Exploring the Mandelbrot Set

The Mandelbrot set is a mathematical mystery. Finding its home somewhere be-<br />tween holomorphic dynamics and complex analysis, the Mandelbrot set showcases<br />its usefulness in fields across the many realms of math—ranging from physics to nu-<br />merical methods and even biology. While typically defined in terms of its bounded<br />sequences, this thesis intends to illuminate the Mandelbrot set as a type of param-<br />eterization of connectivity itself, specifically that of complex-valued rational maps<br />of the form z → z² + c. This fully illustrated guide to the Mandelbrot set merges<br />the worlds of intuition and theory with a series of self-contained arguments found in<br />published texts over the years since the Mandelbrot set’s conception—all to answer<br />one question: is the Mandelbrot set connected? That is, are there any pieces of the<br />Mandelbrot set just ‘hanging off’ ? To answer this, we will appeal to the proof by the<br />now-famous collaborators Adrien Douady and John Hubbard, whose work deep-dives<br />into topologically grounded ideas and makes use of some functional analysis.

Name thesis:degree_name

Master of Science - Mathematical Sciences

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics and Statistics

Year dc:date.available

2024

Author dc:creator

• Shirley, James

Contributors dc:contributor

• Jacob Pratscher
• Roy Joe Harris
• Clint Richardson

• Mandelbrot
• Complex Analysis
• Analysis
• Connected
• Connectivity
• Dynamical Systems

Repository record dc:identifier

https://scholarworks.sfasu.edu/etds/546

OAI identifier oai:identifier

oai:scholarworks.sfasu.edu:etds-1598