#### Publication Date

2018

#### Document Type

Dissertation/Thesis

#### First Advisor

Fletcher, A. (Alastair)

#### Degree Name

Ph.D. (Doctor of Philosophy)

#### Legacy Department

Department of Mathematical Sciences

#### LCSH

Mathematics

#### Abstract

The work presented in this thesis pertains to the function theoretic and geometric aspects of quasiregular mappings. The central theme of this thesis involves the properties of strongly automorphic quasiregular mappings and the properties and dynamics of their associated uniformly quasiregular (uqr) maps arising as solutions to a Schroder functional equation. Of key interest here is when the strongly automorphic mapping is a linearizer of the associated uqr mapping, since one can deduce properties of the uniformly quasiregular mapping via properties of a linearizer. Chapter 1 contains introductions to the key facets of complex analysis and complex dynamics, some of which will be generalized to the setting of quasiregular mappings as new work in later chapters. Chapter 1 also contains the necessary introduction and background of the theory of quasiregular mappings and dynamics, which will be used in later chapters. Chapter 2, which is modified material from published work, presents some results on how linearizers of a uqr map associated with a repelling fixed point are related. It turns out one can pre-compose via a quasiconformal map with one linearizer to get another. Further conditions can be imposed on the uqr map to achieve a stronger equivalence. This provides a quasiregular generalization of the uniqueness portion of Koenig's Linearization Theorem. Chapter 3, which is modified material from submitted work, presents some foundational results on the construction and properties of strongly automorphic quasiregular mappings, and determines the topology of the Julia sets of their associated uqr mappings. A strongly automorphic mapping can be constructed through the action of an orientation-preserving crystallographic group G acting on R^k, n--1 [less than or equal to] k [less than or equal to] n, with spherical orbifold, i.e., R^k/ G [asymptotically equal to] S^k. We obtain a larger class of strongly automorphic mappings by allowing these groups to be conjugated via an entire quasiconformal map. A theorem of Ritt showed that if the linearizer of a rational map is periodic, then the rational map is either a power map, Chebyshev polynomial or Lattes map. The converse of this theorem is true, except for one exceptional case. This result can be generalized to the setting of uqr mappings which have strongly automorphic linearizers associated with a repelling fixed point. The Julia sets of power maps, Chebyshev polynomials and Lattes rational maps are S¹, [-1,1] and C [union] {[infinity]} respectively. Consequently, Ritt's theorem gives sufficient conditions for when the Julia set of a rational map is smooth: a linearizer associated with a repelling fixed point is periodic. This too has a quasiregular analogue. If a uqr map f : R^n [right arrow] R^n has a linearizer associated with a repelling fixed point which is strongly automorphic with respect to a quasiconformal conjugate crystallographic group, then the Julia set is either a quasi-sphere, quasi-disk or R^n. A quasi-sphere is the image of S^[n--1] under an ambient quasiconformal map and a quasi-disk is the image of an n--1 dimensional ball under an ambient quasiconformal map. Chapter 4, which is modified material from submitted work, addresses the converse to the topological portion of extending Ritt's Theorem: if the Julia set of a uqr map f is a quasi-sphere or quasi-disk, what can be said of f? The set of conical points of f contains the set of points where a linearization of the iterates of f can be performed, and by a theorem of Miniowitz, necessarily belongs to the Julia set. If the Julia set of f is a quasi-disk or quasi-sphere and the set of conical points is dense in the Julia set, then f agrees with a Chebyshev-type map or power-type map respectively on its Julia set. Further, a Denjoy-Wol Theorem for proper surjective uqr maps in dimension 3 for parabolic xed points is given. It is shown that such uqr maps can be extended to entire uqr maps, and that the restriction of the map to the unit sphere is itself uqr in the topology of the sphere. If the uqr map has two xed points on the sphere, then the map has a continuum of xed points on the sphere. Since the restriction of the uqr map to the sphere arises as a quasiconformal conjugate of a rational map, then the Identity Theorem gives a contradiction for uqr maps which are non-injective. (Abstract shortened by ProQuest.)

#### Recommended Citation

MacClure, Douglas Malcolm, "On uniformly quasiregular solutions to the Schroder functional equation and strongly automorphic mappings" (2018). *Graduate Research Theses & Dissertations*. 4186.

https://huskiecommons.lib.niu.edu/allgraduate-thesesdissertations/4186

#### Extent

172 pages

#### Language

eng

#### Publisher

Northern Illinois University

#### Rights Statement

In Copyright

#### Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

#### Media Type

Text

## Comments

Advisors: Alastair N. Fletcher.||Committee members: Douglas Bowman; Ilya Krishtal; Jeffrey Thunder.||Includes illustrations.||Includes bibliographical references.