Abstract

Let f be a function holomorphic on the complex plane C, so f is either a polynomial or an entire transcendental map. The objects of main dynamical interest are the Fatou set F(f) of the map f, defined as the maximal open set where the iterates, f_n, form a normal family in the sense of Montel, and its complement the Julia set J(f). Another dynamically relevant set is the escaping set I(f) of f, which consists of all points that tend to infinity under iteration of f. Interesting on its own, the escaping set is also closely related to J(f): in many cases J(f) equals the boundary of I(f).

A powerful tool for the study of Julia sets of polynomials are dynamic rays (also known as external rays) which are certain curves to infinity that lie in the escaping set. Since Douady and Hubbard′s famous work on the Mandelbrot set, dynamic rays, and their landing properties in particular, have provided one of the main ingredients in polynomial dynamics and have been used to establish various important results. While for polynomials dynamic rays arise in a very natural way, this is not the case when we look at transcendental maps. Still, for certain classes of entire transcendental functions there exist curves in the escaping set which can be seen as an analog of dynamic rays of polynomials.

In this talk we will describe some properties of escaping sets and dynamic rays, and explain the difficulties that arise when we try to transfer results and methods from the polynomial to the transcendental case. We will also give some recent results about the topology of escaping sets and Julia sets of certain entire transcendental functions like hyperbolic and subhyperbolic maps.