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.
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.
Accepted:
Published online:
DOI: 10.5802/jep.14
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.
Yves Coudène 1; Barbara Schapira 2
@article{JEP_2014__1__387_0, author = {Yves Coud\`ene and Barbara Schapira}, title = {Generic measures for geodesic flows on nonpositively curved manifolds}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {387--408}, publisher = {\'Ecole polytechnique}, volume = {1}, year = {2014}, doi = {10.5802/jep.14}, mrnumber = {3322793}, zbl = {1341.37004}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.14/} }
TY - JOUR AU - Yves Coudène AU - Barbara Schapira TI - Generic measures for geodesic flows on nonpositively curved manifolds JO - Journal de l’École polytechnique — Mathématiques PY - 2014 SP - 387 EP - 408 VL - 1 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.14/ DO - 10.5802/jep.14 LA - en ID - JEP_2014__1__387_0 ER -
%0 Journal Article %A Yves Coudène %A Barbara Schapira %T Generic measures for geodesic flows on nonpositively curved manifolds %J Journal de l’École polytechnique — Mathématiques %D 2014 %P 387-408 %V 1 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.14/ %R 10.5802/jep.14 %G en %F JEP_2014__1__387_0
Yves Coudène; Barbara Schapira. Generic measures for geodesic flows on nonpositively curved manifolds. Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 387-408. doi : 10.5802/jep.14. https://jep.centre-mersenne.org/articles/10.5802/jep.14/
[ABC11] - “Nonuniform hyperbolicity for -generic diffeomorphisms”, Israel J. Math. 183 (2011), p. 1-60 | DOI | MR | Zbl
[Bab02] - “On the mixing property for hyperbolic systems”, Israel J. Math. 129 (2002), p. 61-76 | DOI | MR | Zbl
[Bal82] - “Axial isometries of manifolds of nonpositive curvature”, Math. Ann. 259 (1982) no. 1, p. 131-144 | DOI | MR
[Bal95] - Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin | DOI | MR | Zbl
[Bil99] - Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999 | DOI | Zbl
[Cou02] - “Une version mesurable du théorème de Stone-Weierstrass”, Gaz. Math. (2002) no. 91, p. 10-17 | MR | Zbl
[Cou04] - “Topological dynamics and local product structure”, J. London Math. Soc. (2) 69 (2004) no. 2, p. 441-456 | DOI | MR | Zbl
[CS10] - “Generic measures for hyperbolic flows on non-compact spaces”, Israel J. Math. 179 (2010), p. 157-172 | DOI | MR | Zbl
[CS11] - “Counterexamples in nonpositive curvature”, Discrete Contin. Dynam. Systems 30 (2011) no. 4, p. 1095-1106 | DOI
[Dal00] - “Topologie du feuilletage fortement stable”, Ann. Inst. Fourier (Grenoble) 50 (2000) no. 3, p. 981-993 | DOI | MR | Zbl
[Ebe80] - “Lattices in spaces of nonpositive curvature”, Ann. of Math. (2) 111 (1980) no. 3, p. 435-476 | DOI | MR | Zbl
[Ebe96] - Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996 | Zbl
[Gro78] - “Manifolds of negative curvature”, J. Differential Geom. 13 (1978) no. 2, p. 223-230 | DOI | MR | Zbl
[Kni98] - “The uniqueness of the measure of maximal entropy for geodesic flows on rank manifolds”, Ann. of Math. (2) 148 (1998) no. 1, p. 291-314 | DOI | MR | Zbl
[Kni02] - “Hyperbolic dynamics and Riemannian geometry”, in Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, p. 453-545 | DOI | MR | Zbl
[Par61] - “On the category of ergodic measures”, Illinois J. Math. 5 (1961), p. 648-656 | MR | Zbl
[Par62] - “A note on mixing processes”, Sankhyā Ser. A 24 (1962), p. 331-332 | MR | Zbl
[Sig70] - “Generic properties of invariant measures for Axiom A diffeomorphisms”, Invent. Math. 11 (1970), p. 99-109 | DOI | MR | Zbl
[Sig72a] - “On mixing measures for Axiom A diffeomorphisms”, Proc. Amer. Math. Soc. 36 (1972), p. 497-504 | DOI | MR
[Sig72b] - “On the space of invariant measures for hyperbolic flows”, Amer. J. Math. 94 (1972), p. 31-37 | DOI | MR | Zbl
[Wal82] - An introduction to ergodic theory, Graduate Texts in Math., vol. 79, Springer-Verlag, New York-Berlin, 1982 | MR | Zbl
Cited by Sources: