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Stably ergodic diffeomorphisms. (English) Zbl 0824.58032

This paper considers the existence of open sets \(U\) in an appropriate space of smooth functions on a compact manifold, with the property that each \(f\in U\) is ergodic. Maps in \(U\) are called stably ergodic. Anosov diffeomorphisms have this property, in addition to the stronger properties of uniform hyperbolicity and structural stability. This paper gives examples of stably ergodic diffeomorphisms that are neither hyperbolic nor structurally stable. The main result is that if \(M\) is the unit tangent bundle of a surface of constant negative curvature and if \(f\) is the time-one map of the geodesic flow on \(M\), then \(f\) is stably ergodic with respect to volume preserving transformations that are \(C^ 2\) small.

MSC:

37A99 Ergodic theory
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
28D10 One-parameter continuous families of measure-preserving transformations
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