Foundations for an iteration theory of entire quasiregular maps

Walter Bergweiler,Daniel A Nicks

doi:10.1007/s11856-014-1081-4

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.

Highlights

Introduction and main resultsIn 1918-20, Fatou [14] and Julia [20] wrote long memoirs on the iteration of rational functions and thereby created the field known as complex dynamics
The analogies that arise in the corresponding theory for transcendental entire functions were studied by Fatou [15] in 1926
A quasiregular map f : Sn → Sn is called uniformly quasiregular if there exists a uniform bound on the dilatation of the iterates f k of f . (We will recall the definition of quasiregularity, in particular the notions of dilatation and inner dilatation, in section 2.) In principle, it would be possible to extend some of Fatou’s results about transcendental entire functions f : C → C to uniformly quasiregular maps f : Rn → Rn where n ≥ 2

Summary

Introduction and main results

In 1918-20, Fatou [14] and Julia [20] wrote long memoirs on the iteration of rational functions and thereby created the field known as complex dynamics. And in the following |y| denotes the Euclidean norm of a point y ∈ Rn. The iteration of entire quasiregular maps of polynomial type was studied in [16]. With this hypothesis the above definition agrees with the classical one for uniformly quasiregular maps and in particular for polynomials These results hold in the current setting: Theorem 1.1. For an open set A ⊂ Rn and a non-empty compact subset C of A, the pair (A, C) is called a condenser and its capacity cap(A, C) is defined by cap(A, C) = inf |∇u|n dm, uA where the infimum is taken over all non-negative functions u ∈ C0∞(A) satisfying u(x) ≥ 1 for all x ∈ C. It follows that cap X > 0 if X contains a non-degenerate continuum

Functions without pits effect

Functions with pits effect

The 2-dimensional case

Examples

Some consequences of Harnack’s inequality