18 - Dynamics of Compactly Nonrecurrent Elliptic Functions

Summary

In this chapter, we define the class of nonrecurrent and, more notably, the class of compactly nonrecurrent elliptic functions. This is the class of elliptic functions that will be dealt with by us from now until the end of the book in greatest detail. Our treatment of nonrecurrent elliptic functions is based on, in fact, is possible at all, an appropriate version of the breakthrough Mañé’s Theorem. The first section of this chapter is entirely devoted to proving this theorem, its first most fundamental consequences, and some other results surrounding it. The next two sections of this chapter, also relying on Mañé’s Theorem, provide us with further refined technical tools to study the structure of Julia sets and holomorphic inverse branches. The last section of this chapter systematically defines and describes various subclasses of the, mainly compactly nonrecurrent, elliptic functions we will be dealing with in the book. Among them are expanding, hyperbolic, topologically hyperbolic, subhyperbolic, and parabolic elliptic functions.