]. R. References1, J. E. Abraham, T. Marsden, and . Ratiu, Manifolds, tensor analysis, and applications, Applied Mathematical Sciences, vol.75, 1988.

M. Arnaud, Le ??closing lemma?? en topologie $C^1$, Mémoires de la Société mathématique de France, vol.1, 1998.
DOI : 10.24033/msmf.387

M. Arnaud, Fibr??s de Green et R??gularit?? des Graphes C 0-Lagrangiens Invariants par un Flot de Tonelli, Annales Henri Poincar??, vol.9, issue.5, pp.881-926, 2008.
DOI : 10.1007/s00023-008-0375-7

M. Arnaud, The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents, Annals of Mathematics, vol.174, issue.3
DOI : 10.4007/annals.2011.174.3.4

M. Arnaud, Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.29, issue.6, 2010.
DOI : 10.1016/j.anihpc.2012.04.007

URL : https://hal.archives-ouvertes.fr/hal-01325834

P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Mathematical Research Letters, vol.14, issue.3, pp.503-511, 2007.
DOI : 10.4310/MRL.2007.v14.n3.a14

URL : https://hal.archives-ouvertes.fr/hal-00014966

P. Bernard, Existence of C1,1C1,1 critical sub-solutions of the Hamilton???Jacobi equation on compact manifolds, Annales Scientifiques de l?????cole Normale Sup??rieure, vol.40, issue.3, pp.445-452, 2007.
DOI : 10.1016/j.ansens.2007.01.004

URL : https://hal.archives-ouvertes.fr/hal-00014965

P. Bernard, On the Number of Mather Measures of Lagrangian Systems, Archive for Rational Mechanics and Analysis, vol.58, issue.1
DOI : 10.1007/s00205-009-0289-7

URL : https://hal.archives-ouvertes.fr/hal-00448263

P. Bernard and G. Contreras, A generic property of families of Lagrangian systems, Annals of Mathematics, vol.167, issue.3, pp.1099-1108, 2008.
DOI : 10.4007/annals.2008.167.1099

URL : https://hal.archives-ouvertes.fr/hal-00294671

P. Cannarsa and L. Rifford, Semiconcavity results for optimal control problems admitting no singular minimizing controls??????This research was completed, in part, while the first author was visiting the Universit?? de Lyon 1 and, in part, while the second author was visiting the Universit?? di Roma Tor Vergata on an INdAM grant. The authors are grateful to the above institutions for their support., Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.25, issue.4, pp.773-802, 2008.
DOI : 10.1016/j.anihpc.2007.07.005

P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications, 2004.

G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points, Ergodic Theory and Dynamical Systems, vol.19, issue.4, pp.901-952, 1999.
DOI : 10.1017/S014338579913387X

G. Contreras, R. Iturriaga, G. P. Paternain, and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values, GAFA, vol.8, issue.5, pp.788-809, 1998.
DOI : 10.1007/s000390050074

J. Coron, Control and nonlinearity, Mathematical Surveys and Monographs, vol.136, 2007.
DOI : 10.1090/surv/136

L. C. Evans and R. F. Gariepy, Measure Theorem and Fine Properties of Functions, Studies in Advanced Mathematics, 1992.

A. Fathi, Th??or??me KAM faible et th??orie de Mather sur les syst??mes lagrangiens, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.324, issue.9, pp.1043-1046, 1997.
DOI : 10.1016/S0764-4442(97)87883-4

A. Fathi, Solutions KAM faibles conjugu??es et barri??res de Peierls, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.325, issue.6, pp.649-652, 1997.
DOI : 10.1016/S0764-4442(97)84777-5

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.327, issue.3, pp.267-270, 1998.
DOI : 10.1016/S0764-4442(98)80144-4

A. Fathi, Regularity of C$^1$ solutions of the Hamilton-Jacobi equation, Annales de la facult?? des sciences de Toulouse Math??matiques, vol.12, issue.4, pp.479-516, 2003.
DOI : 10.5802/afst.1059

A. Fathi, Weak KAM Theorem and Lagrangian Dynamics

A. Fathi, On the existence of smooth subsolution of the Hamilton-Jacobi equation, 2010.

A. Fathi, A. Figalli, and L. Rifford, On the Hausdorff dimension of the mather quotient, Communications on Pure and Applied Mathematics, vol.30, issue.1, pp.445-500, 2009.
DOI : 10.1002/cpa.20250

URL : https://hal.archives-ouvertes.fr/hal-00769107

A. Fathi and E. Maderna, Weak kam theorem on non compact manifolds, Nonlinear Differential Equations and Applications NoDEA, vol.14, issue.1-2, pp.1-27, 2007.
DOI : 10.1007/s00030-007-2047-6

A. Fathi and A. Siconolfi, Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation, Inventiones Mathematicae, vol.155, issue.2, pp.363-388, 2004.
DOI : 10.1007/s00222-003-0323-6

H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, 1969.

A. Figalli and L. Rifford, Closing Aubry Sets II, Communications on Pure and Applied Mathematics, vol.33, issue.1-3, 2010.
DOI : 10.1002/cpa.21512

URL : https://hal.archives-ouvertes.fr/hal-00935970

C. Gutierrez, A counter-example to a C 2 closing lemma. Ergodic Theory Dynam, Systems, vol.7, issue.4, pp.509-530, 1987.

C. Gutierrez, -closing for flows on 2-manifolds, Nonlinearity, vol.13, issue.6, pp.1883-1888, 2000.
DOI : 10.1088/0951-7715/13/6/301

URL : https://hal.archives-ouvertes.fr/hal-00409102

C. Gutierrez, On the C r -closing lemma. The geometry of differential equations and dynamical systems, Comput. Appl. Math, vol.20, issue.12, pp.179-186, 2001.

M. Herman, Exemples de flots hamiltoniens dont aucune perturbation en topologie C ? n'a d'orbites périodiques sur un ouvert de surfaces d'´ energies. (French) [Examples of Hamiltonian flows such that no C ? perturbation has a periodic orbit on an open set of energy surfaces, C. R. Acad. Sci. Paris Sér. I Math, issue.13, pp.312989-994, 1991.

M. Herman, Différentiabilité optimale et contre-exemplesàexemplesà la fermeture en topologie C ? des orbites récurrentes de flots hamiltoniens. (French. English summary) [Optimal differentiability and counterexamples to the C ? closing lemma for Hamiltonian vector fields, C. R. Acad. Sci. Paris Sér. I Math, vol.313, issue.1, pp.49-51, 1991.

S. Lloyd, On the Closing Lemma problem for the torus, Discrete and Continuous Dynamical Systems, vol.25, issue.3, pp.951-962, 2009.
DOI : 10.3934/dcds.2009.25.951

J. Mai, A simpler proof of C 1 closing lemma, Sci. Sinica Ser. A, vol.29, issue.10, pp.1020-1031, 1986.

R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, vol.9, issue.2, pp.273-310, 1996.
DOI : 10.1088/0951-7715/9/2/002

D. Massart, On Aubry sets and Mather???s action functional, Israel Journal of Mathematics, vol.43, issue.1, pp.157-171, 2003.
DOI : 10.1007/BF02787406

J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Mathematische Zeitschrift, vol.98, issue.1, pp.169-207, 1991.
DOI : 10.1007/BF02571383

J. N. Mather, Variational construction of connecting orbits, Annales de l???institut Fourier, vol.43, issue.5, pp.1349-1386, 1993.
DOI : 10.5802/aif.1377

J. N. Mather, Total disconnectedness of the quotient Aubry set in low dimensions, Communications on Pure and Applied Mathematics, vol.43, issue.8, pp.1178-1183, 2003.
DOI : 10.1002/cpa.10091

J. N. Mather, Examples of Aubry sets, Ergodic Theory and Dynamical Systems, vol.24, issue.5, pp.1667-1723, 2004.
DOI : 10.1017/S0143385704000446

C. C. Pugh, The Closing Lemma, American Journal of Mathematics, vol.89, issue.4, pp.956-1009, 1967.
DOI : 10.2307/2373413

C. C. Pugh, An Improved Closing Lemma and a General Density Theorem, American Journal of Mathematics, vol.89, issue.4, pp.1010-1021, 1967.
DOI : 10.2307/2373414

C. C. Pugh and C. Robinson, The C 1 closing lemma, including Hamiltonians. Ergodic Theory Dynam, Systems, vol.3, issue.2, pp.261-313, 1983.

L. Rifford, On Viscosity Solutions of Certain Hamilton???Jacobi Equations: Regularity Results and Generalized Sard's Theorems, Communications in Partial Differential Equations, vol.149, issue.3, pp.517-559, 2008.
DOI : 10.1016/j.anihpc.2005.05.002

L. Rifford, Nonholonomic Variations: An introduction to sub-Riemannian geometry

J. Roquejoffre, Propriétés qualitatives des solutions deséquationsdeséquations de Hamilton-Jacobi (d'après A, Séminaire Bourbaki, 2007.

T. Sakai, Riemannian geometry, Translations of Mathematical Monographs, vol.149, 1996.

A. Sorrentino, On the total disconnectedness of the quotient Aubry set. Ergodic Theory Dynam, Systems, vol.28, issue.1, pp.267-290, 2008.