Geodesic
A Variational Approach to the Study of the Existence of Invariant Lagrangian Graphs
Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 405-440.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

This paper surveys some recent results by the author and some collaborators, on the existence of invariant Lagrangian graphs for Tonelli Hamiltonian systems. 
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Sorrentino, Alfonso. A Variational Approach to the Study of the Existence of Invariant Lagrangian Graphs. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 405-440. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_2_a8/

[1] Ian Agol, Criteria for virtual fibering. J. Topol., 2 (1) (2008), 269-284. | DOI | MR | Zbl

[2] Marie-Claude Arnaud, Fibrés de Green et régularité des graphes $C^0$-Lagrangiens invariants par un flot de Tonelli. Ann. Henri Poincaré, 9 (5) (2008), 881-926. | DOI | MR | Zbl

[3] Marie-Claude Arnaud, The tiered Aubry set for autonomous Lagrangian functions. Ann. Inst. Fourier (Grenoble), 58 (5) (2008), 1733-1759. | fulltext EuDML | MR | Zbl

[4] Vladimir I. Arnol'D, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspehi Mat. Nauk, 18 (5) (1963), 13-40. | MR

[5] Vladimir I. Arnol'D, Mathematical methods of classical mechanics Graduate Texts in Mathematics, 60 (Springer-Verlag, New York, 1989). | DOI | MR

[6] Serge Aubry - P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states. Phys. D, 8 (3) (1983), 381-422. | DOI | MR | Zbl

[7] Florent Balacheff - Daniel Massart, Stable norms of non-orientable surfaces. Ann. Inst. Fourier (Grenoble) 58, no. 4 (2008), 1337-1369. | fulltext EuDML | MR | Zbl

[8] Victor Bangert, Mather sets for twist maps and geodesics on tori. Dyn. Rep. 1 (1988), 1-56. | MR | Zbl

[9] Patrick Bernard, Symplectic aspects of Mather theory. Duke Math. J., 136 (3) (2007), 401-420. | MR

[10] Luitzen E. J. Brouwer, Zur invarianz des n-dimensionalen Gebiets. Math. Ann., 72 (1) (1912), 55-56. | fulltext EuDML | DOI | MR | Zbl

[11] Dmitri Burago - Sergey Ivanov, Riemannian tori without conjugate points are flat. Geom. Funct. Anal, 4, no. 3 (1994), 259-269. | fulltext EuDML | DOI | MR | Zbl

[12] Leo T. Butler - Gabriel P. Paternain, Collective geodesic flows. Ann. Inst. Fourier (Grenoble), 53 (1) (2003), 265-308. | fulltext EuDML | MR | Zbl

[13] Leo T. Butler, A weak Liouville-Arnol'd Theorem. Talk at University of Pittsburgh, January 2011.

[14] Leo T. Butler - Alfonso Sorrentino, Weak Liouville-Arnol'd Theorems and their implications. Comm. Math. Phys., 315 (1) (2012), 109-133. | DOI | MR | Zbl

[15] Constantin Caratheodory, Variationsrechnung und partielle Differentialgleichung erster Ordnung. Leipzig-Berlin: B. G. Teubner, 1935. | MR

[16] Mario J. Dias Carneiro, On minimizing measures of the action of autonomous Lagrangians. Nonlinearity, 8 (6) (1995), 1077-1085. | MR | Zbl

[17] Gonzalo Contreras - Renato Iturriaga, Global minimizers of autonomous Lagrangians. Preprint, 1999. | MR | Zbl

[18] Nathan M. Dunfield - William P. Thurston, The virtual Haken conjecture: experiments and examples. Geom. Topol., 7 (2003), 399-441. | fulltext EuDML | DOI | MR | Zbl

[19] Albert Fathi, The Weak KAM theorem in Lagrangian dynamics. Cambridge University Press (to appear). | Zbl

[20] Albert Fathi - Alessandro Giuliani - Alfonso Sorrentino, Uniqueness of Invariant Lagrangian graphs in a homology or a cohomology class. Ann. Sc. Norm. Super. Pisa Cl. Sci., Vol VIII (4) (2009), 659-680. | MR | Zbl

[21] Gustav A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Ann. of Math. (2), 33 (4) (1932), 719-739. | DOI | MR | Zbl

[22] John Hempel, Homology of coverings. Pacific J. Math., 112 (1) (1984), 83-113. | MR

[23] Andrey N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 572-530. | MR

[24] Ezequiel Maderna, Invariance of global solutions of Hamilton-Jacobi equation. Bull. Soc. Math. France, 130 (4) (2002), 493-506. | fulltext EuDML | DOI | MR | Zbl

[25] Ricardo Mañé, On the minimizing measures of Lagrangian dynamical systems. Nonlinearity, 5 (3) (1992), 623-638. | MR | Zbl

[26] Ricardo Mañé, Lagrangian flows: the dynamics of globally minimizing orbits. Bol. Soc. Brasil. Mat. (N.S.), 28 (2) (1997), 141-153. | DOI | MR | Zbl

[27] Daniel Massart, Aubry sets vs Mather sets in two degrees of freedom. Calculus of Variations and PDE, no. 3-4 (2011), 429-460. | DOI | MR | Zbl

[28] Daniel Massart - Alfonso Sorrentino, Differentiability of Mather's average action and integrability on closed surfaces. Nonlinearity, 24 (2011), 1777-1793. | DOI | MR | Zbl

[29] John N. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology, 21 (4) (1982), 457-467. | DOI | MR | Zbl

[30] John N. Mather, A criterion for the nonexistence of invariant circles. Inst. Hautes Études Sci. Publ. Math., 63 (1986), 153-204. | fulltext EuDML | MR | Zbl

[31] John N. Mather - Giovanni Forni, Action minimizing orbits in Hamiltonian systems. Transition to chaos in classical and quantum mechanics. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, July 6-13, 1991. Berlin: Springer-Verlag. Lect. Notes Math. 1589, 92-186, 1994. | DOI | MR | Zbl

[32] John N. Mather, Variational construction of orbits of twist diffeomorphisms. J. Amer. Math. Soc., 4 (2) (1991), 207-263. | DOI | MR | Zbl

[33] John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z., 207 (2) (1991), 169-207. | fulltext EuDML | DOI | MR | Zbl

[34] John N. Mather, Variational construction of connecting orbits. Ann. Inst. Fourier (Grenoble), 43 (5) (1993), 1349-1386. | fulltext EuDML | MR | Zbl

[35] John N. Mather, Order structure on action minimizing orbits. Symplectic topology and measure preserving dynamical systems. Papers of the AMS-IMS-SIAM joint summer research conference, Snowbird, UT, USA, July 1-5, 2007. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 512, 41-125, 2010.

[36] Jürgen Moser, Convergent series expansions for quasi-periodic motions. Math. Ann., 169 (1967), 136-176. | fulltext EuDML | DOI | MR | Zbl

[37] R. Tyrell Rockafellar, Convex analysis. Princeton Mathematical Series, No. 28, Princeton University Press (1970), xviii+451. | MR

[38] Sol Schwartzman, Asymptotic cycles. Ann. of Math. (2), 66 (1957), 270-284. | DOI | MR

[39] Alfonso Sorrentino, On the integrability of Tonelli Hamiltonians. Trans. Amer. Math. Soc., 363 (10) (2011), 5071-5089. | DOI | MR

[40] Alfonso Sorrentino, Lecture Notes on Mather's theory for Lagrangian systems. Preprint, 2011. | MR | Zbl