Let be a smooth compact surface of nonpositive curvature, with genus . We prove the ergodicity of the geodesic flow on the unit tangent bundle of with respect to the Liouville measure under the condition that the set of points with negative curvature on 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.
Soit une surface lisse compacte de courbure négative ou nulle, de genre . Nous prouvons l’ergodicité du flot géodésique sur la tangente du faisceau unitaire de par rapport à la mesure de Liouville, en supposant que l’ensemble des points à courbure négative sur 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.
Weisheng Wu 1
@article{AFST_2015_6_24_3_625_0,
author = {Weisheng Wu},
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 = {https://afst.centre-mersenne.org/articles/10.5802/afst.1457/}
}
TY - JOUR AU - Weisheng Wu TI - On the ergodicity of geodesic flows on surfaces of nonpositive curvature JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2015 SP - 625 EP - 639 VL - 24 IS - 3 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1457/ DO - 10.5802/afst.1457 LA - en ID - AFST_2015_6_24_3_625_0 ER -
%0 Journal Article %A Weisheng Wu %T On the ergodicity of geodesic flows on surfaces of nonpositive curvature %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2015 %P 625-639 %V 24 %N 3 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1457/ %R 10.5802/afst.1457 %G en %F AFST_2015_6_24_3_625_0
Weisheng Wu. On the ergodicity of geodesic flows on surfaces of nonpositive curvature. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 3, pp. 625-639. doi : 10.5802/afst.1457. https://afst.centre-mersenne.org/articles/10.5802/afst.1457/
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