On the ergodicity of geodesic flows on surfaces of nonpositive curvature

Wu, Weisheng

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 3, pp. 625-639.

Soit $M$ une surface lisse compacte de courbure négative ou nulle, de genre $\ge 2$. Nous prouvons l’ergodicité du flot géodésique sur la tangente du faisceau unitaire de $M$ par rapport à la mesure de Liouville, en supposant que l’ensemble des points à courbure négative sur $M$ a un nombre fini de composantes connexes. Sous la même hypothèse, nous prouvons qu’il n’existe pas de géodésique “plate” non-fermée. De plus, il existe au plus un nombre fini de bandes plates, et au plus un nombre fini de géodésiques fermées “plates” isolées.

Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\ge 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of points with negative curvature on $M$ has finitely many connected components. Under the same condition, we prove that a non-closed “flat” geodesic doesn’t exist, and moreover, there are at most finitely many flat strips, and at most finitely many isolated closed “flat” geodesics.

Mots clés : Ergodicity, Geodesic flow, Nonpositive curvature, Flat geodesic, Expansivity

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