Abstract

In many fields in mathematics averages of functions under the action of a unitary group arise. Flows due to transport operators appearing in Kinetic Theory, such as Vlasov systems, are a particular example due to their Hamiltonian structure. It is well known that, in general, the ergodic theorem states that ``time averages converge to spatial averages’’. However, typically, this convergence is only in the strong sense, not uniform. In this talk we first review Von Neumann’s proof of the (strong) ergodic theorem, and show how to obtain uniform convergence under certain geometric assumptions on the flow and the underlying functional space. The main ingredients of the proof are new estimates on the associated density of states and criteria for the global rectification of vectorfields. This is joint work with Clement Mouhot.