| DOI: 10.1017/S0143385702000172 Printed in the United Kingdom Spheres with positive curvature and nearly dense orbits for the geodesic flow (2000) | |||||||||||||
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| Abstract. For any ε> 0, we construct an explicit smooth Riemannian metric on the sphere S n, n ≥ 3, that is within ε of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is ε-dense in the unit tangent bundle. Moreover, for any ε> 0, we construct a smooth Riemannian metric on S n, n ≥ 3, that is within ε of the round metric and has a geodesic for which the complement of the closure of the corresponding orbit of the geodesic flow has Liouville measure less than ε. It has long been known that the geodesic flow for a Riemannian metric of negative curvature possesses chaotic dynamics with the strongest possible stochastic behavior: the flow is not only ergodic but also has the Bernoulli property. A major open problem in ergodic theory and geometry is whether the geodesic flow of a Riemannian metric with | |||||||||||||
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