Transcendental dynamics studies the iteration of functions of one complex variable that have at least one essential singularity (e.g., non-polynomial analytic self-maps of the complex plane, such as exponential or trigonometric functions). That is, for such a map f and a complex starting value z, we consider the long-term behaviour of \(f^n(z)=f(f(...(f(z))...))\). While the subject is nearly a century old, it has received increasing attention recently, partly due to intriguing connections with other areas of mathematics.
We shall begin with a gentle introduction to the field by discussing elementary properties of an example already studied by Fatou in 1926. We will see how this study led him to ask a question concerning the existence of "hairs" (curves to infinity), which remained unresolved until recently. In the remainder of the talk, I will discuss how work that began with a negative solution to this problem (joint with Rottenfußer, Rückert and Schleicher, Ann. of Math., 2011) has more recently led to a detailed understanding of the dynamical behaviour of a large and important class of functions.
In the course of this exploration, we will discover a number of intriguing and seemingly "pathological" topological phenomena, and see how these arise quite naturally in the dynamics of analytic self-maps of the plane. The talk will be accessible to a general mathematical audience, including postgraduate students.