98-693 Keith Burns and Howard Weiss

Spheres with positive curvature and nearly dense orbits for the geodesic flow (643K, Postscript) Oct 30, 98

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Abstract. For any $\ep > 0$, we construct an explicit smooth Riemannian metric on the sphere $S^n, n \geq 3$, that is within $\ep$ of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is $\ep$-dense in the unit tangent bundle. Moreover, for any $\ep > 0$, we construct a smooth Riemannian metric on $S^n, n \geq 3$, that is within $\ep$ 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 $\ep$.

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