19 - Various Examples of Compactly Nonrecurrent Elliptic Functions

Summary

The purpose of this chapter is to provide examples of elliptic functions with prescribed properties of the orbits of critical points (and values). We are primarily focused on constructing examples of various classes of compactly nonrecurrent elliptic functions. All these examples are either Weierstrass elliptic functions or their modifications. The dynamics of such functions depends heavily on the lattice. The first three sections of this chapter have a preparatory character and, respectively, describe the basic dynamical and geometric properties of all Weierstrass elliptic functions generated by square and triangular lattices. We then provide simple constructions of many classes of elliptic functions discerned in the previous chapter. We essentially cover all of them. All these examples stem from Weierstrass $\wp$ functions. Finally, we also provide some different, interesting on their own, and historically first examples of various kinds of Weierstrass $\wp$ elliptic functions and their modifications coming from a series of papers by Hawkins and her collaborators.