This is an interesting and substantial book which makes a valuable contribution to the theory of iterations of expanding and non-uniformly expanding holomorphic maps, an area with a long tradition as well as a lot of current activities. The book also covers topics from geometric measure theory for the associated invariant fractal sets. Probability measures on such sets have proven to be efficient tools for gaining deeper insight into the fractal nature of these sets, in particular their Hausdorff dimension and various other fractal dimensions. The book begins with a comprehensive chapter on abstract ergodic theory which is then followed by chapters on uniform distance expanding maps and thermodynamical formalism. This type of material is applicable in many other branches of the theory of dynamical systems and related fields. A more precise outline of the book follows.
In Chapters 1 and 2, the authors give some introductory definitions and basic examples. These chapters represent a continuation of the introduction. Also, an introduction to abstract (finite) ergodic theory is given. The reader will find proofs of the Birkhoff Ergodic Theorem and the Shannon-McMillan-Breiman Theorem. Moreover, one finds introductions to entropy, to measurable partitions, to conditional measures on Lebesgue spaces and to the idea of natural extensions. The approach follows Rohlin’s ideas which later were developed more rigorously by Kornfeld, Fomin and Sinai. Subsequently, as a preparation for the later application of Rohlin’s theory to finite-to-one rational maps, Chapter 2 continues by considering countable-to-one endomorphisms, an introduction of the notion of the Jacobian and a discussion of mixing properties such as the K-property, exactness, and Bernoullicity. The chapter closes with the Central Limit Theorem, the Law of Iterated Logarithm and the Almost Sure Invariance Principle for sequences of random variables.
Chapter 3 focuses on ergodic theory and thermodynamical formalism for general continuous maps on compact metric spaces. The main points here are the Variational Principle for the pressure function and the duality between entropy and pressure, expressed in terms of the Legendre transform. These results have interesting applications in the theory of large deviations and multifractal analysis, and they are closely related to the uniqueness of Gibbs states.
Chapters 4 and 5 consider distance expanding maps and give outlines of their topological properties and thermodynamical formalism. This includes spectral decompositions, specification, Markov partitions and a primer on bounded distortion for Hölder continuous functions. Here, the chosen approach is analogous to the one of Bowen and Smale for Axiom A diffeomorphisms and of Przytycki for endomorphisms. This is followed by a study of multifractal aspects such as mixing properties of Gibbs measures associated to Hölder continuous potentials as well as the existence and uniqueness of invariant Gibbs probability states. For the latter, three different proofs are given.
Chapter 6 studies compact metric spaces with an action of some distance expanding map which are embedded in a smooth manifold such that the map extends to a \(C^{1+\epsilon}\)-map on a certain neighbourhood. Similarly to hyperbolic sets, it turns out that the intrinsic property of this map of being open is equivalent to the fact that the underlying compact metric space is the repellor for the extension. Also, in order to prepare for things to come, the authors discuss some distortion theorems for holomorphic maps.
Chapter 7 starts with a detailed account of Sullivan’s classification of Cantor sets in the reals via scaling functions. Also, there is a discussion of applications to Cantor-like closures of postcritical sets for infinitely renormalizable Feigenbaum quadratic-like maps. The infinitesimal geometry of these sets turns out to be independent of the map, which is one of the famous Collet-Tresser-Feigenbaum universalities.
In Chapter 8, the authors give definitions of various fractal dimensions. These include Hausdorff dimension, box-dimension and packing dimension. Moreover, they give an interesting discussion of the Besicovitch Covering Theorem, Vitali’s Theorem and the concept of density points.
Chapter 9 develops the theory of conformal expanding repellors and shows how to relate pressure functions to Hausdorff dimensions. This is followed by a crash course on Pesin’s multifractal analysis of Gibbs measures for Hölder continuous potential functions. The remainder of the chapter deals with boundary behaviour of Riemann mappings. Using some fundamental results of Przytycki, Urbański and Zdunik, the authors derive a dichotomy for harmonic measures on Jordan curves which bound repellors of conformal expanding maps. This result represents a dynamical counterpart of Makarov’s theory on boundary behaviour for general simply connected domains. The chapter closes with demonstrations of how to apply this analysis to the von Koch snowflake curve and to more general Carleson fractals.
Chapter 10 is devoted to Sullivan’s rigidity theorem, which states that if two nonlinear expanding repellors are Lipschitz conjugate (or more generally, if there exists a measurable conjugacy that transforms a geometric measure on the one space to a geometric measure on the other space), then the conjugacy extends to a conformal map. The philosophy of this is, that measures classify non-linear conformal repellors. The authors give a rigorous proof of this type of rigidity result.
Chapter 11 deals with non-uniformly expanding phenomena. The heart of this chapter is to show that in this non-uniformly expanding situation the Ledrappier-Young Formula holds for an arbitrary invariant ergodic measure of positive Lyapunov exponent.
Chapter 12 is devoted to the introduction and study of conformal measures. The idea of this type of measures goes back to Patterson (for Fuchsian groups) and to Sullivan (for rational maps). The authors give a very good account on how to employ conformal measures in various different settings and how to relate them to certain fractal dimensions. Also, there is a discussion of how to gain information about continuity for some of these dimensions in certain cases.
Motivational and historical material is given throughout the book, and there are many general examples to illustrate the results. Every ergodic theorist or graduate student with some background in ergodic theory, conformal dynamics and geometric measure theory will enjoy this book as a rich source of ideas and problems in recent developments of these prominent areas of modern mathematics. An extensive bibliography and a useful index complete this essential reference in ergodic theory for conformal fractals.