Chapter 47 - Open problems in complex dynamics and “complex” topology

Abstract

This chapter discusses open problems in complex dynamics and “complex” topology. Complex dynamics is a field in which a large number of captivating structures from planar topology occur quite naturally. Of primary interest in complex dynamics is the Julia set of a complex analytic function. These sets are often quite interesting from a topological point of view. Examples of functions whose Julia sets (or invariant subsets of the Julia sets) are Cantor bouquets, indecomposable continua, and Sierpiński curves are also elaborated. Because both the topology of and the dynamics on these Julia sets is so rich, it is little wonder that there are many open problems in this field. The goal of this chapter is to describe several of these problems. To keep the exposition accessible, attention is restricted to two very special families of functions, namely the complex exponential function and a particular family of rational maps.