This lecture addresses two central problems in classical ergodic theory: the classification problem and the realization problem. An historical focus of ergodic theory has been the structure and properties of single transformations. Perhaps the most prominent is the Furstenberg-Zimmer structure theorem which describes ergodic transformations in terms of limits of compact and weakly mixing extensions.
This lecture discusses a new phenomenon, Global Structure Theorems. We define two categories: the Odometer Based Systems (finite entropy transformations that have an odometer factor) and Circular Systems (those diffeomorphisms built using a version of the Anosov-Katok technique.) The morphisms in each category are measure-theoretic joinings.
The main result is that these two categories are isomorphic by a composition-preserving bijection that that takes conjugacies to conjugacies, extensions to extensions, weakly mixing extensions to weakly mixing extensions, compact extensions to compact extensions, distal towers to distal towers (and more). In short, all of the structure present in the odometer based systems is exactly reflected in the Circular Systems and vice versa.
The lecture will conclude with a provocative conjecture.
This is joint work with B. Weiss.