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 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.

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.

We deal, in this chapter, with refined stochastic laws for dynamical systems preserving an infinite measure. This is primarily the Darling–Kac Theorem. We make use of some recent progress on this theorem and related issues, mainly due to Zweimüller, Thaler, Theresiu, Melbourne, Gouëzel, Bruin, Aaronson, and others, but we do not go into the most recent subtleties and developments of this branch of infinite ergodic theory. We do not need them for our applications to elliptic functions.

This chapter is devoted to some selected topics of geometric function theory. Its is entirely classical, meaning that no dynamics is involved. We deal here with Riemann surfaces, normal families and Montel's Theorems, extremal lengths, and moduli of topological annuli. The central theme is the various versions of the Koebe Distortion Theorems. These theorems form a beautiful, elegant, and powerful tool of complex analysis. We prove them and provide their many versions of analytic and geometric character. These theorems form an indispensable tool for nonexpanding holomorphic dynamics and their applications very frequently occur throughout the book, most notably when dealing with holomorphic inverse branches, conformal measures, and Hausdorff and packing measures. The version of the Riemann–Hurwitz Formula, appropriate in the context of transcendental meromorphic functions, which we treat at length in Volume 2, is a very helpful tool to prove the existence of holomorphic inverse branches and an elegant and probably the best tool to control the topological structure of connected components of inverse images of open connected sets under meromorphic maps. Our approach to the Riemann–Hurwitz Formula stems from that of Beardon's book on rational functions. We modify it to fit our context of transcendental meromorphic functions.

As has already been mentioned when we discussed the previous chapter, in the current one we introduce and thoroughly study the objects related to the powerful concept of nice sets, which will be our indispensable tool in the last part of the second volume of the book, leading, along with the theory of conformal graph directed Markov systems, the first return map method, and the techniques of Young towers, to such stochastic laws as the exponential decay of correlations, the Central Limit Theorem, and the Law of the Iterated Logarithm follows for large classes of elliptic functions constituted by subexpanding and parabolic ones. However, the main objective of this chapter is to prove the existence of nice and pre-nice sets.

In this chapter, we collect and prove basic concepts and theorems of classical thermodynamic formalism. This includes topological pressure, variational principle, and equilibrium states. We provide several illustrating examples.