Daniel A. Nicks
Hollow quasi-Fatou components of quasiregular maps
Nicks, Daniel A.; Sixsmith, David J.
David J. Sixsmith
We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in Rd is called hollow if it has a bounded complementary component. We show that for each d≥2 there exists a quasiregular map of transcendental type f:Rd→Rd 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.
|Journal Article Type||Article|
|Publication Date||May 31, 2017|
|Journal||Mathematical Proceedings of the Cambridge Philosophical Society|
|Publisher||Cambridge University Press (CUP)|
|Peer Reviewed||Peer Reviewed|
|APA6 Citation||Nicks, D. A., & Sixsmith, D. J. (2017). Hollow quasi-Fatou components of quasiregular maps. Mathematical Proceedings, 162(3), doi:10.1017/S0305004116000840|
|Related Public URLs||http://arxiv.org/abs/1505.08114|
|Copyright Statement||Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf|
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf
You might also like
The dynamics of quasiregular maps of punctured space
The bungee set in quasiregular dynamics
Baker's conjecture for functions with real zeros
Periodic domains of quasiregular maps
The size and topology of quasi-Fatou components of quasiregular maps