Spheres with positive curvature and nearly dense orbits for the geodesic flow

Keith Burns; Howard Weiss

doi:10.1017/S0143385702000172

Spheres with positive curvature and nearly dense orbits for the geodesic flow

Keith Burns, Howard Weiss

Biology

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

For any ε > 0, we construct an explicit smooth Riemannian metric on the sphere Sⁿ, 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 Sⁿ, 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.

Original language	English (US)
Pages (from-to)	329-348
Number of pages	20
Journal	Ergodic Theory and Dynamical Systems
Volume	22
Issue number	2
DOIs	https://doi.org/10.1017/S0143385702000172
State	Published - Aug 26 2002

All Science Journal Classification (ASJC) codes

Mathematics(all)
Applied Mathematics

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abstract = "For any ε > 0, we construct an explicit smooth Riemannian metric on the sphere Sn, 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 Sn, 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.",

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AB - For any ε > 0, we construct an explicit smooth Riemannian metric on the sphere Sn, 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 Sn, 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.

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Spheres with positive curvature and nearly dense orbits for the geodesic flow

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