Date of Award
Spring 5-11-2024
Degree Type
Thesis
Degree Name
Master of Science - Mathematical Sciences
Department
Mathematics and Statistics
First Advisor
Jacob Pratscher
Second Advisor
Roy Joe Harris
Third Advisor
Clint Richardson
Fourth Advisor
Jeremy Becnel
Abstract
The Mandelbrot set is a mathematical mystery. Finding its home somewhere be-
tween holomorphic dynamics and complex analysis, the Mandelbrot set showcases
its usefulness in fields across the many realms of math—ranging from physics to nu-
merical methods and even biology. While typically defined in terms of its bounded
sequences, this thesis intends to illuminate the Mandelbrot set as a type of param-
eterization of connectivity itself, specifically that of complex-valued rational maps
of the form z → z² + c. This fully illustrated guide to the Mandelbrot set merges
the worlds of intuition and theory with a series of self-contained arguments found in
published texts over the years since the Mandelbrot set’s conception—all to answer
one question: is the Mandelbrot set connected? That is, are there any pieces of the
Mandelbrot set just ‘hanging off’ ? To answer this, we will appeal to the proof by the
now-famous collaborators Adrien Douady and John Hubbard, whose work deep-dives
into topologically grounded ideas and makes use of some functional analysis.
Repository Citation
Shirley, James, "Exploring the Mandelbrot Set" (2024). Electronic Theses and Dissertations. 546.
https://scholarworks.sfasu.edu/etds/546
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.
