Topology and dynamics of the escaping set

Abstract

The set of points that escape to infinity under iteration is a fundamental object in complex dynamics, often providing valuable insight into the global behavior of iterates and their relationship with the Fatou and Julia sets. The aim of this thesis is to study the topological and dynamical properties of the escaping set for polynomials, as well as transcendental entire and meromorphic functions.

As an original contribution, we show that for the function $f(z) = z - \tan(z)$, the Julia set becomes connected upon adjoining infinity and contains the escaping set, which is totally disconnected. This is the first known nontrivial example exhibiting such behavior.