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.

In this chapter, we provide a very detailed qualitative and quantitative description of the local behavior of iterates of locally and globally defined analytic functions around their rationally indifferent periodic points. We also examine the structure of corresponding Leau–Fatou flower petals, including the Fatou Flower Petal Theorem. These will be frequently used in further chapters of the book devoted to the study of compactly nonrecurrent parabolic elliptic functions.

We analyze the structure of Fatou components and the structure of their boundaries in greater detail. In particular, we study the simple connectedness of such components. We also bring up the definitions of Speiser class $\cS$ and Eremenko–Lyubich class $\cB$ and we prove some structural theorems about their Fatou components. In particular, we prove no existence of Baker domains and wandering domains (Sullivan Nonwandering Theorem) for class $\cS$, the latter in Appendix B.

The results of the previous chapter are not the last word about Sullivan conformal measures. Left alone, these measures would be a kind of curiosity. Their true power, meaning, and importance come from their geometric characterizations and their usefulness, one could even say indispensability, in understanding geometric measures on Julia sets, i.e., their Hausdorff and packing $h$-dimensional measures, where, we recall, $h=\HD(J(f))$. This is fully achieved in the present chapter. Having said that, this chapter can be viewed from two perspectives. The first is that we provide therein a geometrical characterization of the $h$-conformal measure $m_h$, which, with the absence of parabolic points, turns out to be a normalized packing measure, and the second is that we give a complete description of geometric, Hausdorff, and packing measures of the Julia sets $J(f)$. Owing to the fact that the Hausdorff dimension of the Julia set of an elliptic function is strictly larger than $1$, this picture is even simpler than for nonrecurrent rational functions.

This chapter, interesting on its own, is devoted to presenting some account of the classical theory of elliptic functions. Almost no dynamics is involved here. We follow classical book expositions. We will actually not need this chapter anywhere else in the book except in one chapter where we provide a lot of examples of elliptic functions, mainly Weierstrass $\wp$ functions, but not only them.

In this chapter, we deal with the dynamical rigidity of compactly nonrecurrent regular elliptic functions. The issue at stake is whether a weak conjugacy such as a Lipschitz one on Julia sets can be promoted to a much better one such as affine on the whole complex plane $\C$. This topic has a long history and goes back at least to the seminal paper by Sullivan, treating among others the dynamical rigidity of conformal expanding repellers. Our approach in this chapter stems from this article by Sullivan.