Abstract
For any ! > 0, we construct an explicit smooth Riemannian metric on the sphere S n , n 3, that is within ! of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is ! -dense in the unit tangent bundle. Moreover, for any ! > 0, we construct a smooth Riemannian metric on S n ,n 3, that is within ! of the round metric and has a geodesic for which the complement of the closure of the correspondingorbit of the geodesic flow has Liouville measure less than ! .
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