DYNAMICS, HOMOTOPY AND RIGIDITY

9:00 am Wednesday, February 17, 2010
Anatole Katok (Penn State)

Interaction between their global topological (homological and homotopical ) properties of group actions and their dynamics takes place on four levels of increasing complexity: 1. Existence of fixed points and periodic orbits for maps and vector fields e.g. Brower Fixed point theorem, Lefschetz Formula, Nielsen theory, various index-based theories. 2. Persistence of major elements of global topological orbit structures for maps and flows when an appropriate model is hyperbolic such as a hyperbolic automorphism of the torus or geodesic flow on a manifold of negative curvature 3. Persistence of invariant geometric structures such as absolutely continuous measures or i flat affine structures smooth in the Whitney sense for actions of higher rank abelian groups 4. Complete global rigidity of smooth orbit structure for actions of ``large'' and ``rigid'' groups such as simple Lie groups of real rank greater than one or lattices in such groups The first two topics are quite classical and will appear in the talk only as a matter of background and as preliminary stages for various constructions. The third is quite new; it has been pursued since 2006 by the speaker in collaboration with Boris Kalinin and Federico Rodriguez Hertz. An overview of principal results so far and outline of some ingredients that go into proofs will be presented. The fourth has been formulated by Robert Zimmer in his 1986 ICM talk and modified in the 1990s to accommodate certain new examples and phenomena that made the original ``rigidity program'' too rigid. Despite considerable efforts by many mathematicians until now the results have been either negative (such as non-existence of actions in low dimension), or perturbative (local differentiable rigidity of algebraic actions), or subject to dynamical restrictions such as existence of Anosov elements. Very recently in collaboration with Federico Rodriguez Hertz we obtained the first positive result free of such restrictions. This result is based on results from the previous group will also be discussed in the talk.

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