Discussions in Analysis and Geometry

In this talk, I will define the Ruelle invariant of a measure preserving flow equipped with a symplectic cocycle. This is a natural dynamical quantity that measures a spacetime averaged rotation of the cocycle along the flow. I will then discuss the examples that show up in Hamiltonian and Reeb dynamics. Finally, for the Reeb flow on a convex hypersurface, I will derive estimates for the Ruelle invariant in terms of the total mean curvature of the convex hypersurface.