Abstract. 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.

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 ∂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.2000 MSC: 30C65 (primary), 30D05, 37F10 (secondary).

Abstract. It is shown that for quasiregular maps of positive lower order the Julia set coincides with the boundary of the fast escaping set.

Abstract. We consider the iteration of quasiregular maps of transcendental type from R d to 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 R 3 to 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 R 3 to R 3 which is equal to the identity map in a half-space.

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

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.

This article studies the iterative behaviour of a quasiregular mapping S : R d → 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 [3] who proved that the escaping set {x : S k (x) → ∞ as k → ∞} is also dense in R d .

Baker's conjecture states that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits consisting of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1. Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1.