Permutable Quasiregular Maps

Abstract

Let $f$ and $g$ be two quasiregular maps in ${\mathbb{R}}^d$ that are of transcendental type and also satisfy $f\circ g=g\circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form $f$ and $g=\varphi \circ f$, where $\varphi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.

arXiv comment15 pages, final version, To appear in Math. Proc. Cambridge Philos. Soc

References in corpus (9)

• Iteration of meromorphic functions
• Fast escaping points of entire functions
• The escaping set of a quasiregular mapping
• Foundations for an iteration theory of entire quasiregular maps
• The fast escaping set for quasiregular mappings
• The size and topology of quasi-Fatou components of quasiregular maps
• Hollow quasi-Fatou components of quasiregular maps
• The Julia set and the fast escaping set of a quasiregular mapping
• On permutable meromorphic functions

Cited by in corpus (1)

• The escaping set in transcendental dynamics