Foundations for an iteration theory of entire quasiregular maps

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.

31 pages

References in corpus (8)

• Iteration of meromorphic functions
• Topological Dynamics of Exponential Maps on their Escaping Sets
• The escaping set of a quasiregular mapping
• Karpińska's paradox in dimension three
• Dynamics of a higher dimensional analog of the trigonometric functions
• Fatou-Julia theory for non-uniformly quasiregular maps
• On the set where the iterates of an entire function are bounded
• Julia sets of uniformly quasiregular mappings are uniformly perfect

Cited by in corpus (7)

• Hollow quasi-Fatou components of quasiregular maps
• The size and topology of quasi-Fatou components of quasiregular maps
• The bungee set in quasiregular dynamics
• The escaping set in transcendental dynamics
• Hausdorff dimension in quasiregular dynamics
• Permutable Quasiregular Maps
• On slow escaping and non-escaping points of quasimeromorphic mappings