Stably ergodic diffeomorphisms
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.
scientific article; zbMATH DE number 742784