Abstract
Iff is a homeomorphism of the annulus andp/q is a rational in lowest terms that is contained in the rotation set off thenf has a (p, q)-topologically monotone periodic orbit. In addition, iff has ap/q-period orbit that is not topologically monotone then the Farey interval ofp/q is contained in the rotation set off.
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References
[B+H]Boyland, P. andHall, G. R.,Invariant circles and the order structure of periodic orbits in monotone twist maps, Topology,26 (1987), pp. 21–35.
[Bd1]Boyland, P.,Rotation sets and Morse decompositions for twist maps, Ergod. Th. and Dynam. Sys.,8 * (1988), pp. 33–61.
[Bd2]Boyland, P.,An analog of Sharkovski's theorem for twist maps, Proceedings of the Joint AMS-SIAM Summer Research Conference on Hamiltonian Dynamical Systems, Contemporary Math.,81 (1988), pp. 119–133.
[Bd3]Boyland, P.,Braid types and a topological method of proving positive entropy, preprint (1984).
[Bd4]Boyland, P.,The rotation set as a dynamical invariant, Proceedings of the IMA Workshop on Twist Maps, to appear. IMA Volumes in Mathematics and its applications, Vol. 44, Springer-Verlag, 1992.
[Bw]Brouwer, L. E. J.,Über die periodischen Transformationen der Kugel, Math. Ann.,80 (1919), pp. 39–41.
[Br]Brunovsky, P.,On one-parameter families of diffeomorphisms, I and II, Comment. Math. Univ. Caroline,11 (1970), pp. 559–582;12 (1970), pp. 765–784.
[E]Eilenberg, S.,Sur les transformations periodiques de la surface de sphere, Fund. Math.,22 (1934), pp. 28–44.
[FLP]Fathi, A., Lauderbach, F. andPoenaru, V.,Travaux de Thurson sur les surfaces, Asterique (1979), pp. 66–67.
[F1]Franks, J.,Recurrence and fixed points of surface homeomorphisms, Ergod. Th. and Dynam. Sys.,8 *(1988), pp. 99–107.
[F2]Franks, J.,Generalizations of the Poincaré—Birkhoff theorem, Ann. of Math.,128 (1988), pp. 139–151.
[F3]Franks, J. Knots, Links and Symbolic Dynamics, Ann of Math.,113 (1981), pp. 529–552.
[Fr1]Fried, D.,The geometry of cross sections to flows, Topology,24 (1983), pp. 353–371.
[Fr2]Fried, D.,Flow equivalence, hyperbolic systems and a new zeta function for flows, Comm. Math. Helv.,57 (1982), pp. 237–359.
[Fr3]Fried, D.,Growth rate of surface homeomorphisms and flow equivalence, Ergod. Th. and Dynam. Sys.,5 (1985), pp. 539–563.
[Fr4]Fried, D.,Fibrations over S 1 with pseudo-Anosov monodromy, in [FLP].
[GK]Gerber, M. andKatok, A.,Smooth models of Thurston's speudo-Anosov maps, Ann. Scient. Ec. Norm. Sup.,15 (1982), pp. 173–204.
[H1]Hall, G. R.,A topological version of a theorem of Mather on twist maps, Ergod. Th. & Dynam. Sys.,4 (1984), pp. 585–603.
[H2]Hall, G. R.,Some problems on dynamics of annulus maps, Cont. Math.,81 (1988), pp. 135–152.
[TH]Hall, T.,Unremovable periodic orbits of homeomorphisms, Math. Proc. Camb. Philos. Soc.,110 (1991), 523–531.
[Hg]Hagelin, J.,Restructuring physics from its foundation in light of Maharishi's Vedic Science, Mod. Sci. and Vedic Sci.,3,1 (1989), pp. 3–72.
[Hn1] Handel, M.,A Pathological area preserving C ∞ diffeomorphism of the plane, Proc. A.M.S.,86 (1982), pp. 163–168.
[Hn2]Handel, M.,Zero entropy surface homeomorphism (1986), preprint.
[Hn3]Handel, M.,The rotation set of a homeomorphism of the annulus is closed, Commoun. Math. Phys.,127 (1990), pp. 339–349.
[Hn4]Handel, M.,The entropy of orientation reversing homeomorphisms of surfaces, Topology,21 (1982), pp. 291–296.
[HT]Handel, G. andThurston, W.,New proofs of some results of Nielsen, Adv. in Math.,2 (1985), pp. 173–191.
[HW]Hardy, G. & Wright, E., An Introduction to the Theory of Numbers, Oxford University Press, 5th ed. (1979).
[K]Katok, A.,Some remarks on theBirkhoff and Mather twist theorems, Ergod. Th. and Dynam. Sys.,2 (1982), pp. 183–194.
[Kj]Kerekjarto, B.,Sur la structure des topologiques des surfaces en elles-memes, L'Enseignement Math.,35 (1936), pp. 297–316.
[LC]Le Calvez, P.,Existence d'orbites de Birkhoff generalisées pour les difféomorphisme conservatifs de l'anneau (1989), preprint.
[M]Miller, R.,Geodesic laminations from Nielsen's viewpoint, Adv. in Math., 45 (1982), pp. 189–212.
[T]Thurston, W.,On the geometry and dynamics of diffeomorphisms of surfaces, Bull. A.M.S.,19 (1988), pp. 417–431.
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Boyland, P. Rotation sets and monotone periodic orbits for annulus homeomorphisms. Commentarii Mathematici Helvetici 67, 203–213 (1992). https://doi.org/10.1007/BF02566496
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DOI: https://doi.org/10.1007/BF02566496