Publication Date

2024

Document Type

Dissertation/Thesis

First Advisor

Fletcher, Alastair N.

Degree Name

Ph.D. (Doctor of Philosophy)

Legacy Department

Department of Mathematical Sciences

Abstract

In this dissertation, we study the set of Julia limiting directions of quasiregular maps. The work combines the study of dynamics of quasiregular maps and applications of nonlinear potential theory to quasiregular maps. Our main result shows that the set of Julia limiting directions of a transcendental-type $K$-quasiregular map $f:\R^n\to \R^n$ must contain a component of a certain measure, depending on the dimension $n$, the maximal dilatation $K$, and the order of growth of $f$. In particular, we show that if the order of growth is small enough, then every direction is a Julia limiting direction. The main tool in proving these results is a new version of a Phragm\'en-Lindel\"of principle for sub-$F$-extremals in sectors, where we allow for boundary growth of the form $O( \log |x| )$ instead of the previously considered $O(1)$ bound. In addition, we discuss some potential problems for future work.

Extent

121 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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