New developments in differentiable rigidity of orbit structure for actions
of higher rank abelian groups


Anatole Katok

Penn. State University


Thursday, March 10, 2005

Talk at 4:30 p.m. in Room 2-190

Tea from 4:00 - 4:30 p.m. in Room 2-290
Refreshments afterwards, in Room 2-290


Abstract:   There is a striking difference between the orbit structure of classical smooth dynamical systems (diffeomorphisms and flows on compact differentiable manifolds) and their higher-rank counterparts. In the former case the differentiable orbit structure is never stable under small perturbations. In contrast to that, for the actions of higher rank abelian groups rigidity of the differentiable orbit structure does appears. During the 1990's it was understood that the circumstances which lead to structural stability for the classical case, i.e. global hyperbolic behavior, typically produce differentiable rigidity in the higher rank case.
    Recently a new phenomenon has been discovered. Certain partially hyperbolic higher rank actions, whose rank one counterparts are highly unstable, have been shown to be differentially rigid. The first class, which has been treated successfully, may at the end turn out to be the most subtle one. It consists of actions by partially hyperbolic automorphisms of a torus. The new technical devise is a KAM-type iteration scheme.
    A totally different method has been introduced in the last few months which provided partial results for such actions. This method uses descriptions of generators and relations in simple Lie groups provided by algebraic K-theory and is quite robust in its analytical aspects.
    This is joint work, partly still in progress, with my recent Ph. D. student Danijela Damjanovic, currently a fellow in the European Post-Doctoral Institute.