17 - Geometry and Dynamics of (All) Elliptic Functions

Summary

In this chapter, we deal with general nonconstant elliptic functions, i.e., we impose no constraints on a given nonconstant elliptic function. We first deal with the forward and backward images of open connected sets, especially with connected the components of the latter. We mean to consider such images under all iterates $f^n$, $n\ge 1$, of a given elliptic function $f$. We do 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. We also provide sufficient conditions for the restrictions of iterates $f^n$ to such components to be proper or covering maps. Both of these methods are our primary tools to study the structure of the connected components backward images of open connected sets. In particular, they give the existence of holomorphic inverse branches if "there are no critical points.’’ Holomorphic inverse branches will be one of the most common tools used throughout the rest of the book. We then apply these results to study images and backward images of connected components of the Fatou set.