# American Institute of Mathematical Sciences

November  2011, 30(4): 1095-1106. doi: 10.3934/dcds.2011.30.1095

## Counterexamples in non-positive curvature

 1 Université de Bretagne Occidentale, 6 av. Le Gorgeu, 29238 Brest cedex, France 2 LAMFA, Université Picardie Jules Verne, 33 rue St Leu 80000 Amiens, France

Received  April 2010 Revised  August 2010 Published  May 2011

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.
Citation: Yves Coudène, Barbara Schapira. Counterexamples in non-positive curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1095-1106. doi: 10.3934/dcds.2011.30.1095
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