A criterion for the non-existence of invariant circles

Mather, John N.

Publications Mathématiques de l'IHÉS, Tome 63 (1986), pp. 153-204.

[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

[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

[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 | Zbl

[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 | Zbl

[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 | MR

[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 | MR

[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. | EuDML | JFM | Numdam | MR | Zbl

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

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

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

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

[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 | Zbl

[18] Kelley, J. L., General Topology, New York, Van Nostrand, 1955. | MR | Zbl

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

[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 | Zbl

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

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

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

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

[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 | Zbl

[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

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

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

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