Kolmogorov–Bernoulli Equivalence for Ergodic Automorphisms of \(\mathbb T^2\)
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.
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