Amie WILKINSON presented new results towards the Pugh-Shub Stable ergodicity conjecture. In particular, with A. AVILA and J. BOCHI, she proved that ergodicity is generic in C^1 partially hyperbolic symplectomorphisms. She noted that, by a result of SAGHIN and Z. XIA, a stably ergodic symplectomorphism is automatically partially hyperbolic (which fails for conservative diffeomorphisms by an example of A. TAHZIBI).
The ingredients of the proof are:
- Bochi’s alternative: a symplectomorphism which is not Anosov has only zero exponents Lebesgue almost everywhere.
- A new criterion for ergodicity and saturation for C^2 partially hyperbolic diffeomorphisms involving a new, non-uniform variant of the center bunching property (used to prove that a set is simultaneously saturated wrt to the stable and unstable foliations)
- A perturbation technique introduced by M.-C. ARNAUD in her proof of Mañé’s ergodic closing lemma
- The ergodic diffeomorphisms of the disk arbitrarily close to the identity, built by ANOSOV and KATOK
- A Baire argument