Fundamental Properties of Meromorphic Dynamical Systems

In this chapter, we provide a relatively short and condensed account of the topological dynamics of all meromorphic functions with an emphasis on Fatou domains, including Baker domains that are exclusive for transcendental functions and do not occur for rational functions. We also carry out a thorough analysis of the singular set of the inverse of a meromorphic function and all its iterates. In particular, we study at length asymptotic values and their relations to transcendental tracts. The results of this analysis will be used very frequently to study the topological structure of connected components (and their boundaries) of Fatou sets in this part of the book and a countless number of times when we move on to dealing with elliptic functions.