Lai-Sang Young (NYU)
Observable events and typical trajectories in finite and
infinite dimensional dynamical systems
Some of the words in the title are obviously subject to interpretation.
For dynamical systems on finite dimensional spaces, one often
equates observable events with positive Lebesgue measure sets,
and invariant measures that reflect the large-time behaviors of
positive Lebesgue measure sets of initial conditions are considered
to be of special importance. I will begin by reviewing these concepts
for general dynamical systems, describing a simple dynamical picture
that one might hope to be true. Reality is a little messier, but a small
amount of random noise will bring this picture 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".