A criterion for the non-existence of invariant circles
Mather, John N.
Publications Mathématiques de l'IHÉS, Tome 64 (1986), p. 153-204 / Harvested from Numdam
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Publié le : 1986-01-01
@article{PMIHES_1986__63__153_0,
     author = {Mather, John},
     title = {A criterion for the non-existence of invariant circles},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     volume = {64},
     year = {1986},
     pages = {153-204},
     mrnumber = {88b:58115},
     zbl = {0603.58028},
     language = {en},
     url = {http://dml.mathdoc.fr/item/PMIHES_1986__63__153_0}
}

Mather, John N. A criterion for the non-existence of invariant circles. Publications Mathématiques de l'IHÉS, Tome 64 (1986) pp. 153-204. http://gdmltest.u-ga.fr/item/PMIHES_1986__63__153_0/

[1] Aubry, S., The devil's staircase transformation in incommensurate lattices, in The Riemann problem, complete integrability and arithmetic applications, Lecture Notes in Math., 925, Berlin-New York, Springer, 1982, 221-245. | MR 659871 | MR 83k:82012

[2] Aubry, S., On the Dynamics of Structured Phase Transition, Lattice Locking and Ergodic Theory, Preprint.

[3] Aubry, S., On Modulated Crystallographic Structures, II : Exact Results on the Classical Ground States of a One-dimensional Model, Preprint, 1978.

[4] Aubry, S., Many Defect Structures, Stochasticity and Incommensurability, in Les Houches Session XXXV, 1980, Physics of Defects, R. Baltan et al. eds., North-Holland Publ. Co., 1981.

[5] Aubry, S., The New Concept of Transition of Breaking of Analyticity in a Crystallographic Model, in Solitons in Condensed Matter, A. Bishop and T. Schneider eds., Springer Series in Solid State Sci. 8, Berlin, Springer, 1978, 264-277. | MR 522493 | MR 80d:82034

[6] Aubry, S. and André, G., Analyticity Breaking and Anderson Localization in Incommensurate Lattices, Annals of Israel Physical Society, Vol. 3, Group Theoretical Methods in Physics, L. P. Horowitz and Y. Ne'eman eds., 1980, 133-164. | MR 626837 | MR 83b:82076 | Zbl 0943.82510

[7] Aubry, S., Le Daeron, P. Y. and André, G., Classical Ground-States of a One-dimensional Model for Incommensurate Structures, Preprint. (A revised version of the first part of this preprint was published as Aubry, S. and Le Daeron, P. Y., The Discrete Frenkel-Kontorova Model and its Extensions, Physica, 8D (1983), 381-422. However this does not include the material referred to in §25.). | MR 719634 | Zbl 1237.37059

[8] Axel, F. and Aubry, S., Spatially modulated phases of a one-dimensional lattice model with competing interactions, J. Phys. C. : Solid State Physics, 14 (1981), 5433-5451.

[9] Birkhoff, G. D., On the Periodic Notions of Dynamical Systems, Acta Math., 50 (1928), 359-379. Reprinted in Collected Mathematical Papers, Vol. II, Amer. Math. Soc., New York, 1950, 333-353. | JFM 53.0733.03 | MR 1555257

[10] Birkhoff, G. D., Surface Transformations and their Dynamical Applications, Acta Math., 43 (1922), 1-119. Reprinted in Collected Mathematical Papers, Vol. II, Amer. Math. Soc., New York, 1950, III-229. | JFM 47.0985.03 | MR 1555175

[11] Birkhoff, G. D., Sur quelques courbes fermées remarquables, Bull. Soc. Math. de France, 60 (1932), 1-26. Reprinted in Collected Mathematical Papers, Vol. II, Amer. Math. Soc., New York, 1950, 418-443. | JFM 58.0633.01 | Numdam | MR 1504983 | Zbl 0005.22002

[12] Bourbaki, N., Topologie générale, Chapitres 1 à 4, Paris, Hermann, 1971. | Zbl 0249.54001

[13] Brisac, R., Sur les fonctions multiformes, C. R. Acad. Sci., Paris, 244 (1947), 92-94. | MR 18720 | MR 8,321d | Zbl 0029.32102

[14] Bruhat, F. et Cartan, H., Sur la structure des sous-ensembles analytiques réels, C. R. Acad. Sci., Paris, 244 (1957), 988-990. | MR 86108 | MR 19,125f | Zbl 0081.17201

[15] Cartan, H., Théorie des filtres ; Filtres et ultrafiltres, C. R. Acad. Sci., Paris, 205 (1937), 595-598 et 777-779. | JFM 63.0569.03 | JFM 63.0569.02 | Zbl 0017.24305

[16] Herman, M. R., Contre-exemples de classe C3-Ɛ et à nombre de rotation fixé au théorème des courbes invariantes, handwritten notes from lectures given in Brazil (1979), published in Astérisque, 103-104 (1983).

[17] Katok, A., Some remarks on Birkhoff and Mather twist map theorems, to appear in Ergodic theory and dynamical systems. | MR 693974 | Zbl 0521.58048

[18] Kelley, J. L., General Topology, New York, Van Nostrand, 1955. | MR 70144 | MR 16,1136c | Zbl 0066.16604

[19] Lojasiewicz, S., Sur les ensembles semi-analytiques, International Congress of Mathematicians, Vol. II, Nice (1970), Paris, Gauthier-Villars, 1971, 237-241. | MR 425152 | MR 54 #13109 | Zbl 0241.32005

[20] Lojasiewicz, S., Ensembles semi-analytiques, Preprint, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France, 1964.

[21] Mather, J. N., Existence of Quasi-Periodic Orbits for Twist Homeomorphisms of the Annulus, Topology, 21 (1982), 457-467. | MR 670747 | MR 84g:58084 | Zbl 0506.58032

[22] Mather, J. N., Concavity of the Lagrangian for Quasi-Periodic Orbits, Comment. Math. Helv., 57 (1982), 356-376. | MR 689069 | MR 84m:58043 | Zbl 0508.58037

[23] Mather, J. N., Glancing Billiards, Ergodic theory and dynamical systems, 2 (1982), 397-403. | MR 721731 | MR 85c:58065 | Zbl 0525.58021

[24] Mather, J. N., Non-existence of invariant circles, Ergodic theory and dynamical systems, 4 (1984), 301-309. | MR 766107 | MR 86i:58112 | Zbl 0557.58019

[25] Michael, E., Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 151-182. | MR 42109 | MR 13,54f | Zbl 0043.37902

[26] Newman, R. P. A. C. and Percival, I. C., Definite Paths and Upper Bounds on Regular Regions of Velocity Phase Space, Physica, D6 (1982-1983), 249-259. | MR 698194 | MR 84i:58048 | Zbl 1194.37005

[27] Percival, I. C., Variational Principles for Invariant Tori and Cantori, in Symposium on Nonlinear Dynamics and Beam-Beam Interactions, Amer. Inst. of Physics, Conf. Proc., No. 57, M. Month and J. C. Herrara eds., 1980, 302-310. | MR 624989 | MR 82m:58040

[28] Percival, I. C., A rational principal for invariant tori of fixed frequency, J. Phys. A., 12 (1979), 157-160. | MR 524167 | MR 80c:70021 | Zbl 0394.70018

[29] Sinai, Ya. G., Introduction to Ergodic Theory, Math. Notes, Princeton Univ. Press, 1977 (translated from Russian by V. Scheffer). | MR 584788 | Zbl 0375.28011

[30] Titchmarsh, E. C., The Theory of Functions, Clarendon Press, Oxford, 1932. | JFM 58.0297.01 | Zbl 0005.21004