Dynamics Seminar-Overview of Spectral Ergodic Theory

Time: 11 am,  January 21
Speaker: Robbie Robinson
Title: Overview of Spectral Ergodic Theory

Abstract: In this occasional series of talks I plan to describe the parts of ergodic theory that are invariants of spectral isomorphism. These include ergodicity, weak and strong mixing, discrete and Lebesgue spectrum, and especially spectral multiplicity. For the classical case of invertible measure preserving transformations, the “Spectral Theorem for Unitary Operators” provides a complete invariant. I will begin by describing this case, while not proving the spectral theorem. The long goal is to show how to generalize to (once) locally compact abelian groups, for the most part Z^d and R^d, and prove the spectral theorem in that more general context. In particular, it is a classification of the unitary representations of the groups. While this is by no means new, I’ve never seen in the literature, a suitable proof of this in this level of generality. In the end we can use it to study the ergodic theory of Z^d and R^d actions. Some of what I cover will be from my joint work with Ayse Sahin.

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