We construct a smooth metic on the sphere $S^2$, aritrarily close to the round metric, with a point $p$, not conjugate to itself along any geodesic, for which the number of geodesic loops based at $p$ with length at most $T$ grows as fast as we wish with $T$.
This article is in the proceedings of the International Congress on Dynamcial Systems held in Montevideo, Uruguay during March 1995. The reference is "International Congress on Dynamical Systems", edited by J. Lewowicz, F. Ledrappier and S. Newhouse, Pitman Lecture Notes in Mathematics 362, 7--20. The publisher is Addison Wesley Longman.
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Authors' addresses: Keith Burns Department of Mathematics Northwestern University Evanston, IL 60208-2730 U.S.A. Gabriel Paternain DPMMS Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB England