×

Inégalités “a priori” pour des tores lagrangiens invariants par des difféomorphismes symplectiques. (‘A priori’ inequalities for Langrangian tori invariant under symplectic diffeomorphisms). (French) Zbl 0717.58020

The author’s result is the generalization of the first Birkhoff theorem on Lipschitz inequalities and the second Birkhoff theorem to the case, when invariant curves are graphs. The method of proving this result is very difficult and interesting. A part of this result was announced in Sémin. Equations Dériv. Partielles 1987/88, Exp. No.14, 24 p. (1988; Zbl 0664.58005)].
Reviewer: D.Nguyen Huu

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37C80 Symmetries, equivariant dynamical systems (MSC2010)

Citations:

Zbl 0664.58005
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] V. I. Arnold, Characteristic class entering in quantization conditions,Funct. Anal. Appl.,1 (1967), 1–13. · Zbl 0175.20303 · doi:10.1007/BF01075861
[2] V. I. Arnold etA. Avez,Problèmes ergodiques de la mécanique classique, Paris, Gauthier Villars, 1967.
[3] M. C. Arnaud, travail en préparation annoncé dans: Sur les points fixes des difféomorphismes exacts symplectiques deT n {\(\times\)}R n ,C.R. Acad. Sci. Paris, t.309 (1989), 191–194.
[4] S. Aubry andP. Y. Daeron, The discrete Frenkel-Kantova model and its extensions I. Exact results for ground states,Physica,8D (1983), 381–422. · Zbl 1237.37059
[5] M. Audin, Fibrés normaux d’immersions en dimension double, points doubles d’immersions lagrangiennes et plongements totalement réels,Comment. Math. Helv.,63 (1988), 593–623. · Zbl 0666.57024 · doi:10.1007/BF02566781
[6] V. Bangert,Minimal geodesics, Prepint, Univ. de Berne, 1987. · Zbl 0645.58017
[7] D. Bernstein andA. B. Katok, Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians,Invent. Math.,88 (1987), 225–241. · Zbl 0642.58040 · doi:10.1007/BF01388907
[8] G. D. Birkhoff, Surface transformations and their dynamical applications,Acta Math.,43 (1920), 1–119;Collected Math. Papers, vol. 2, p. 111–229. · JFM 47.0985.03 · doi:10.1007/BF02401754
[9] J.-B. Bost, Tores invariants des systèmes dynamiques hamiltoniens,Séminaire Bourbaki, Exposé no 639,Astérisque,133–134 (1986), 113–157.
[10] N. Bourbaki,Intégration, chap. 3, § 1, no 6, 2e éd., Paris, Hermann, 1965. · Zbl 0136.03404
[11] M.-L. Byalyi andL. V. Polterovich, Geodesic flows on the two-dimensional torus and phase transitions ”commensurability-non commensurability”,Funct. Anal. Appl.,20 (1986), 260–266. · Zbl 0641.58032 · doi:10.1007/BF01083491
[12] M.-L. Byalyi andL. V. Polterovich, Lagrangian singularities of invariant tori of Hamiltonian systems with two degrees of freedom,Invent. Math.,97 (1989), 291–303. · Zbl 0675.58016 · doi:10.1007/BF01389043
[13] M.-L. Byalyi,Aubry Mather sets and Birkoff’s theorem for geodesic flows on the two-dimensional torus, Preprint 1988, Weizmann Inst.
[14] A. Chencincer, La dynamique au voisinage d’un point fixe elleptique conservatif: de Poincaré et Birkhoff à Aubry et Mather,Séminaire Bourbaki, exposé no 622,Astérisque,121–122 (1985), 147–170.
[15] C. C. Conley, The gradient structure of a flow: I,Erg. Th. Dyn. Syst.,8* (1988), 11–31. · Zbl 0687.58033 · doi:10.1017/S0143385700009305
[16] R. Douady, Stabilité ou instabilité des points fixes elliptiques,Ann. Sci. Ec. Norm. Sup., 4e série,21 (1988), 1–46. · Zbl 0656.58020
[17] A. Fathi, “ Une interprétation plus topologique de la démonstration du théorème de Birkhoff{”, Appendice du Chap. I de [H3].}
[18] W. H. Gottschalk andG. A. Hedlund, Topological Dynamics,Am. Math. Soc. (1955), § 14.
[19] M. R. Herman, Existence et non-existence de tores invariants par des difféomorphismes symplectiques,Séminaire Equations aux dérivées partielles, Exposé XIV, 1987–1988, Ecole Polytechnique.
[20] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,Publ. Math. I.H.E.S.,49 (1979), 5–233. · Zbl 0448.58019
[21] M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l’anneau, vol. 1,Astérisque,103–104 (1983).
[22] M. R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser,Comment. Math. Helv.,58 (1983), 453–502. · Zbl 0554.58034 · doi:10.1007/BF02564647
[23] R. A. Horn etC. A. Johnson,Matrix analysis, New York, Cambridge Univ. Press (1985).
[24] R. A. Johnson,m-functions and Floquet exponents for linear differential systems,Annali di Mat. Pura ed Appl.,147 (1987), 211–248. · Zbl 0652.34016 · doi:10.1007/BF01762419
[25] A. B. Katok,Minimal orbits for small perturbations of completely integrable Hamiltonian systems, Preprint 1989, Cal. Tech. · Zbl 0762.58024
[26] P. Le Calvez, Propriétés dynamiques des zones d’instabilités,Ann. Sci. Ec. Norm. Sup., 4e série,20 (1987), 443–464. · Zbl 0653.58014
[27] P. Le Calvez, Propriétés générales des applications déviant la verticale,Bull. Soc. Math. France,117 (1989), 69–102.
[28] J. Mather, The existence of quasi-periodic orbits for twist homeomorphisms of the annulus,Topology,21 (1982), 457–467. · Zbl 0506.58032 · doi:10.1016/0040-9383(82)90023-4
[29] J. Mather, Minimal measures,Comm. Math. Helv.,64 (1989), 375–394. · Zbl 0689.58025 · doi:10.1007/BF02564683
[30] J. Mather,Letter to R. Mac Kay, Feb. 21, 1984.
[31] J. Mather, Minimal action measures for positive definite lagrangian systems, to appear, inProc. IXth Int. Conf. Math. Phys.; version détaillée, Preprint, ETH, Zürich, 1989.
[32] J. Mather, Exposés au Séminaire de Systèmes dynamiques au Centre de Mathématiques de l’Ecole Polytechnique (1985);Variational construction of orbits of twist diffeomorphisms, Preprint, ETH, Zürich, 1990.
[33] J. Mather, Communications personnelles, 1988–1989.
[34] M. Mohsin, Formes cobordables,Ann. Sci. Ec. Norm. Sup., 3e série,83 (1966), 201–213. · Zbl 0153.13701
[35] M. Misiurewiz etK. Zieman,Rotation sets for maps of tori, Preprint, Univ. Warwick (1988).
[36] D. Salamon,The Kolmogorov-Arnold-Moser theorem, Preprint, ETH Zürich (1986). · Zbl 1136.37348
[37] D. Salamon andE. Zehnder, KAM theory in configuration space,Comment. Math. Helv.,64 (1989), 84–142. · Zbl 0682.58014 · doi:10.1007/BF02564665
[38] J. J. Schwartz,Nonlinear functional analysis, Courant Inst. of Math. Sci., 1965.
[39] L. Schwartz,Théories des distributions, t. I et II, Paris, Hermann, 1950.
[40] M. Shub, Stabilité globale des systèmes dynamiques,Astérisque,56 (1978). · Zbl 0396.58014
[41] E. M. Stein,Singular integrals and differentiability properties of functions, Princeton, Princeton Univ. Press, 1970. · Zbl 0207.13501
[42] C. Viterbo, Lettre 10 mai 1989.
[43] C. Viterbo,A new obstruction to embedding Lagrangian tori, Preprint, Berkeley, 1989, à paraître dansInventiones Math. · Zbl 0727.58015
[44] E. Zehnder, Generalized implicit function theorems with applications to small divisor problems, II,Comm. Pure Appl. Math.,29 (1976), 49–113. · Zbl 0334.58009 · doi:10.1002/cpa.3160290104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.