The escaping set of a quasiregular mapping
Walter Bergweiler, Alastair Fletcher, Jim Langley, Janis Meyer
TL;DR
This paper investigates the escaping set of quasiregular mappings, showing that rapid growth ensures the existence of points tending to infinity under iteration, with unique topological properties compared to entire functions.
Contribution
It establishes conditions for the existence of an escaping set in quasiregular mappings and explores its topological structure, contrasting it with entire functions.
Findings
Existence of an escaping set for rapidly growing quasiregular mappings
The escaping set has an unbounded component
The closure of the escaping set may have a bounded component
Abstract
We show that if the maximum modulus of a quasiregular mapping f grows sufficiently rapidly then there exists a non-empty escaping set I(f) consisting of points whose forward orbits under iteration tend to infinity. This set I(f) has an unbounded component but, in contrast to the case of entire functions on the complex plane, the closure of I(f) may have a bounded component.
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