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References
[An1] S.Angenent: Monotone recurrence relations, their Birkhoff orbits, and their topological entropy, Ergodic Th. Dynam. Sys. (to appear)
[An2] S.Angenent: A remark on the topological entropy and invariant circles of an area preserving twist map, in Twist mappings and their applications, R. McGehee and K.R. Meyer editors, New York, Springer-Verlag, 1992.
[Au] S.Aubry: The twist map, the extended Frenkel-Kontorova model and the devil's staircase, Physica 7D (1983), 240–258.
[Au-LeD] S.Aubry—P.Y.LeDaeron: The discrete Frenkel-Kontorova model and its extensions I: exact results for the ground states, Physica 8D (1983), 381–422.
[B-M] J.Ball—V.Mizel: One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Ration. Mech. Anal. 90 (1988), 325–388.
[Ba] V.Bangert: Mather sets for twist maps and geodesics on tori, Dynamics Reported 1 (1988), 1–45.
[Bi1] G.D. Birkhoff: Surface transformations and their dynamical applications, Acta Math. 43 (1922), 1–119. Reprinted in Collected Mathematical papers, American Math. Soc., New York, 1950, Vol. II, 111–229.
[Bi2] G.D. Birkhoff: On the periodic motion of dynamical systems. Acta Math. 50 (1927), 359–379. Reprinted in Collected Mathematical papers, American Math. Soc., New York, 1950, Vol. II, 333–353.
[Bi3] G.D. Birkhoff: Sur quelques courbes fermées remarquables, Bull. Soc. Math. de France 60 (1932), 1–26. Reprinted in Collected Mathematical papers, American Math. Soc., New York, 1950, Vol. II, 418–443.
[B1] S. Bullet: Invariant circles for the piece-wise linear standard map, Comm. Math. Phys. 107 (1986), 241–262.
[De] A. Denjoy: Sur les courbes définies par les équations differentielles à la surface du tore, J. Math. Pures Appl. 11 (1932), 333–375.
[F] G.Forni: Construction of invariant measures and destruction of invariant curves for twist maps of the annulus, Ph. D. Thesis, Princeton University, October 1993.
[Gr] J.M. Greene: A method for determining stochastic transition, J. Math. Phys. 20 (1979), 1183–1201.
[Hd] G.A. Hedlund: Godesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math. 33 (1932), 719–739.
[He] M.R. Herman: Sur les courbes invariantes par les difféomorphismes de l'anneau, Vol. I & II, Asterisque 103–104 (1983) & 144 (1986).
[La] V.F. Lazutkin: The existence of caustics for a billiard problem in a convex domain, Math. USSR Izvestija 7 (1973), 185–214. Translation of Izvestija, Mathematical series, Academy of Sciences of the USSR, 37, 1973.
[L-L] A.J. Lichtenberg—M.A. Liebermann: Regular and Chaotic Dynamics, Springer-Verlag, New-York 1983 (Second Edition 1992)
[MK-P] R.S. MacKay— I.C. Percival: Converse KAM: Theory and Practice, Comm. Math. Phys. 98 (1985), 469–512.
[Mñ] R. Mañé: Properties and Problems of Minimizing Measures of Lagrangian Systems, preprint, 1993.
[Ma1] J.N. Mather: Existence of quasi-periodic orbits for twist homeomorphism of the annulus, Topology 21 (1982), 457–467.
[Ma2] J.N. Mather: Glancing billiards, Ergod. Th. Dynam. Sys. 2 (1982), 397–403.
[Ma3] J.N. Mather: letter to R.S. MacKay, February 1984.
[Ma4] J.N. Mather: Non-existence of invariant circles, Ergod. Th. Dynam. Sys. 4 (1984), 301–309.
[Ma5] J.N. Mather: Non-uniqueness of solutions of Percival's Euler-Lagrange equations, Commun. Math. Phys. 86 (1983), 465–473.
[Ma6] J.N. Mather: More Denjoy invariant sets for area preserving diffeomorphisms, Comment. Math. Helv. 60 (1985), 508–557.
[Ma7] J.N. Mather: A criterion for the non existence of invariant circles, Publ. Math. I.H.E.S. 63 (1986), 153–204.
[Ma8] J.N. Mather: Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian systems and Related topics, ed. P.H. Rabinowitz et al. NATO ASI Series C 209. D. Reidel, Dordrecht (1987), 177–202.
[Ma9] J.N. Mather: Destruction of invariant circles, Ergod. Th. Dynam. Sys. 8 (1988), 199–214.
[Ma10] J.N. Mather: Minimal measures, Comment. Math. Helv. 64 (1989), 375–394.
[Ma11] J.N. Mather: Variational construction of orbits for twist diffeomorphisms, J. Amer. Math. Soc. 4 (1991), no. 2, 203–267.
[Ma12] J.N. Mather: Action minimizing invariant measures for positive definite Lagrangian systems, Mah. Z. 207 (1991), 169–207.
[Ma13] J.N. Mather: Variational construction of orbits of twist diffeomorphisms II, to Bernard Malgrange on his 65th Birthday, preprint (to appear in the Proceedings of the Malgrange Fest).
[Mo1] J. Moser: Stable and Random motions in Dynamical Systems, Princeton Univ. Press, Princeton, 1973.
[Mo2] J. Moser: Monotone twist mappings and the calculus of variations, Ergod. Th. Dynam. Sys. 6 (1986), 401–413.
[P-deM] J. Palis-W.de Melo: Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, New York-Heidelberg-Berlin, 1982.
[P-M] R. Perez-Marco: Solution complète au problème de Siegel de linéarization d'une application holomorphe au voisinage d'un point fixe (d'après J.-C. Yoccoz), Séminaire Bourbaki, 44ième année, 753 (1991–92), 273–309.
[Pe1] I.C. Percival: A variational principle for invariant tori of fixed frequency, J. Phys. A: Math. and Gen. 12 (1979), No. 3, L. 57.
[Pe2] I.C. Percival: Variational principles for invariant tori and cantori, in Symp. on Nonlinear Dynamics and Beam-Beam Interactions, (Edited by M. Month and J.C. Herrara), No. 57 (1980), 310–320.
[Po] H. Poincaré: Oeuvres, Vol. I., Gauthier-Villars, Paris, 1928–1956.
[Rc] R.T. Rockafellar: Convex Analysis, Princeton Math. Ser., vol. 28, Princeton University Press, Princeton, 1970.
[Rs] H. Rüssmann: On the frequencies of quasi-periodic solutions of nearly integrable Hemiltonian systems, Preprint, Euler International Mathematical Institute, St. Peterburg, Dynamical Systems, 14–27 October 1991.
[S-Z] D. Salamon— E. Zehnder: KAM theory in configuration space, Comm. Math. Helvetici 64 (1989), 84–132.
[Sw] S. Schwartzman: Asymptotic cycles, Ann. Math. II Ser., 66 (1957), 270–284.
[Yo] J.-C. Yoccoz: Conjugaison Différentiable des Difféomorphismes du Cercle dont le Nombre de Rotation Vérifie une Condition Diophantienne, Ann. Scient. Éc. Norm. Sup., 17 (1984), 333–359.
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Mather, J.N., Forni, G. (1994). Action minimizing orbits in hamiltomian systems. In: Graffi, S. (eds) Transition to Chaos in Classical and Quantum Mechanics. Lecture Notes in Mathematics, vol 1589. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074076
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