| Résume | The talk centers around the question of what invariant sets for Anosov flows and diffeomorphisms can look like. It is known that closed invariant sets can be arbitrarily bad and for example be fractals of arbitrary Hausdorff dimension. An old and interesting question, first raised by Hirsch in 1968 is what closed invariant submanifolds can look like. There are many surprising rigidity results in this direction, some quite old, and recently Rose Eliott Smith and I proved a new one, giving a dynamical characterization of totally geodesic submanifolds in negative curvature. I will explain the theorem and both the classical motivation and some more contemporary motivation. In the first hour I hope to give some hints about the proof and how it uses some very old ideas of Man\~{e} and Smale and how those ideas bear a strong resemblance to contemporary ideas in homogeneous dynamics. |
