Preliminaries in Ergodic Theory

Abstract

In this chapter we situate the context in which we will work along the book and we recall some central theorems in ergodic theory and entropy theory which will be crucial to the development of the results in the subsequent chapters. This chapter has no intention of being an introductory approach to ergodic theory or entropy theory, but to provide an account of results which will be necessary for the subsequent chapters, therefore proofs of the cited results are omitted and can be found in standard ergodic theory books such as Glasner (Ergodic Theory via Joinings. Mathematical Surveys and Monographs, vol. 101. American Mathematical Society, Providence 2003) and Kechris (Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, New York, 1995). We start this chapter with Sect. 2.1 by fixing some notation and recalling some standard definitions and results on the existence of invariant measures for a continuous map. In Sect. 2.2 we state the Birkhoff ergodic theorem and recall the definitions of ergodicity and mixing, two of the properties commonly cited in the ergodic hierarchy. In Sect. 2.3 we fix several notations for the operations among partitions, such as join and intersection, of a certain measurable space which will be used all along the book. In Sect. 2.4 we recall the classical Fubbini’s theorem and the much more general Rohklin disintegration theorem. As the Fubbini’s theorem is enough for the study of the Bernoulli property for the linear automorphisms of $\mathbb T^2$, the much more general Rohklin disintegration theorem is crucial to the study of the Kolmogorov and Bernoulli properties in the uniformly and non-uniformly hyperbolic context of Chaps. 4 and 5. Section 2.5 contains some basic definitions and results on Lebesgue spaces and Sect. 2.6 provides an account of results in entropy theory which will be necessary mainly in the development of Chap. 4. Although we assume the reader to have a working knowledge in ergodic theory and entropy theory, since the main goal of the book is to study the relation between the Kolmogorov and the Bernoulli properties, in Sects. 2.7 and 2.8 we provide carefully the definitions of the Bernoulli and the Kolmogorv properties, as well as proofs to some of the specific resultswhich are related to these properties. In particular we recall the structure of the Bernoulli shift more carefully (see Sect. 2.7.1), we prove that Bernoulli automorphisms are Kolmogorov and that Kolmogorov automorphisms are mixing (see Theorems 2.21 and 2.22).