| Stable Ergodicity and Anosov Flows (1998) | |||||||||||||||
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| In this note we prove that if M is a 3-manifold and ' t : M ! M is a C 2 , volume-preserving Anosov flow, then the time-1 map ' 1 is stably ergodic if and only if ' t is not a suspension of an Anosov diffeomorphism. 1 Introduction A volume-preserving diffeomorphism is stably ergodic if it and all sufficiently C 2 -close volume-preserving diffeomorphisms are ergodic. Until recently, the only known examples were hyperbolic --- namely Anosov diffeomorphisms [1]. Grayson, Pugh and Shub found the first example of a nonhyperbolic stably ergodic diffeomorphism. They proved in [4] that if S is a surface of constant negative curvature and ' t is the geodesic flow on the unit tangent bundle of S, then the time-1 map ' 1 is stably ergodic (with respect to Liouville measure). Wilkinson later generalized this result to the case where S has variable negative curvature [14]. Pugh and Shub proved it for higher dimensional manifolds of constant, or nearly constant, negative curvature [11]. In a... | |||||||||||||||
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