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October  2006, 14(4): 845-855. doi: 10.3934/dcds.2006.14.845

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

Charles Pugh 1, , Michael Shub 2, and  Alexander Starkov 3,

1. 

Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada
2. 

Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada
3. 

All-Russian Institute of Electrotechnics, Istra, Moscow region, Russian Federation

Received  October 2004 Revised  August 2005 Published  January 2006

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.
Mathematics Subject Classification: 53D25, 37A2.
Citation: Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845
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