An Introduction to the Kolmogorov–Bernoulli Equivalence

In this chapter we briefly motivate the main theme of this book: the Kolmogorov–Bernoulli equivalence problem. We introduce the idea behind the ergodic hierarchy of measure preserving transformations and quickly discuss the problem of detecting conditions under which the Kolmogorov property is promoted to the Bernoulli property. In particular the method introduced by Ornstein and Weiss is of particular interest for our context (smooth dynamics).

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).

The main goal of this chapter is to show that the class of ergodic automorphisms of the 2-torus are Bernoulli. The proof summarized in this chapter was originally given by Ornstein and Weiss in 1973 in the article entitled “Geodesic flows are Bernoullian” (Ornstein and Weiss, Isr J Math 14:184–198, 1973). The method introduced by Ornstein–Weiss uses the geometric structures associated to the ergodic automorphisms of \(\mathbb T^2\) to obtain a sequence of refining partitions which are Very Weak Bernoulli (VWB) so that, by Ornstein Theory, one concludes that the automorphism is actually Bernoulli. The same method is used in Ornstein and Weiss (Isr J Math 14:184–198, 1973) to show that geodesic flows on negatively curved Riemannian surfaces are Bernoulli. Posteriorly many authors used the tools introduced by Ornstein and Weiss to show that the Kolmogorov property implies Bernoulli property for a larger class of dynamics such as the Anosov diffeomorphisms which will be treated later in this book. Until now, the ideas introduced by Ornstein and Weiss are still in the core of the arguments used to obtain Bernoulli property from the Kolmogorov property.

In Sect. 3.1 we introduce the concept of very weak Bernoulli partitions and state two theorems of Ornstein theory (Theorems 3.2 and 3.3) which are crucial in the proof of the main theorems of this chapter and Chap. 4. In Sect. 3.2 we show that ergodic automorphisms of \(\mathbb T^2\) are Kolmogorov by referring to a more general result proven in Chap. 3. Finally, in Sect. 3.3 we show in detail the main result of this chapter, namely we show that ergodic automorphisms of \(\mathbb T^2\) are Bernoulli. This section may be considered as the most important section of this book since the argument showed in this section is essentially the same argument used multiple times in the theory to show that Kolmogorov diffeomorphisms with certain hyperbolic structure are Bernoulli.

In the previous chapter we have proved that linear ergodic automorphisms of \(\mathbb T^2\) are Kolmogorov and, furthermore, they are Bernoulli. The main goal of this chapter is to show that Kolmogorov and Bernoulli property can be obtained for a much more general class of dynamical systems, namely those admitting a global uniform hyperbolic behavior, i.e., the Anosov systems (Definition 4.1). Anosov systems play a crucial role in smooth ergodic theory being the model for a huge variety of dynamical properties. In the first part of this chapter we make a quick introduction to the basic definitions and properties of uniformly hyperbolic systems and we will briefly present the geometric structures which are invariant by the dynamics of an Anosov diffeomorphism (Theorem 4.1). Linear ergodic automorphisms of \(\mathbb T^2\) are very particular examples of Anosov diffeomorphisms. In light of this fact we will show how to obtain the Kolmogorov property for C ^1+α Anosov diffeomorphisms (Theorem 4.8) and how we can use it to obtain the Bernoulli property (Theorem 4.9) in parallel to the argument used in Chap. 3.

This chapter is devoted to present some other results concerning the equivalence of the Kolmogorov and the Bernoulli property for systems which preserve a smooth measure and admit some level of hyperbolicity. We define the class of non-uniformly hyperbolic diffeomorphisms (resp. flows), the class of smooth maps (resp. flows) with singularities, and the class of partially hyperbolic diffeomorphisms derived from Anosov, and present the state of art of the problem inside each of this classes. In each case we briefly comment on the similarities with the Anosov case as well as the central difficulties that appear along the arguments. The class derived from Anosov diffeomorphisms is the one for which the results differ the most from the results for Anosov diffeomorphisms, therefore we go deeper in this particular case and prove the key results which allow us to overcome the absence of complete hyperbolicity along the center direction.