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