Ergodic theory has blossomed within mathematics over the last half century, due in no small part to a variety of rich interactions with a number of disparate areas. Today, ergodic-theoretic techniques and methodology are part and parcel of advances in fields ranging from additive combinatorics and Lie theory to metric Diophantine approximation and meromorphic dynamics in one and several complex variables.

It was probably clear to those who foresaw “the dawning of the age of stochasticity” that probabilistic reasoning was destined to become core methodology in a variety of fields. Cornfeld, Fomin and Sinai prefaced their now-classic monograph *Ergodic Theory* (Grundlehren der Mathematischen Wissenschaften, Springer, 1982) with the following remarks:

Ergodic theory is one of the few branches of mathematics which has changed radically during the last two decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily with averaging problems and general qualitative questions, while now it is a powerful amalgam of methods used for the analysis of statistical properties of dynamical systems. For this reason, the problems of ergodic theory now interest not only the mathematician, but also the research worker in physics, biology, chemistry, etc.

Cornfeld-Fomin-Sinai’s monograph, which continues to be an excellent resource, was followed by a variety of textbooks reflecting different emphases while highlighting distinct areas of application. A representative sample of such books should include

- Walters:
*An introduction to ergodic theory*. Springer GTM 79, 1982.
- Petersen:
*Ergodic theory*. Cambridge Studies in Advanced Mathematics, 2. Cambridge, 1989.
- Arbieto-Matheus-Moreira:
*The remarkable effectiveness of ergodic theory in number theory.* Ensaios Matemáticos, 17. Sociedade Brasileira de Matemática, 2009. Available here.
- Przytycki-Urbański:
*Conformal fractals: ergodic theory methods.* LMS Lecture Note Series, 371. Cambridge, 2010.
- Einsiedler-Ward:
*Ergodic Theory: with a View Towards Number Theory*. Springer GTM 259, 2011.
- Barreira-Pesin:
*Introduction to smooth ergodic theory*. Graduate Studies in Mathematics, 148. American Mathematical Society, 2013.

So, why another tome on ergodic theory? Well, simply put, because Viana and Oliveira have written yet another excellent textbook! It may be fruitfully used to guide a graduate course in dynamical systems, or a topics seminar at either advanced undergraduate or early graduate levels. The book is designed so that the instructor may cull a variety of courses from its contents.

This book is an updated version in English of an earlier Portuguese edition *Fundamentos da Teoria Ergódica* (SBM, 2014), and has been tested (among other settings) in a number of graduate courses at IMPA (Instituto de Matemática Pura e Aplicada) in Rio de Janeiro and at UFAL (Universidade Federal de Alagoas) in Maceió, Brasil.

The authors deserve special kudos for their collection of over 400 exercises, many with hints and solutions at the end of the book. As a further bonus, if only to pique the reader’s interest, a number of recent research results and open problems are sprinkled throughout the book. I hope a newer edition will highlight more such material that could help a neophyte reader connect with the latest advances and guide future research directions. As a first step, having a webpage dedicated to such additions to the text would certainly enhance the material and make it more attractive and alive for students.

The first chapter, *Recurrence*, introduces invariant measures and the basic recurrence theorems due to Poincaré, Kac, and Birkhoff. This is followed by a few key examples: decimal expansions, continued fractions via the Gauss map, toral rotations, and conservative flows. The chapter ends with a description of using first-return maps to construct induced systems, and a brief section on multiple recurrence theorems. There is an innocuous nod to Furstenberg’s beautiful Porter lectures *Recurrence in ergodic theory and combinatorial number theory*, Princeton, 1981 for a proof of Furstenberg’s generalization of Poincaré’s recurrence theorem, which is employed at the close of the next chapter to prove Szemerédi’s classic result on the existence of arithmetic progressions of arbitrary length in sets of integers with positive (upper Banach) density.

The second chapter, *Existence of invariant measures*, proves the fundamental existence result that guarantees a Borel probability measure invariant under the action of a continuous map on a compact metric space. The authors give three approaches to prove this theorem, the slickest of which is an application of the non-trivial Schauder-Tychonoff fixed-point theorem. They also present the more intuitive and elegant averaging argument that is usually credited to N. Kryloff and N. Bogoliouboff via their paper “La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire” (*Ann. of Math.* (1937), 65–113).

The third and fourth chapters begin with the three classic ergodic theorems that would form a staple in any ergodic theory course: Kingman’s subadditive ergodic theorem, Birkhoff’s pointwise/individual ergodic theorem, and von Neumann’s mean ergodic theorem. This is followed by various characterizations of ergodicity, and a useful loopback to the examples discussed in the first chapter with a view to proving ergodicity. There is a brief discussion of the ergodic measures forming the extremal points of the convex set of invariant probability measures for a fixed dynamical system, which could have been moved to the start of the next short chapter on the *Ergodic decomposition*. The authors prove the fundamental ergodic decomposition theorem (which guarantees the representation of an invariant measure as an (integral) convex combination of the extremal ergodic measures) via the Rokhlin disintegration theorem.

The authors take the pedagogically sound strategy of presenting more than a single proof of fundamental results, when available. In that vein, it would have been useful to discuss a well-known alternate route {Greschonig-Schmidt, Ergodic decomposition of quasi-invariant probability measures, *Colloq. Math.* 84/85 (2000), 495–514) via the Choquet-Bishop-de Leeuw representation theorem: every point in \(K\) (a compact convex subset of a Hausdorff locally convex topological vector space) is the barycenter of a probability measure supported on the set of extreme points of \(K\).

The sixth chapter, *Unique ergodicity*, describes systems that admit exactly one invariant probability measure. The authors sketch a proof of Furtsenberg’s beautiful example of a minimal (all orbits are dense) but *not* uniquely ergodic dynamical system on the 2-torus. On the positive side, they prove that every transitive translation on a compact metrizable topological group is uniquely ergodic with the Haar measure as the unique invariant probability. The chapter closes with another gem, Weyl’s equidistribution theorem for polynomial sequences on the circle.

The seventh chapter, *Correlations*, studies the evolution of the correlations between observables (which could measure temperature, or spatial position via a characteristic function) as time tends to infinity. This begins with the notions of strong and weak mixing, and their characterization via the Koopman operator. The authors then move to a section on finite-memory processes, a.k.a. *Markov shifts*, which generalize the class of Bernoulli shifts (which model I.I.D. sequences of outcomes), and characterize when such systems are ergodic and strong mixing.

The next class of examples, *interval exchange transformations* (IETs), generalize one-dimensional rotations/translations. An IET is a bijection of a compact interval with a finite number of discontinuities and whose restriction to every interval of continuity is a translation. The authors provide a proof of Katok’s result that such systems are never strong mixing. However, they fail to mention that it was only a decade ago that Avila-Forni proved that a typical IET is either weakly mixing or it is an irrational rotation, see Weak Mixing for Interval Exchange Transformations and Translation Flows, *Ann. of Math.* (2007), 637–664. The chapter closes with a proof that strong mixing Markov shifts experience exponential decay of correlations when dealing with Hölder continuous observables.

The eighth chapter introduces the famous *isomorphism problem* that aims to classify ergodic systems and motivates the next chapter on the seminal concept of (Kolmogorov-Sinai, a.k.a. measure-theoretic) *entropy*. The definition of entropy is highly non-trivial and the authors spend the next few sections on theorems that help calculate entropy in pleasant circumstances, via the Kolmogorov-Sinai and Shannon-McMillan-Breiman theorems. The reader is then returned to computations in the realms of Markov shifts, the Gauss map and linear endomorphisms of tori. The tenth chapter, *Variational principle*, introduces the notion of *topological entropy* and its generalization known as *pressure*, and is concerned with a careful proof that the topological entropy of a continuous map acting on a compact metric space equals the supremum of measure-theoretic entropies with respect of all invariant probability measures.

The last two chapters, *Expanding maps* and *Thermodynamical formalism*, form a playground “to exhibit a concrete (and spectacular!) application of many of the general ideas presented in the text”. The focus is on the specific, but fundamental, class of expanding dynamical systems. The authors refer to the 1975 edition of Rufus Bowen’s lucid notes *Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms*, though readers may want to study the excellent revised edition edited by Jean-René Chazottes (Springer Lect. Notes in Mathematics vol. 470, 2008). Students interested in how these ideas are applicable to the investigation of conformal dynamical systems should study Przytycki-Urbański’s text referred to above.

Tushar Das is an Associate Professor of Mathematics at the University of Wisconsin–La Crosse.