Iterations on infinite-dimensional Teichmüller spaces and transcendental dynamics

In complex dynamics it is usually important to understand the dynamical behavior of critical (or singular) orbits. For quadratic polynomials, this leads to the study of the Mandelbrot set. In our talk we present a classification of some explicit families of transcendental entire functions for which all singular values escape, i.e. functions belonging to the complement of the "transcendental analogue" of the Mandelbrot set.
A key ingredient is a generalization of the famous Thurston's Topological Characterization of Rational Functions. Analogously to Thurston's theorem, we define a dynamical system on a specially chosen Teichmüller space and investigate its properties. But unlike the classical case, the underlying Teichmüller space is infinite-dimensional which leads to a completely different framework.
We also discuss potential further generalizations and some of the interesting open questions which arose during the research.