Robust mechanisms for chaos, II: Stable ergodicity and partial hyperbolicity

Kolmogorov introduced in the 1950’s a robust mechanism for {\em non-ergodicity}, which is now known as the KAM phenomenon (named for Kologorov, Arnol’d and Moser). A current, pressing problem in smooth dynamics is to understand the interplay between KAM and Hopf phenomena in specific classes of dynamical systems. I will describe a class of dynamical systems, called the {\em partially hyperbolic systems}, in which the two phenomena can in some sense be combined. I’ll also explain recent results that give strong evidence for the truth of a modified ergodic hypothesis in this setting, known as the Pugh-Shub stable ergodicity conjecture