On the iteration of quasimeromorphic mappings

Warren, Luke (2020) On the iteration of quasimeromorphic mappings. PhD thesis, University of Nottingham.

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Abstract

This thesis is concerned with the iterative behaviour of quasimeromorphic mappings of transcendental type, which form higher-dimensional analogues of transcendental meromorphic functions on the complex plane. We extend classical Julia theory and results on escaping points from complex dynamics to the new setting. This complements recent dynamical advancements for quasiregular mappings, which are higher-dimensional analogues of holomorphic functions on the complex plane.



First, we define the Julia set for quasimeromorphic mappings of transcendental type and investigate its properties through two cases based on the cardinality of the backward orbit of infinity. To this end, we construct an example of a quasiregular mapping in dimension 3 with exactly one zero, subsequently showing that both cases arise. We then generalise an important growth result by Bergweiler to quasiregular mappings defined near an essential singularity. From this we show that many classical properties of the Julia set hold in our case; this includes proving a cardinality conjecture that remains open for general quasiregular mappings.



Next, we study the existence of escaping and non-escaping points in the new Julia set. In particular, following work by Nicks, we show that there exist points that escape arbitrarily slowly to infinity under iteration. Moreover we prove some basic relationships between the Julia set, the escaping set, the set of points whose orbit is bounded, and the set of points whose orbit is neither bounded nor tends to infinity.



Finally, motivated by the work of Bolsch, we consider a class of mappings that is closed under composition and contains all quasimeromorphic mappings. Adapting previous methods, we show that the above results for quasimeromorphic mappings of transcendental type continue to hold for their iterates in a natural way. We also define a generalised escaping set, consisting of points whose orbits accumulate to some essential singularities or their pullbacks, and prove some existence results regarding points with specified accumulation sets.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Nicks, Daniel
Edjevet, Martin
Keywords: Quasimeromorphic, quasiregular, iteration, Julia set, slow escape, meromorphic, escaping set
Subjects: Q Science > QA Mathematics > QA299 Analysis
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 60523
Depositing User: Warren, Luke
Date Deposited: 24 Jul 2020 04:40
Last Modified: 24 Jul 2020 04:40
URI: http://eprints.nottingham.ac.uk/id/eprint/60523

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