DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 14, Number 4, April 2006 pp. 845–855 UNIQUE ERGODICITY, STABLE ERGODICITY, AND THE MAUTNER PHENOMENON FOR DIFFEOMORPHISMS Charles Pugh Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada, M5S 2E4 Michael Shub Department of Mathematics, University of Toronto, 40 St. George Street Toronto, ON, Canada, M5S 2E4 Alexander Starkov All-Russian Institute of Electrotechnics, Istra, Moscow region, Russia, 143500 (Communicated by Marcelo Viana) Abstract. In 1954, F. Mautner gave a simple representation theoretic argu- ment that for compact surfaces of constant negative curvature, invariance of a function along the geodesic flow implies invariance along the horocycle flows (these are facts which imply ergodicity of the geodesic flow itself), [M]. Many generalizations of this Mautner phenomenon exist in representation theory, [St1]. Here, we establish a new generalization, Theorem 2.1, whose novelty is mostly its method of proof, namely the Anosov-Hopf ergodicity argument from dynamical systems. Using some structural properties of Lie groups, we also show that stable ergodicity is equivalent to the unique ergodicity of the strong stable manifold foliations in the context of affine diffeomorphisms. 1. Introduction. Beginning with [GPS] the first two authors have been studying stable ergodicity of volume preserving partially hyperbolic diffeomorphisms 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 difficult to verify and even more difficult to prove stable under perturbation. A stronger assumption is full accessibility 1 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. The second author was supported in part by an NSERC Discovery Grant. The third author was supported by the Leading Scientific School Grant, No 457.2003.1. 1 In previous papers, we referred to full accessibility as us-accessibility, and to a stronger condition as homotopy accessibility. The latter is always stable under perturbation and is often a consequence of the former. 845