Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, p. 989-1007
Dans cet article, on étudie les mesures minimisantes de Hamiltoniens de Tonelli. Plus précisément, on explique quelles relations existent entre les fibrés de Green et différentes notions comme :•les exposants de Lyapunov des mesures minimisantes ;•les solutions KAM faibles. On en déduit par exemple que si tous les exposants de Lyapunov dʼune mesure minimisante μ sont nuls, alors le support de cette mesure est C 1 -régulier en μ-presque tout point.
In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as:•the Lyapunov exponents of minimizing measures;•the weak KAM solutions. In particular, we deduce that the support of every minimizing measure μ, all of whose Lyapunov exponents are zero, is C 1 -regular μ-almost everywhere.
DOI : https://doi.org/10.1016/j.anihpc.2012.04.007
Classification:  37J50,  35D40,  37C40,  34D08,  35D65
Mots clés: Orbites et mesures minimisantes, Exposants de Lyapunov, Théorie KAM faible, Fibrés de Green, Régularité des solutions de lʼéquation de Hamilton–Jacobi
@article{AIHPC_2012__29_6_989_0,
     author = {Arnaud, M.-C.},
     title = {Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {29},
     number = {6},
     year = {2012},
     pages = {989-1007},
     doi = {10.1016/j.anihpc.2012.04.007},
     zbl = {1269.37031},
     mrnumber = {2995103},
     language = {en},
     url = {http://http://www.numdam.org/item/AIHPC_2012__29_6_989_0}
}
Arnaud, M.-C. Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 989-1007. doi : 10.1016/j.anihpc.2012.04.007. http://www.numdam.org/item/AIHPC_2012__29_6_989_0/

[1] M.-C. Arnaud, Fibrés de Green et régularité des graphes C 0 -Lagrangiens invariants par un flot de Tonelli, Ann. Henri Poincaré 9 no. 5 (2008), 881-926 | MR 2438501

[2] M.-C. Arnaud, Three results on the regularity of the curves that are invariant by an exact symplectic twist map, Publ. Math. Inst. Hautes Études Sci. 109 (2009), 1-17 | Numdam | MR 2511585 | Zbl 1177.53070

[3] M.-C. Arnaud, The link between the shape of the Aubry–Mather sets and their Lyapunov exponents, Ann. of Math. 174 no. 3 (2011), 1571-1601 | MR 2846487 | Zbl 1257.37028

[4] P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc. 21 no. 3 (2008), 615-669 | MR 2393423 | Zbl 1213.37089

[5] G.D. Birkhoff, Surface transformations and their dynamical application, Acta Math. 43 (1920), 1-119 | JFM 47.0985.03 | MR 1555175

[6] J. Bochi, M. Viana, Lyapunov exponents: how frequently are dynamical systems hyperbolic?, Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge (2004), 271-297 | MR 2090775 | Zbl 1147.37315

[7] G. Bouligand, Introduction à la géométrie infinitésimale directe, Librairie Vuibert, Paris (1932) | JFM 58.0086.03

[8] G. Contreras, R. Iturriaga, Convex Hamiltonians without conjugate points, Ergodic Theory Dynam. Systems 19 no. 4 (1999), 901-952 | MR 1709426 | Zbl 1044.37046

[9] P. Eberlein, When is a geodesic flow of Anosov type? I, J. Differential Geom. 8 (1973), 437-463 P. Eberlein, When is a geodesic flow of Anosov type? II, J. Differential Geom. 8 (1973), 565-577 | MR 380891 | Zbl 0285.58008

[10] A. Fathi, Weak KAM Theorems in Lagrangian Dynamics, in preparation.

[11] A. Fathi, Regularity of C 1 solutions of the Hamilton–Jacobi equation, Ann. Fac. Sci. Toulouse Math. (6) 12 no. 4 (2003), 479-516 | Numdam | MR 2060597 | Zbl 1059.37047

[12] A. Freire, R. Mañé, On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math. 69 no. 3 (1982), 375-392 | MR 679763 | Zbl 0476.58019

[13] L.W. Green, A theorem of E. Hopf, Michigan Math. J. 5 (1958), 31-34 | MR 97833 | Zbl 0134.39601

[14] M. Herman, Sur les courbes invariantes par les difféomorphismes de lʼanneau, vol. 1, Asterisque 103–104 (1983) | MR 728564

[15] M. Herman, Inégalités “a priori” pour des tores lagrangiens invariants par des difféomorphismes symplectiques, vol. I, Inst. Hautes Études Sci. Publ. Math. 70 (1989), 47-101 | Numdam | MR 1067380 | Zbl 0717.58020

[16] R. Iturriaga, A geometric proof of the existence of the Green bundles, Proc. Amer. Math. Soc. 130 no. 8 (2002), 2311-2312 | MR 1896413 | Zbl 1067.37037

[17] R. Mañé, Global Variational Methods in Conservative Dynamics, 18 Coloquio Brasileiro de Matematica, IMPA (1991)

[18] R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977), 351-370 | MR 482849 | Zbl 0356.58009

[19] J.N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 no. 2 (1991), 169-207 | MR 1109661 | Zbl 0696.58027