Daniel A. Nicks

Quasiregular dynamics on the n-sphere

In this paper, 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 ∂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.

Foundations for an iteration theory of entire quasiregular maps

The Fatou-Julia iteration theory of rational functions has been extended to uniformly quasiregular mappings in higher dimension by various authors, and recently some results of Fatou-Julia type have also been obtained for non-uniformly quasiregular maps. The purpose of this paper is to extend the iteration theory of transcendental entire functions to the quasiregular setting. As no examples of uniformly quasiregular maps of transcendental type are known, we work without the assumption of uniform quasiregularity. Here the Julia set is defined as the set of all points such that the 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.

Julia sets of uniformly quasiregular mappings are uniformly perfect

It is well known that the Julia set J(f) of a rational map f: ℂ → ℂ is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this paper we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f: ℝn → ℝn is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

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.

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.

Chaotic dynamics of a quasiregular sine mapping

This article studies the iterative behaviour of a quasiregular mapping that is an analogue of a sine function. We prove that the periodic points of S form a dense subset of . We also show that the Julia set of this map is in the sense that the forward orbit under S of any non-empty open set is the whole space . The map S was constructed by Bergweiler and Eremenko (Ann. Acad. Sci. Fenn. Math. 36 (2011), pp. 165–175) who proved that the escaping set is also dense in .

Iteration of quasiregular tangent functions in three dimensions

We define a new quasiregular mapping

Hollow quasi-Fatou components of quasiregular maps

T:\mathbb{R}^3\to \mathbb{R}^3 \cup \{\infty \}

Superattracting fixed points of quasiregular mappings

that generalizes the tangent function on the complex plane and shares a number of its geometric properties. We investigate the dynamics of the family

Rational deficient functions of derivatives of mappings in the classes S and B

\{\lambda T:\lambda >0\}