In this chapter, we provide a classical account of Kolmogorov–Sinai metric entropy for measure-preserving dynamical systems. We prove the Shannon–McMillan–Breimann Theorem and, based on Abramov's Formula, define the concept of Krengel's Entropy of a conservative system preserving a (possibly infinite) invariant measure.

In this chapter we encounter for the first time in the book holomorphic dynamics. Its settings are somehow technical and it has, on the one hand, a very preparatory character serving the needs of constructing and controlling Sullivan's conformal measures in various subsequent parts of the book; already in the next chapter. On the other hand, this chapter is important and interesting on its own. Indeed, its hypotheses are very general and flexible, and under such weak assumptions it establishes in the context of holomorphic dynamics such important results as Pesin's Theory, Ruelle's Inequality, and Volume Lemmas.

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.

In this chapter, we encounter the elegant and powerful concept of conformal measures, which is due to Patterson for Fuchsian groups and due to Sullivan for all Kleinian groups and rational functions of the Riemann sphere. We deal, in this chapter, with conformal measures in the settings of the previous chapter. Sullivan conformal measures and their invariant versions will form the central theme of Volume 2. In fact, the current chapter is the first and essential step for construction of Sullivan conformal measures for elliptic functions. It deals with holomorphic maps defined on some open neighborhood of a compact invariant subset of a parabolic Riemann surface. We provide a fairly complete account of Sullivan conformal measures in such a setting. We also introduce several dynamically significant concepts and sets such as radial or conical points and several fractal dimensions defined in dynamical terms. We relate them to exponents of conformal measures. However, choosing the most natural, at least in our opinion, framework, we do not restrict ourselves to conformal dynamical systems only but present, in the first section of this chapter, a fairly complete account of the theory of general conformal measures.

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.

This chapter deals with conformal graph directed Markov systems, its special case of iterated function systems, and thermodynamic formalism of countable alphabet subshifts of finite type, frequently also called topological Markov chains. This theory started in the mid-1990s with the papers and a book by the second named author and Mauldin. It was there where the concept of conformal measures due to Patterson and Sullivan was adapted to the realm of conformal graph directed Markov systems and iterated function systems. We present here some elements of this theory, primarily those related to conformal measures and Bowen's Formula for the Hausdorff dimension of limit sets of such systems. In particular, we get a cost-free, effective, lower estimate for the Hausdorff dimension of such limit sets. More about conformal graph directed Markov systems can be found in many papers and books. In the second volume of the book, we apply these techniques, by means of nice sets in the next chapter, to get a good, explicit estimate from below of Hausdorff dimensions of Julia sets of elliptic functions and to explore stochastic properties of invariant versions of conformal measures for parabolic and subexpanding elliptic functions.