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Rotation sets and Morse decompositions in twist maps

Published online by Cambridge University Press:  10 December 2009

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Abstract

Positive tilt maps of the annulus are studied, and a correspondence is developed between the rotation set of the map and certain of its Morse decompositions. The main tool used is a characterization of fixed point free lifts of positive tilt maps. As an application, some alternative hypotheses under which the conclusions of the Aubry-Mather theorem hold are given, and it is also shown that the rotation band of a chain transitive set is always in the rotation set of the map.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1988

References

Arnol'd, V. I.. Small denominators I; On the mappings of the circumference onto itself. Transl. AMS 46, Series 2 (1965), 213284.Google Scholar
Aronson, D., Chory, M., Hall, G. R. & McGehee, R.. Bifurcation from an invariant circle for two parameter families of maps of the plane: a computer assisted study. Comm. Math. Phys. 83 (1982), 303354.10.1007/BF01213607CrossRefGoogle Scholar
Aubry, S. & Le Daeron, P. Y.. The discrete Frenkel-Kontorova model and its extensions I. Physica D 8 (1983), 381422.Google Scholar
Barkmeijer, J.. The speed interval of maps of the circle. Preprint.Google Scholar
Bernstein, D.. Birkhoff periodic orbits for twist maps with the circle intersection property. Ergod. Th. & Dynam. Sys. 5 (1985), 531537.10.1017/S014338570000314XCrossRefGoogle Scholar
Birkhoff, G. D.. Surface transformations and their dynamical applications. Acta Math. 43 (1920), 4474.Google Scholar
Reprinted in Collected Mathematical Papers of G. D. Birkhoff, Vol. II. Dover, New York (1968), 195202.Google Scholar
Birkhoff, G. D.. Dynamical Systems. AMS Colloquium Publications IX. AMS, Providence, RI (1927).Google Scholar
Birkhoff, G. D.. Sur quelques courbes fermées remarquables. Bull. SMF 60 (1932).Google Scholar
Reprinted in Collected Mathematical Papers of G. D. Birkhoff, Vol. II. Dover, New York (1968), 418443.Google Scholar
Boyland, P.. Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals. Comm. Math. Phys. 106 (1986), 353381.10.1007/BF01207252CrossRefGoogle Scholar
Boyland, P.. Braid types and a topological method of proving positive entropy. Preprint.Google Scholar
Boyland, P.. Invariant circles and rotation bands in twist maps. Comm. Math. Phys. 113 (1987), 6777.10.1007/BF01221397CrossRefGoogle Scholar
Boyland, P. & Hall, G. R.. Invariant circles and the order structure of periodic orbits in monotone twist maps. Topology 26 (1987), 2135.10.1016/0040-9383(87)90017-6CrossRefGoogle Scholar
Casdagli, M.. Periodic orbits for dissipative twist maps. Ergod. Th. & Dynam. Sys. 7 (1987), 165173.10.1017/S0143385700003916CrossRefGoogle Scholar
Charpentier, M.. Sur quelques propriétés des courbes de M. Birkhoff. Bull. SMF 62 (1934), 193224.Google Scholar
Choquet, G.. Lectures on Analysis, Vol. 1. Benjamin Inc., New York (1969).Google Scholar
Conley, C.. Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Mathematics 38. AMS, Providence, RI (1978).10.1090/cbms/038CrossRefGoogle Scholar
Conley, C.. The gradient structure of a flow: I. Ergod. Th. & Dynam. Sys. 8* (1988), s1126.10.1017/S0143385700009305CrossRefGoogle Scholar
Douady, R.. Application du théorème des tores invariants. Thése de 3e cycle. Université Paris VII (1982).Google Scholar
Franks, J.. Recurrence and fixed points of surface homeomorphisms. Ergod. Th. & Dynam. Sys. 8* (1988), 99107.10.1017/S0143385700009366CrossRefGoogle Scholar
Hall, G. R.. A topological version of a theorem of Mather on twist maps. Ergod. Th. & Dynam. Sys. 4 (1984), 585603.10.1017/S0143385700002662CrossRefGoogle Scholar
Hall, G. R.. Birkhoff orbits for non-monotone twist maps. Preprint.Google Scholar
Handel, M.. Zero entropy surface diffeomorphisms. Preprint.Google Scholar
Harrison, J.. C 2-counterexample to the Seifert conjecture. Preprint.CrossRefGoogle Scholar
Herman, M.. Sur les courbes invariantes par les difféomorphismes de l'anneau. Astérisque 103–104 (1983).Google Scholar
Hockett, K. & Holmes, P.. Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets. Ergod. Th. & Dynam. Sys. 6 (1986), 205211.10.1017/S0143385700003412CrossRefGoogle Scholar
Katok, A.. Some remarks on Birkhoff and Mather twist map theorems. Ergod. Th. & Dynam. Sys. 2 (1982), 185192.10.1017/S0143385700001504CrossRefGoogle Scholar
Le Calvez, P.. Existence d'orbites quasi-periodiques dans les attracteurs de Birkhoff. Comm. Math. Phys. 106 (1986), 383394.CrossRefGoogle Scholar
MacKay, R.. Rotation interval from a time series. J. Phys. A: Math. Gen. 20 (1987), 587592.10.1088/0305-4470/20/3/020CrossRefGoogle Scholar
Mather, J.. Non-existence of invariant circles. Ergod. Th. & Dynam. Sys. 4 (1984), 301309.10.1017/S0143385700002455CrossRefGoogle Scholar
Mather, J.. More Denjoy minimal sets for area preserving diffeomorphisms. Comm. Math. Helv. 60 (1985), 508557.10.1007/BF02567431CrossRefGoogle Scholar
Moeckel, R.. Morse decompositions and connection matrices. Ergod. Th. & Dynam. Sys. 8* (1988), 227249.CrossRefGoogle Scholar
Pugh, C.. An improved closing lemma and a general density theorem. Am. J. Math. 89 (1967), 10101021.10.2307/2373414CrossRefGoogle Scholar
Yoccoz, J. C.. Bifurcations de points fixes elliptiques (d'après A. Chenciner). Astérisque 145–146 (1987), 313334.Google Scholar

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