International research in smooth ergodic theory is increasingly focusing on the ergodic and mixing properties of continuous time dynamical systems, that is measure-preserving flows. This includes traditional examples such as geodesic flow on surfaces of negative (or variable) curvature, horocycle flows, billiard flows (such a Lorentz gases or other models describing the behaviour of moving and bouncing particles in some environment) and systems arising from a variety of mechanical models. The basic question is how fast initial information disperses over time, more specifically the rate of mixing which implies many other statistical properties of the flow as well.
Mixing properties of dynamical systems can be studied via transfer operators; these were initially applied to hyperbolically expanding systems. Young's application of coupling methods to towers brought non-uniformly hyperbolic systems within reach, and at the same time demonstrated the importance of tail estimates of the return time (height function). Operator theory introduced into dynamical systems by Sarig and Gouezel proved that rates obtained by Young towers are sharp. Later on, this powerful technique was adapted to obtain finer statistical properties of Young towers. This machinery offered also an answer to the question of mixing in infinite ergodic theory, which has been adapted to deal with certain invertible systems as well.
Recent advances on the mixing properties of chaotic flows opened up the possibilities for studying the separation of time scales or spatially extended dynamical phenomena, and thus for the applications to models related to statistical physics. One of the key tools here is averaging theory, with the fast motion often modeled by a strongly chaotic flow.
The thematic programme's tentative schedule includes the following events:
(Please note: The activities on Saturday, May 21, 2016 will take place at the Faculty of Mathematics, Sky Lounge.)