Generic measures for geodesic flows on nonpositively curved manifolds

Yves Coudène (1) , Barbara Schapira (2)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique
2 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set. In the case of a compact surface, we get the following sharp result: ergod- icity is a generic property in the space of all invariant measures defined on the unit tangent bundle of the surface if and only if there are no flat strips in the universal cover of the surface. Finally, we show under suitable assumptions that generically, the invariant probability measures have zero entropy and are not strongly mixing.

Mots clés: nonpositive curvature; geodesic flow; ergodic measures; generic properties

Domaines: Systèmes dynamiques [math.DS]; Géométrie différentielle [math.DG]

Yves Coudène, Barbara Schapira. Generic measures for geodesic flows on nonpositively curved manifolds. Journal de l'École polytechnique — Mathématiques, 2014, 1, pp.387-408. ⟨10.5802/jep.14⟩. ⟨hal-00933903⟩