**Abstract:**

For any $\ep > 0$, we construct a smooth Riemannian metric on the sphere $S^3$ 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 we construct an orbit of the geodesic flow such that
the complement of the orbit closure has Liouville measure less than $\ep$.

This article is available in the following formats:

The paper will be appearing in Ergodic Theory and Dynamical Systems. A closely related article is

- A surface with positive curvature and positive topological entropy by Gerhard Knieper and Howard Weiss. This paper is published in Journal of Differential Geometry 39(1994), 229-249.

Authors' addresses:Keith Burns Department of Mathematics Northwestern University Evanston, IL 60208-2730 burns followed by math.northwestern.edu Howard Weiss Mathematics Department Pennsylvania State University University Park, PA 16802 weiss followed by math.psu.edu.

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