Cantor and Sierpinski, Julia and Fatou:
Crazy Topology meets Complex Dynamical Systems

Abstract

Topologists are used to looking at such complicated planar sets as Cantor bouquets, indecomposable continua, and Sierpinski curves. Each of these spaces has some very interesting and almost counter-intuitive properties. Often these spaces are considered as "exceptions" or counterexamples to specific topological constructions. In this talk we will describe how each of these sets arise naturally and very often as the Julia sets of complex dynamical systems. We give specific examples of how infinitely many types of each of these sets arise in certain families of complex exponentials and rational maps. We also describe the rich dynamics that occur on each of these sets.