Lecture 2 - Ergodicity and partial hyperbolicity Globally hyperbolic systems (also called Anosov systems) are stably ergodic. For decades these were the only known examples of stable ergodicity. This was unfortunate, because Anosov systems are relatively rare. Then, in 1994, Grayson, Pugh, and Shub discovered a non-Anosov example. Further examples were found, leading to the Pugh-Shub conjectures, which state that stable ergodicity is dense among partially hyperbolic systems, and that this ergodicity is due to a (slightly) technical condition called accessibility." In this talk, Ill define partial hyperbolicity and accessibility and show how the proof of ergodicity for Anosov systems can be extended, in some cases, to the partially hyperbolic setting.