The general case?

In the early 1930's, the Ergodic theorems of von Neumann and Birkhoff put Boltzmann's Ergodic Hypothesis in mathematical terms, and the natural question was born: is ergodicity the "general case" among conservative dynamical systems? Oxtoby and Ulam tackled this question early on and showed that the answer to this question is "yes" for continuous dynamical systems. The work of Kolmogorov Arnol'd and Moser beginning in the 1950's showed that the answer to this question is "no" for $C^\infty$ dynamical systems. I will discuss recent work with Artur Avila and Sylvain Crovisier that addresses what happens for $C^1$ dynamical systems.