Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms

Beginning with [GPS] the first two authors have been studying stable ergodicity of volume preserving partially hyperbolic difieomorphisms on a compact manifold M. The most recent survey on the subject is [PS3]. A key issue is the way in which the strong stable and strong unstable manifolds foliate M. To prove ergodicity one assumes essential accessibility, namely that every Borel set S ? M which consists simultaneously of whole strong stable leaves and whole strong unstable leaves has measure zero or one. Such a set S is said to be us-saturated. As essential accessibility is a measure theory concept, it is di卤cult to verify and even more di卤cult to prove stable under perturbation. A stronger assumption is full accessibility, in which it is required that M and the empty set are the only us-saturated sets. In many cases full accessibility is stable under perturbation, and this leads to stable ergodicity. We have conjectured that the stably ergodic difieomorphisms are open and dense among C, volume preserving partially hyperbolic difieomorphisms. Our plan of attack was to prove that an open and dense subset of partially hyperbolic difieo-morphisms are fully accessible. As already noted, this seems far easier than the similar assertion for essential accesssibility, so we focused on the full accessibility property. The following recent developments, however, caused us to reconsider our position and to shift our attention more in the direction of essential accessibility. (a) Among a卤ne difieomorphisms of finite volume, compact homogeneous spaces those which are stably ergodic among left translations are precisely those with the essential accessibility property [St3]. In other words, a卤ne stable ergodicity is equivalent to essential accessibility. The proof relies signifi-cantly on the structural properties of Lie groups. (b) As was shown by Federico Rodgriguez Hertz, essential accessibility without full accessibility sometimes leads to (nonlinear) stable ergodicity, [RH]. (c) The Mautner phenomenon from representation theory leads to a proof of half of the a卤ne stable ergodicity result mentioned in (a), while in [PS2] we establish a nonlinear version of the Mautner phenomenon, which we apply to nonlinear stable ergodicity. Below, we give a proof of the Mautner phenomenon in the case it is used for (a), but instead of structural properties of Lie groups or their representation theory,