DCDS
Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms
Charles Pugh Michael Shub Alexander Starkov
Discrete & Continuous Dynamical Systems - A 2006, 14(4): 845-855 doi: 10.3934/dcds.2006.14.845
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.
keywords: manifold foliations affine diffeomorphisms. stable ergodicity Representation theory the Anosov-Hopf ergodicity Lie groups unique ergodicity

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