Under a non-nilpotency condition in free loop space homology with respect to the Chas-Sullivan product, the number of simple Reeb orbits on star-shaped hypersurfaces grows at least like T/log(T).
On the growth of the number of closed geodesics on the two-sphere
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On the growth rate of Reeb orbit on star-shaped hypersurfaces
Under a non-nilpotency condition in free loop space homology with respect to the Chas-Sullivan product, the number of simple Reeb orbits on star-shaped hypersurfaces grows at least like T/log(T).