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.

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.

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.

In this chapter, we use the fruits of the, already proven, existence of Sullivan conformal measures with a minimal exponent and its various dynamical characterizations. Having compact nonrecurrence, we are able to prove in this chapter that this minimal exponent is equal to the Hausdorff dimension $\HD(J(f))$ of the Julia set $J(f)$, which we denote by $h$. We also obtain here some strong restrictions on the possible locations of atoms of such conformal measures. In the last section of this chapter, we culminate our work on Sullivan conformal measures for elliptic functions treated on their own. There and from then onward, we assume that our compactly nonrecurrent elliptic function is regular (we define this concept). For this class of elliptic functions, we prove the uniqueness and atomlessness of $h$-conformal measures along with their first fundamental stochastic properties such as ergodicity and conservativity.

This chapter is in a sense a core of our book. Using what has been done in all previous chapters, we prove here the existence and uniqueness, up to a multiplicative constant, of a $\sg$-finite $f$-invariant measure $\mu_h$ equivalent to the $h$-conformal measure $m_h$. Furthermore, still heavily relying on what has been done in all previous chapters, particularly on conformal graph directed Markov systems, nice sets, first return map techniques, and Young towers, we provide here a systematic account of ergodic and refined stochastic properties of the dynamical system $(f,\mu_h)$ generated by such subclasses of compactly nonrecurrent regular elliptic functions as normal subexpanding elliptic functions of finite character and parabolic elliptic functions. By stochastic properties, we mean the exponential decay of correlations, the Central Limit Theorem, and the Law of the Iterated Logarithm for subexpanding functions, the Central Limit Theorem for those parabolic elliptic functions for which the invariant measure $\mu_h$ is finite and an appropriate version of the Darling–Kac Theorem establishing the strong convergence of weighted Birkhoff averages to Mittag–Leffler distributions for those parabolic elliptic functions for which the invariant measure $\mu_h$ is infinite.