Generic measures for geodesic flows on nonpositively curved manifolds

Coudène, Yves; Schapira, Barbara

Journal de l’École polytechnique — Mathématiques, Tome 1 (2014), pp. 387-408.

Nous étudions les propriétés génériques des mesures de probabilité invariantes par le flot géodésique sur des variétés connexes à courbure négative ou nulle. Sous une hypothèse technique assez faible, nous démontrons que l’ergodicité est une propriété générique dans l’ensemble des mesures de probabilité sur le fibré unitaire tangent de la variété dont le support est constitué de trajectoires qui ne bordent pas de ruban plat. Pour cela, nous démontrons que les mesures portées par les orbites périodiques sont denses dans cet ensemble. Dans le cas d’une surface compacte, nous obtenons le résultat optimal suivant : l’ergodicité est générique dans l’espace de toutes les probabilités invariantes sur le fibré unitaire tangent si et seulement s’il n’y a pas de ruban plat sur le revêtement universel de la surface.

Finalement nous démontrons que sous les hypothèses adéquates, génériquement, les mesures de probabilité invariantes sont d’entropie nulle et ne sont pas fortement mélangeantes.

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: ergodicity 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.

Classification : 37B10, 37D40, 34C28
Keywords: Geodesic flow, hyperbolic dynamical systems, nonpositive curvature, ergodicity, generic measures, zero entropy, mixing.
Mot clés : Flot géodésique, systèmes dynamiques hyperboliques, courbure négative ou nulle, ergodicité, mesures génériques, entropie nulle, mélange.

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