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

Charles Pugh; Michael Shub; Alexander Starkov

doi:10.3934/dcds.2006.14.845

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2006, Volume 14, Issue 4: 845-855. Doi: 10.3934/dcds.2006.14.845

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

1.
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON
2.
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4
3.
All-Russian Institute of Electrotechnics, Istra, Moscow region

Received: September 30, 2004

Revised: July 31, 2005

Published: September 2006

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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.

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Mathematics Subject Classification: 53D25, 37A25.

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