Abstract
In 1954, F. Mautner gave a simple representation theoretic argument 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.
Highlights
Beginning with [GPS] the first two authors have been studying stable ergodicity of volume preserving partially hyperbolic diffeomorphisms on a compact manifold M
(a) Among affine diffeomorphisms of finite volume, compact homogeneous spaces, those which are stably ergodic among left translations are precisely those with the essential accessibility property [St3]. (These are precisely those which are K-automorphisms.) In other words, affine stable ergodicity is equivalent to essential accessibility
The version of the Mautner phenomenon that we prove is: Theorem 2.1
Summary
Beginning with [GPS] the first two authors have been studying stable ergodicity of volume preserving partially hyperbolic diffeomorphisms on a compact manifold M. (a) Among affine diffeomorphisms of finite volume, compact homogeneous spaces, those which are stably ergodic among left translations are precisely those with the essential accessibility property [St3].
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