Descriptive Set Theory Seminar
Kostya Slutsky
UIC
Lebesgue Orbit Equivalence of multidimensional flows
Abstract: Let's say we have two free actions of (possibly distinct) locally compact groups on standard Borel spaces. Let us also fix a Haar measure on each of the groups. Since actions are assumed to be free,
each orbit can be identified with the group and therefore the Haar measure can naturally be transferred onto each orbit separately.
These two actions are said to be Lebesgue Orbit Equivalent if there exists a Borel bijection between the phase spaces which maps orbits onto orbits and which preserves
the Haar measure with restricted onto any orbit. In the talk we shall study the actions of R^n, the n-dimensional Euclidean spaces, with respect to this notion of orbit equivalence.
Our main result is that non-smooth free actions of R^n are classified up to Lebesgue Orbit Equivalence by the number of invariant ergodic probability measures. This classification is similar to the
classification of hyperfinite Borel equivalence relations obtained by Dougherty, Jackson, and Kechris.
Wednesday January 28, 2015 at 2:00 PM in SEO 1227