Colloque des sciences mathématiques du Québec

6 mai 2022 de 15 h 30 à 16 h 30 (heure de Montréal/HNE)

Generic measure preserving transformations and Descriptive Set Theory

Colloque par Slawomir Solecki (Cornell University)

The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$---the non-locally compact, topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this was the case. 

We will describe the background touched on above, including the connections with Descriptive Set Theory. Further, we will indicate a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is {\bf not} topologically isomorphic to $L^0$.