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Actions of higher rank abelian groups: from measure rigidity to arithmeticity to topology

4:00 pm Thursday, February 16, 2012
Anatole Katok (Penn State)

While topology of a compact manifold allows to make inferences about periodic points of maps and, to a lesser extent, closed orbit of vector fields supported by the manifold, there is notoriously little relation between topology of a manifold and ergodic properties of diffeomorphisms and smooth flows supported by it. A somewhat related fact is that ergodic properties of such dynamical systems tell little about their geometric structure.

All this changes in a dramatic way when one considers dynamical systems with multi-dimensional time, i.e. actions of higher rank abelian groups by diffeomorphisms of compact manifolds. Here is a representative case that will be discussed in the talk.

Assume that k>1. Consider k commuting diffeomorphisms of a (k+1)- dimensional manifold M and assume that they preserve a measure that is sufficiently ``rich'' or ``stochastic''. This assumption will be explained in the talk, but for a person with some dynamical background one form of it states that all non-zero elements of the suspension action have positive entropy with respect to the measure.

A striking topological conclusion is that the manifold is a ``slightly modified'' (finite factor of the) torus: there is homeomorphism from an open subset in finite cover of M that contains support of the lifted measure to the( k+1)-dimensional torus with a finite set removed. In particular the fundamental group of a finite cover of M contains a free abelian subgroup or rank (k+1).

On the ergodic side we obtain precise information about the structure of the action. It is fully ``arithmetic'', i.e. exactly the same as an action by commuting hyperbolic matrices with integer entries and determinant 1 or -1 on the torus or its finite factor. Moreover, the correspondence is smooth the sense of Whithey on a set whose complement has arbitrary small measure.

THis is the joint work with Federico Rodriguez Hertz based on our joint work with Boris Kalinin.

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