Date of Award
8-2025
Document Type
Thesis
Degree Name
Master of Science (MS)
Department
Mathematical Sciences
Committee Chair/Advisor
Michael Burr
Committee Member
Ryann Cartor
Committee Member
Martin Schmoll
Abstract
The chaotic and fractal nature of Julia sets makes them difficult to graph. This research aims to provide graphical approximations of Julia sets with known and guaranteed levels of accuracy. We implement three methods to approximate Julia sets with c values chosen from the main cardioid of the Mandelbrot set. Each method utilizes different properties of these Julia sets. The exclusion method makes use of the fact that a Julia set of this type is topologically a circle. Attracting and repelling fixed points are used to find a region on the interior of the Julia set and a region on the exterior of the Julia set. The region between these is guaranteed to contain the Julia set. The backward solving method uses the inverse of the function used to define Julia sets. The inverse function is applied iteratively to points on the complex plane. The Julia set attracts these points, so they move closer to the Julia set with each iteration. The homotopy continuation method uses the fact that holomorphic motion deforms one Julia set into another. Periodic points are tracked from a simple Julia set to a more complicated one to create a polygon approximation of the more complicated Julia set.
Recommended Citation
Kaufman, Hannah, "Certified Computation of Julia Sets via Numerical Methods" (2025). All Theses. 4599.
https://open.clemson.edu/all_theses/4599