Publication Date

2025

Document Type

Dissertation/Thesis

First Advisor

Fletcher, Alastair N.

Degree Name

Ph.D. (Doctor of Philosophy)

Legacy Department

Department of Mathematical Sciences

Abstract

This dissertation is on the dynamics of planar quasiregular maps of degree 2. These maps have three parameters. One of these parameters determines the distortion, one of them determines the orientation, and the last one is a shift parameter. Our first objective is to construct a new dynamical version of Green's functions for these maps. These functions provide information about the escaping sets of our quasiregular maps and are used to prove topological results about them. We show that the boundary of the escaping set is either connected or consists of infinitely many components. Additionally, we prove that if one of these maps has 2^n+1 periodic points of period n, then the boundary of the escaping set is not totally disconnected, in contrast to the classical case in complex dynamics. We give examples of combinations of parameters in which this occurs. Our second main objective is to determine the behavior of the fixed points of these maps depending on their location in the plane, dependent on the distortion and orientation parameters. We give a complete classification of the regions in the plane where the fixed points of our map have real eigenvalues or complex conjugate eigenvalues based on the derivative of our map, and we also determine if they have attracting, repelling, or saddle behavior.

Extent

108 pages

Language

en

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

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Mathematics Commons

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