Hyperbolic Structures and the Kolmogorov–Bernoulli Equivalence

Abstract

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.