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

Published online by Cambridge University Press:  07 May 2002

KEITH BURNS  and
HOWARD WEISS
KEITH BURNS
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA (e-mail: burns@math.nwu.edu)
HOWARD WEISS
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (e-mail: weiss@math.psu.edu)
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Abstract

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

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 2 , April 2002 , pp. 329 - 348
Copyright
2002 Cambridge University Press

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