Graphs and entire functions

In this talk, I will discuss how graph-theoretic ideas can be used to obtain results in one-dimensional holomorphic dynamics. In particular, I will explain how a large and dynamically important class of entire functions can be approximated by complex polynomials in a way that preserves much of the underlying dynamical behavior, making these approximations dynamically meaningful. This is joint work with M. Mukundan and B. Reinke. I will also describe how the questions that arise connect to certain dynamical systems in several complex variables and to Teichmüller theory. Finally, time permitting, I will say a few words about the so-called Thurston theory in holomorphic dynamics and some ongoing work in this area.