Matricial Frameworks for the Mandelbrot and Filled Julia Sets
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.
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Analytic and geometric function theory
Holomorphic and Operator Theory
Publication details
- Journal / source
- Mathematics Exchange
- Volume
- 17
- Issue
- 1
- Pages
- 118–133
- Publication date
- 3 March 2026
- Language
- EN