Geometry, Topology and Dynamics Seminar
Victor Kleptsyn
Institut de Recherche Mathematiques de Rennes
Group actions on the circle
Abstract: A well-known conjecture concerning smooth actions on the circle, says that if the action is minimal, then it is Lebesgue-ergodic.
That is, there are no non-trivial measurable invariant subsets.
It was proven in the 1980's by Herman and by Katok that this conjecture holds for one circle diffeomorphism.
Sullivan proved the conjecture for groups that allow local expansion at some point, and Hurder proved it for groups that have positive expansion exponent.
An obstacle to the application of Sullivan's strategy is the presence of "non-expandable points".
Two examples of actions possessing such points are PSL(2,Z) and the Ghys-Sergisecu smooth realization of the Thompson group acting on the circle.
In this talk, we give a sufficient condition for actions possessing non-expandable points to be ergodic.
Moreover, under this assumption we deduce some surprising structure properties for the action.
Monday October 4, 2010 at 3:00 PM in SEO 636