Analysis Seminar

Some words in the title are between quotation marks because it is a matter of interpretation. For dynamical systems on finite dimensional spaces, one often equates observable events with positive Lebesgue measure sets, and invariant distributions that reflect the large-time behaviors of positive Lebesgue measure sets of initial conditions (such as Liouville measure for Hamiltonian systems) are considered to be especially important. I will begin by introducing these concepts for general dynamical systems, describing a simple dynamical picture that one might hope to be true. This picture does not always hold, unfortunately, but a small amount of random noise will bring it about. In the second part of my talk I will consider infinite dimensional systems such as semi-flows arising from dissipative evolutionary PDEs, and discuss the extent to which the ideas above can be generalized to infinite dimensions, proposing a notion of ``typical solutions".