Title: Hyperbolic Dynamical Systems, Higher Rank and Classification
Speaker: Ralf Spatzier
Speaker Info: U. Michigan
Brief Description: Dept. colloquium joint with Midwest Dynamical Systems Conference
Dynamical systems with extra symmetry turn out to be surprisingly rigid. For instance generically a diffeomorphism cannot have infinite index in its centralizer in the diffeomorphism group, as asserted by Smale and proved by Bonatti, Crovisier and Wilkinson. For hyperbolic systems one can hope to go one step further, and conjecture that such are always smoothly conjugate to an action on a homogeneous space (Katok-Spatzier).Date: Friday, November 12, 2021
For a special class of hyperbolic actions, the so-called Cartan actions, this was proved recently (Vinhage-S). Most importantly, we introduce a novel way of providing a homogeneous structure to a system coming from actions of free products.
As we will explain, this particular conjecture was motivated in part by the Zimmer program on actions of higher rank semisimple Lie groups and their lattices. And indeed (Butler-Damjanovic-S-Vinhage-Xu) proved a classification result for (totally) Anosov volume preserving actions of such groups, using related tools.
These ideas are closely related to superrigidity, rank rigidity in Riemannian geometry. This will be the setting for the talk, and also the promise and outlook for future work.