The dynamics of quasiregular maps of punctured space

Abstract

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 Martí-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.

On the iteration of quasimeromorphic mappings

“…We summarise some of the results found in [13] in the following theorem. Here, any closure is taken with respect toR d \ S, unless stated otherwise.…”

“…Motivated by this, the relationship between cardinality and capacity in the new setting has also been established. Combining this result with [11,Proposition 3.4] gives the following improved theorem. Firstly in Section 2 we shall provide some important definitions, alongside some key results and observations regarding quasimeromorphic mappings of transcendental type with at least one pole.…”

“…To analyse the case when 2 ≤ card(O − f (∞)) < q, we will need to consider the behaviour of quasiregular mappings in punctured space. The iterative theory of such mappings has been studied by Nicks and Sixsmith in [11], thus we shall only state the definition and a few key results here.…”

“…Since then, the Julia set of other types of quasiregular mappings have been established and studied using a similar method. Examples of these include quasiregular self-maps ofR d , d 2, by Bergweiler [5], entire quasiregular mappings of transcendental type by Bergweiler and Nicks [6] and quasiregular mappings of punctured space by Nicks and Sixsmith [13]. Here we say that a quasiregular or quasimeromorphic mapping f is of transcendental type if lim |x|→∞ f (x) does not exist.…”

“…[11]). Let S ⊂R d be a finite set with ∞ ∈ S and suppose that g :R d \ S →R d \ S is a quasiregular map of S-transcendental type.…”