Riemannian spheres. The proof is based on a method for building conformal traps on sub-Riemannian spheres using quasiconformal flows, and an adaptation of this approach to quotients of spheres. One may then study the quasiregular semigroup generated by a UQR mapping. In the second part of the paper we follow Tukia to prove the existence of a measurable conformal structure which is invariant under such a semigroup. Here, the conformal structure is specified only on the horizontal distribution, and the pullback is defined using the Margulis–Mostow derivative (which generalizes the classical and Pansu derivatives).

Poletskiĭ Type Inequality for Mappings from the Orlicz-Sobolev Classes

We study the distortion of $$p$$ -module under non-homeomorphic mappings $$f$$ from Orlicz-Sobolev classes $$W^{1,\varphi }_\mathrm{loc}$$ and established a strengthened form of Poletskii’s inequality. This inequality was known for quasiregular mappings and conformal moduli. In addition, our estimates involve the $$p$$ -outer dilatation (instead of the classical inner dilatation) and the multiplicity function. In the case of the planar domains, the condition $$f\in W^{1,\varphi }_\mathrm{loc}$$ can be replaced by $$f\in W^{1,1}_\mathrm{loc}$$ .

The geometry of planar p-harmonic mappings: convexity, level curves and the isoperimetric inequality

We discuss various representations of planar $p$-harmonic systems of equations and their solutions. For coordinate functions of $p$-harmonic maps we analyze signs of their Hessians, the Gauss curvature of $p$-harmonic surfaces, the length of level curves as well as we discuss curves of steepest descent. The isoperimetric inequality for the level curves of coordinate functions of planar $p$-harmonic maps is proven. Our main techniques involve relations between quasiregular maps and planar PDEs. We generalize some results due to P. Lindqvist, G. Alessandrini, G. Talenti and P. Laurence.

Poincaré linearizers in higher dimensions

It is well-known that a holomorphic function near a repelling fixed point may be conjugated to a linear function. The function which conjugates is called a Poincar\'e linearizer and may be extended to a transcendental entire function in the plane. In this paper, we study the dynamics of a higher dimensional generalization of Poincar\'e linearizers. These arise by conjugating a uniformly quasiregular mapping in $\R^m$ near a repelling fixed point to the mapping $x\mapsto 2x$. In particular, we show that the fast escaping set of such a linearizer has a spider's web structure.