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An introduction to infinite ergodic theory. (English) Zbl 0882.28013
Mathematical Surveys and Monographs. 50. Providence, RI: American Mathematical Society (AMS). xii, 284 p. (1997).
Infinite ergodic theory is the study of measure-preserving transformations of infinite measure spaces. Perhaps the simplest examples to write down are certain unsuspicious maps of the real line such as \(Tx=x-{1\over x}\) or \(Tx= \tan x\), the first of which preserves Lebesgue measure while the second has the nonintegrable invariant density \(h(x)={1\over x^2}\). (These observations were made by G. Boole and J. W. L. Glashier more than a hundred years ago but, of course, stated in different terms.) A class of very natural examples is that of null-recurrent Markov chains (resp. their shifts) such as the symmetric coin-tossing random walk on the integers. Others will be mentioned below. There is a great variety of ergodic behaviour infinite measure-preserving transformations (i.m.p.t.s) can exhibit, and they have undergone some intense research within the last twenty years, much of which is associated with the name of the book’s author.
The first chapter introduces some basic non-singular ergodic theory including recurrence behaviour, ergodicity, dual (Perron-Frobenius) operators, existence of invariant measures, and induced transformations. Chapter 2 contains a proof of Hurewicz’s ratio ergodic theorem (a special case of the Chacon-Ornstein theorem) which plays a fundamental role guaranteeing that the pointwise behaviour of ergodic sums both of the Koopman and its dual operator is the same for all integrable functions with nonzero mean. Converses to Birkhoff’s theorem are discussed and Aaronson’s ergodic theorem showing that for i.m.p.t.s. there is no absolutely normalized pointwise convergence of Birkhoff sums is established. Further sections are devoted to spectral theory and ergodicity of Cartesian products. Chapter 3 contains the core of abstract infinite ergodic theory, starting with a discussion of structural aspects such as adequate notions of factors and isomorphisms, then turning to diverse phenomena which occur under additional assumptions. Topics include intrinsic normalizing constants, laws of large numbers weaker than pointwise ergodic theorems, rational ergodicity, asymptotic distributional behaviour and related mixing properties such as pointwise dual ergodicity, and wandering rates. Chapter 5 introduces some concepts of classification theory and culminates in the proof of an isomorphism theorem for random walks on \(\mathbb{Z}^d\) with centered jump distributions subject to certain moment conditions.
The remainder of the book is devoted to examples: Chapter 4 develops an abstract framework, based on distortion properties, for the treatment of various Markov maps. Within that Markov shifts and Markovian interval maps with indifferent fixed points are studied in some detail. Chapter 6 is about inner functions, and Chapter 7 about geodesic flows on hyperbolic surfaces. Chapter 8 on cocycles and skew products contains some of the more pathological examples in the field.
The book is on advanced graduate level, and it should be accessible to readers with a firm background in measure-theoretic probability and preferably some standard (i.e., finite) ergodic theory. It is carefully organized and well written. Still even the initiated reader might save some time by first having a look at the list of corrections to the regrettable number of typing errors, which can be found at the author’s homepage (www.math.tau.ac.il/\(\sim\)aaro). The book will then be invaluable both as an introduction and as a reference work on its subject, and this definitely is not just because it is the only one at the moment.

28D05 Measure-preserving transformations
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
37A99 Ergodic theory
37E99 Low-dimensional dynamical systems
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60G50 Sums of independent random variables; random walks