Entropy and the structure of measure-preserving transformations

Tim Austin
University of California, Los Angeles (UCLA)

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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