On the ergodicity of geodesic flows on surfaces of nonpositive curvature
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 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 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.

DOI : https://doi.org/10.5802/afst.1457
Mots clés : Ergodicity, Geodesic flow, Nonpositive curvature, Flat geodesic, Expansivity
@article{AFST_2015_6_24_3_625_0,
     author = {Wu, Weisheng},
     title = {On the ergodicity of geodesic flows on surfaces of nonpositive curvature},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {625--639},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 24},
     number = {3},
     year = {2015},
     doi = {10.5802/afst.1457},
     mrnumber = {3403734},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1457/}
}
Wu, Weisheng. On the ergodicity of geodesic flows on surfaces of nonpositive curvature. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 3, pp. 625-639. doi : 10.5802/afst.1457. http://www.numdam.org/articles/10.5802/afst.1457/

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