# Matthew Foreman (UC IRVINE), Global Structure Theorems for the space of measure preserving transformations

Abstract: This talk describes two classes of symbolic topological systems, the odometer based and the circular systems. The odometer based systems are ubiquitous--when equipped with invariant measures they form an upwards closed cone in the space of ergodic transformations (in the pre-ordering induced by factor maps). The circular systems are a small class, but represent the diffeomorphisms of the 2-torus built using the Anosov-Katok technique of approximation by conjugacy.

We form two categories by taking as morphisms factor maps that preserve the underlying timing factor (the odometer or the circle). The main result is that these two categories are isomorphic by an isomorphism that preserves all relevant ergodic structure (compact extensions, simplices of invariant measures and so forth).

As a consequence the space of diffeomorphisms has all of the ergodic theoretic structure that the odometer based systems do. This solves several well-known problems such as the existence of ergodic Lebesgue measure preserving diffeomorphisms with an arbitrary compact Choquet simplices of invariant measures and the existence of measure-distal diffeomorphisms of the two torus of height greater than 2. (In fact we give examples of arbitrary countable ordinal height.) We conclude the talk with a discussion of a novel phenomena: Gödels Diffeomorphisms.

This is joint work with B. Weiss.

## Date:

Tue, 18/12/2018 - 14:15 to 15:15