Matricial Frameworks for the Mandelbrot and Filled Julia Sets

Authors

  • Eric Babcock Embry-Riddle Aeronautical University
  • Dawson Brindle Embry-Riddle Aeronautical University Prescott
  • Mitch Hamidi Embry-Riddle Aeronautical University
  • Lara Ismert Embry-Riddle Aeronautical University

DOI:

https://doi.org/10.33043/7EAyzVQVyF

Abstract

Both the Mandelbrot set and filled Julia sets are subsets in the complex plane derived by studying iterations of complex polynomials. We develop a matricial framework to establish an alternate form of iteration by complex polynomials using a sequence of affine transformations. Using this framework, we are able to check membership in a filled Julia set and the Mandelbrot set by studying boundedness of sequences of matrices. Specifically, we show that a complex number belongs to the Mandelbrot set if and only if a particular sequence of matrices is bounded in the operator norm, and a complex number belongs to a filled Julia set if and only if a particular sequence of matrices is bounded in operator norm.

Author Biographies

Dawson Brindle, Embry-Riddle Aeronautical University Prescott

Dawson Brindle earned a bachelor’s degree in aerospace engineering from Embry-Riddle Aeronautical University Prescott in 2023 and is currently working as a specialty test engineer. During his free time, he likes to try making different types of coffee and learning new concepts.

Mitch Hamidi, Embry-Riddle Aeronautical University

Mitch Hamidi earned his Ph.D. in mathematics from the University of Nebraska-Lincoln in 2019 and is currently an assistant professor of mathematics at Embry-Riddle Aeronautical University in Prescott, Arizona. His research interests lie in operator algebras and quantum information theory. In his free time, he enjoys playing music and hanging out with his wife, 3-month old son, and his four cats.

Lara Ismert, Embry-Riddle Aeronautical University

Lara Ismert earned her Ph.D. in mathematics from the University of Nebraska-Lincoln in 2019 and is currently an assistant professor of mathematics at Embry-Riddle Aeronautical University in Prescott, Arizona. Her research interests lie in operator algebras and quantum information theory. When she finds spare time, she loves performing in musical theatre and spending time with her husband, 3-month old son, and her four cats.

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Published

2026-03-03

How to Cite

Babcock, E., Brindle, D., Hamidi, M., & Ismert, L. (2026). Matricial Frameworks for the Mandelbrot and Filled Julia Sets. Mathematics Exchange, 17(1), 118–133. https://doi.org/10.33043/7EAyzVQVyF

Issue

Vol. 17 No. 1 (2023): Mathematics Exchange, Fall 2023

Section

Articles

License

Copyright (c) 2026 Eric Babcock, Dawson Brindle, Mitch Hamidi, Lara Ismert

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.