For more information about this meeting, contact Sergei Tabachnikov.

Title: | Dynamics, homotopy and rigidity |

Seminar: | Department of Mathematics Colloquium |

Speaker: | Anatole Katok, Penn State |

Abstract: |

Our goal is to present some recent advances in the following general problem: how global algebraic topology information about a group action by diffeomorphisms of a compact manifold (such as induced action on homology groups or homotopy types of its elements) influences geometric and dynamical properties of the action?
Classical methods such as Brouwer Fixed Point Theorem Lefschetz formula or Nielsen theory allow to deduce from this kind of global algebraic topology information existence of periodic orbits of various periods and estimate from below numbers of those orbits.
Modern methods based on variational calculus and hyperbolic dynamics allow to deduce from similar global data existence of large invariant sets with complicated behavior modeled usually on symbolic systems such as topological Markov chains. However, in this setting the correspondence with a model is only continuous, virtually never differentiable, and those invariant sets have zero volume.
This changes dramatically when instead of a single differentiable map one considers several commuting maps. In this case global algebraic topology information may force invariance of a real geometric structure e.g. absolutely continuous invariant measure, or a flat affine structure defined on an invariant set of positive volume. Furthermore, there may be a smooth correspondence in the sense of Whitney on an invariant set of positive volume with a standard algebraic model.
These results, that appeared in a series of joint papers with Boris Kalinin and Federico Rodriguez Hertz in various combinations, are based on the approach that we call Nonuniform measure rigidity (NUMR for short) that combines insights from two principal sources:
(i) earlier work on measure rigidity of algebraic actions that, among other things, is used in number theory applications such as a partial solution of the Littelwood conjecture in Diophantine approximation, (joint work with M. Einsiedler and E. Lindenstrauss that was mentioned in the Lindenstrauss's Fields medal quotation) and
(ii) smooth ergodic theory, aka nonuniformly hyperbolic dynamics, or Pesin theory. |

### Room Reservation Information

Room Number: | MB114 |

Date: | 02 / 03 / 2011 |

Time: | 04:00pm - 05:00pm |