Since Kolmogorov and Sinai brought the notion of entropy into ergodic theory, it has been found to have many important consequences for the structure of a measure-preserving transformation. Very loosely, transformations can be divided into (i) `deterministic' transformations, which have zero (or at least very low) entropy; and (ii) K-automorphisms, which are in a sense `orthogonal' to all zero-entropy transformations. The best-known examples of K-automorphisms are the Bernoulli shifts, the `most random' of all.
Some of the most substantial results aim to describe a general ergodic transformation in terms of those two special cases. I will give a quick overview of this area and some of its main results: Sinai's factor theorem, Ornstein's isomorphism theorem, and some of the basic theory of K-automorphisms and Bernoulli shifts. I will work towards stating a new result in this vein, the weak Pinsker theorem. I will conclude by sketching some of the new phenomena in the proof of the weak Pinsker theorem, as far as time allows.
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