Robust mechanisms for chaos, I: Geometry and the birth of stable ergodicity

The first general, robust mechanism for ergodicity was developed by E. Hopf in the 1930’s in the context of Riemannian geometry. Loosely put, Hopf showed that for a negatively curved, compact surface, the “typical infinite geodesic fills the manifold in a very uniform way, a property called equidistribution. I will discuss Hopf’s basic idea in both topological and measure-theoretic settings and how it has developed into a widely applicable mechanism for chaotic behavior in smooth dynamics.