The stable ergodicity conjecture
by
Amie Wilkinson
Northwestern University

A volume-preserving diffeomorphism is stably ergodic if all nearby volume-preserving diffeomorphisms are ergodic. Until 1995, when Grayson-Pugh-Shub found a non-hyperbolic example, the only known examples of stably ergodic diffeomorphisms were Anosov diffeomorphisms, which are globally hyperbolic. Since then, other examples have been found; all are partially hyperbolic. A conjecture of Pugh and Shub states that among the partially-hyperbolic, volume-preserving diffeomorphisms, stable ergodicity is open and dense. I will discuss recent work with Keith Burns that shows evidence for this conjecture among certain classes of partially-hyperbolic diffeomorphisms (skew extensions of Anosov diffeomorphisms). I will also discuss how related techniques can be used to show that nonergodic diffeomorphisms with very complicated center behavior (e.g. elliptic islands) can be approximated by stably ergodic ones (joint with Mike Shub). Both of these results rely on powerful new techniques developed by Pugh and Shub.

Date received: February 26, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caas-95.