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.
@article{JEP_2014__1__387_0,
author = {Coud\`ene, Yves and Schapira, Barbara},
title = {Generic measures for geodesic flows on nonpositively curved manifolds},
journal = {Journal de l'\'Ecole polytechnique - Math\'ematiques},
volume = {1},
year = {2014},
pages = {387-408},
doi = {10.5802/jep.14},
language = {en},
url = {http://dml.mathdoc.fr/item/JEP_2014__1__387_0}
}
Coudène, Yves; Schapira, Barbara. Generic measures for geodesic flows on nonpositively curved manifolds. Journal de l'École polytechnique - Mathématiques, Tome 1 (2014) pp. 387-408. doi : 10.5802/jep.14. http://gdmltest.u-ga.fr/item/JEP_2014__1__387_0/
[ABC11] Nonuniform hyperbolicity for -generic diffeomorphisms, Israel J. Math., Tome 183 (2011), pp. 1-60 | Article | MR 2811152 | Zbl 1246.37040
[Bab02] On the mixing property for hyperbolic systems, Israel J. Math., Tome 129 (2002), pp. 61-76 | Article | MR 1910932 | Zbl 1053.37001
[Bal82] Axial isometries of manifolds of nonpositive curvature, Math. Ann., Tome 259 (1982) no. 1, pp. 131-144 | Article | MR 656659 | Zbl 0487.53039
[Bal95] Lectures on spaces of nonpositive curvature, Birkhäuser Verlag, Basel, DMV Seminar, Tome 25 (1995), pp. viii+112 (With an appendix by Misha Brin) | Article | MR 1377265
[Bil99] Convergence of probability measures, John Wiley & Sons, Inc., New York, Wiley Series in Probability and Statistics: Probability and Statistics (1999), pp. x+277 | Article | MR 1700749 | Zbl 0172.21201
[Cou02] Une version mesurable du théorème de Stone-Weierstrass, Gaz. Math. (2002) no. 91, pp. 10-17 | MR 1896063 | Zbl 1026.46021
[Cou04] Topological dynamics and local product structure, J. London Math. Soc. (2), Tome 69 (2004) no. 2, pp. 441-456 | Article | MR 2040614 | Zbl 1055.37017
[CS10] Generic measures for hyperbolic flows on non-compact spaces, Israel J. Math., Tome 179 (2010), pp. 157-172 | Article | MR 2735038 | Zbl 1229.53078
[CS11] Counterexamples in nonpositive curvature, Discrete Contin. Dynam. Systems, Tome 30 (2011) no. 4, pp. 1095-1106 http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=6195 | Article | MR 2812955 | Zbl 1257.37020
[Dal00] Topologie du feuilletage fortement stable, Ann. Inst. Fourier (Grenoble), Tome 50 (2000) no. 3, pp. 981-993 | Numdam | MR 1779902 | Zbl 0965.53054
[Ebe80] Lattices in spaces of nonpositive curvature, Ann. of Math. (2), Tome 111 (1980) no. 3, pp. 435-476 | Article | MR 577132 | Zbl 0401.53015
[Ebe96] Geometry of nonpositively curved manifolds, University of Chicago Press, Chicago, IL, Chicago Lectures in Mathematics (1996), pp. vii+449 | MR 1441541 | Zbl 0883.53003
[Gro78] Manifolds of negative curvature, J. Differential Geom., Tome 13 (1978) no. 2, pp. 223-230 http://projecteuclid.org/euclid.jdg/1214434487 | MR 540941 | Zbl 0433.53028
[Kni02] Hyperbolic dynamics and Riemannian geometry, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam (2002), pp. 453-545 | Article | MR 1928523 | Zbl 1049.37020
[Kni98] The uniqueness of the measure of maximal entropy for geodesic flows on rank manifolds, Ann. of Math. (2), Tome 148 (1998) no. 1, pp. 291-314 | Article | MR 1652924 | Zbl 0946.53045
[Par61] On the category of ergodic measures, Illinois J. Math., Tome 5 (1961), pp. 648-656 | MR 148850 | Zbl 0103.28101
[Par62] A note on mixing processes, Sankhyā Ser. A, Tome 24 (1962), p. 331-332 | MR 169283 | Zbl 0108.30702
[Sig70] Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., Tome 11 (1970), pp. 99-109 | MR 286135 | Zbl 0193.35502
[Sig72a] On mixing measures for Axiom A diffeomorphisms, Proc. Amer. Math. Soc., Tome 36 (1972), pp. 497-504 | MR 309155 | Zbl 0225.28013
[Sig72b] On the space of invariant measures for hyperbolic flows, Amer. J. Math., Tome 94 (1972), pp. 31-37 | MR 302866 | Zbl 0242.28014
[Wal82] An introduction to ergodic theory, Springer-Verlag, New York-Berlin, Graduate Texts in Math., Tome 79 (1982), pp. ix+250 | MR 648108 | Zbl 0958.28011