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DCDS

We give examples of rank one compact surfaces on which there exist recurrent
geodesics that cannot be shadowed by periodic geodesics. We build rank one
compact surfaces such that ergodic measures on the unit tangent bundle of the
surface are not dense in the set of probability measures invariant by the
geodesic flow. Finally, we give examples of complete rank one surfaces for
which the non wandering set of the geodesic flow is connected, the periodic
orbits are dense in that set, yet the geodesic flow is not transitive in
restriction to its non wandering set.

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