This is a survey article concerning mainly the classical case of an action of a single automorphism on a standard probability space equipped with a finite generating partition. It focusses on the Ornstein isomorphism theory and the Kakutani equivalence for Bernoulli, and loosely Bernoulli systems. The survey contains all necessary definitions and formulations of many key theorems in the area. Some theorems are provided with complete proofs, but for most, the proofs are only outlined. This article can be recommneded as a handbook for quick reference in classical ergodic theory.
Section 3 contains the definitions and basics, such as joinings, factors, d-bar distance and the \(\delta\)-distance between joinings.
Section 4 discusses entropy, the Pinsker factor and K-automorphisms. We find here the Krieger generator theorem, the Shannon-McMillan-Breiman theorem (with a proof) and the Ornstein-Weiss theorem on return times.
Section 5 contains the Rokhlin lemma and a sketch of the Ornstein copying lemma.
Section 6 summarizes the Ornstein theory. It starts with the definitions of three properties: very weakly Bernoulli, finitely determined and extremal, and eqivalence between them is formulated. The proof is sketched for one implication. Then the Ornstein isomorpism theorem follows (with a proof). We also find here the Sinai theorem about the existence of a Bernoulli factor of full entropy, and the facts that all factors of Bernoulli systems are Bernoulli and that a d-bar limit of Bernoulli systems is Bernoulli. Provided are some natural examples of systems isomorphic to Bernoulli shifts.
Section 7 discusses the notion of “Bernoullicity” for flows and the extension of the Ornstein theory.
Section 8 mentions the extension to systems of infinite entropy and countable partitions.
Section 9 quotes the Keane-Smorodinsky finitary isomorphism theorem.
Section 10 is a remark on some cases of splitting (of a system as a direct product with a Bernoulli shift).
Section 11 contains a theorem by Rudolph: a weakly mixing isometric extension of a Bernoulli shift is Bernoulli.
Section 12 contains some K but not Bernoulli examples, in particular the (\(T\), \(T\)-inverse) transformation is defined here.
Section 13 discusses the Kakutani equivalence. We find here the definition of special flows and cross sections, and a theorem of Ambrose and Kakutani: every flow is special, every two cross sections of the same flow are Kakutani equivalent. The definitions of the f-bar distance and of loosely Bernoulli systems are provided. Theorems are stated that the loosely Bernoulli property passes to induced maps and to isomorphic extensions (Rudolph). Feldman’s example is mentioned, of a not loosely Bernoulli zero entropy system (this shows that not all zero entropy systems are Kakutani equivalent).
Section 14 contains some “natural”: examples of K systems, a K not Bernoulli \(C^\infty\) example, and a geodesic K but not loosely Bernoulli system. A theorem of Katok and Satayev is stated: interval exchanges are loosely Bernoulli. The horocycle is loosely Bernoulli but its cartesian square is not. The Ratner theorem is formulated.
Section 15 mentions about actions of amenable groups. We learn how to define entropy, Pinsker algebra and the K property using Fölner sequences. An extension of the Ornstein isomorphism theorem is formulated.
For the entire collection see [Zbl 1013.00016].