Robert Kozma : Chaotic dynamics of Perturbed Quadratic Maps

Posted by Trevor Leslie , part of the Graduate Analysis Seminar.

At

Feb. 25, 2015, 2 p.m.

In

SEO 427

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.