Dynamical Systems Seminar

The Hopf argument has been employed for (almost) all proofs of ergodicity of systems with some hyperbolicity. It has been noticed for quite a while that this argument only works for smooth measure preserving systems. A new approach has been introduced to study the metric properties of dissipative dynamical systems. In particular, we prove the following dichotomy for some partially hyperbolic systems: (1) Either there is no ACIP, or there exists some smooth invariant measure; (2) Either the system is completely dissipative, or it is ergodic.