Logic Seminar
David McClendon
Northwestern University
Discontinuous identifications in measure-preserving semiflows
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.
Tuesday March 17, 2009 at 4:00 PM in SEO 612