David McClendon : Discontinuous identifications in measure-preserving semiflows

Posted by , part of the Logic Seminar.

At

March 17, 2009, 4 p.m.

In

SEO 612

Abstract

A theorem of Becker and Kechris guarantees the existence of "nice topologies" for jointly Borel Polish group actions on Polish spaces (a "nice topology" is a Polish topology on the phase space with the same Borel sets as the original topology for which the action is jointly continuous). The same result holds for countably generated actions of Polish semigroups. But for semiflows (actions of the semigroup $[0,\infty)$ of non-negative real numbers), this theorem is false. Let $\{T_t : t \geq 0\}$ be a semiflow and consider two points $x \neq y$ in the phase space which map to the same point under all $T_t, t > 0$. We say $x$ and $y$ are "instantaneously and discontinuously identified" by the semiflow; the presence of any such pair of points ensures that no "nice topology" can exist for the semiflow. In this talk, we will discuss some results related to this phenomenon and explain why this behavior is worth studying, from the perspective of ergodic theory.

seminar begins with tea.