Graduate Analysis Seminar
Robert Kozma
UIC
Chaotic dynamics of Perturbed Quadratic Maps
Abstract: Even the simplest nonlinear function $z^2+c$ gives rise to chaotic dynamics resulting in very
intricate mathematical objects such as the Mandelbrot set, and Julia sets. In this talk we first
go over some basic properties of the Mandelbrot set, and Julia sets. We then go on to
describe novel results on the behavior of Julia sets of certain families of rational maps
arising from $z^2+c$ by adding a small perturbation term $\lambda/z^2$ and taking the limit as $\lambda \to 0$.
We will see that for certain $c$ values the resulting Julia sets have some astonishing
geometric and topological properties. Using symbolic dynamics and Cantor necklaces, we
prove that as $\lambda\to 0$, the Julia for this family set evolves into a space-filling fractal curve.
Wednesday February 25, 2015 at 2:00 PM in SEO 427