Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-17T18:20:46.308Z Has data issue: false hasContentIssue false
  • English
  • Français

On reversible maps and symmetric periodic points

Published online by Cambridge University Press:  08 November 2016

JUNGSOO KANG
JUNGSOO KANG*
Affiliation:
Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Münster, Germany email jkang_01@uni-muenster.de
Get access Rights & Permissions [Opens in a new window]

Abstract

In reversible dynamical systems, it is of great importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks’ theorem on a dichotomy of the number of periodic points of area-preserving maps on the annulus to symmetric periodic points of area-preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area-preserving reversible maps isotopic to the identity, which can be applied to dynamical systems with double symmetries. Our approach is simple, elementary, and far from Franks’ proof. We also show that a reversible map has a symmetric fixed point if and only if it is a twist map which generalizes a boundary twist condition on the closed annulus in the sense of Poincaré–Birkhoff. Applications to symmetric periodic orbits in reversible dynamical systems with two degrees of freedom are briefly discussed.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 4 , June 2018 , pp. 1479 - 1498
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albers, P., Fish, J., Frauenfelder, U., Hofer, H. and van Koert, O.. Global surfaces of section in the planar restricted 3-body problem. Arch. Ration. Mech. Anal. 204 (2012), 273–284.Google Scholar
Bangert, V.. On the existence of closed geodesics on two-spheres. Internat. J. Math. 4(1) (1993), 1–10.CrossRefGoogle Scholar
Bramham, B. and Hofer, H.. First Steps Towards a Symplectic Dynamics (Surveys in Differential Geometry, 17) . International Press, Boston, MA, 2012, pp. 127–178.Google Scholar
Bing, R. H.. The Cartesian product of a certain manifold and a line is E 4 . Ann. of Math. (2) 70(2) (1959), 399–412.Google Scholar
Birkhoff, G. D.. Proof of Poincaré’s last geometric theorem. Trans. Amer. Math. Soc. 14 (1913), 14–22.Google Scholar
Birkhoff, G. D.. The restricted problem of three bodies. Rend. Circ. Mat. Palermo 39 (1915), 265–334.CrossRefGoogle Scholar
Birkhoff, G. D.. Dynamical systems with two degrees of freedom. Trans. Amer. Math. Soc. 18 (1917), 199–300.Google Scholar
Birkhoff, G. D.. An extension of Poincaré’s last geometric theorem. Acta Math. 47 (1925), 297–311.CrossRefGoogle Scholar
Birkhoff, G. D.. Dynamical Systems (Colloquium Publications, 9) . American Mathematical Society, Providence, RI, 1927.Google Scholar
Brown, M. and Neumann, W. D.. Proof of the Poincaré–Birkhoff fixed point theorem. Michigan Math. J. 24 (1977), 21–31.CrossRefGoogle Scholar
Braun, M.. Particle motions in a magnetic field. J. Differential Equations 8 (1970), 294–332.CrossRefGoogle Scholar
Brouwer, L. E. J.. Beweiss des ebenen Translationssatzes. Math. Ann. 72 (1912), 37–54.Google Scholar
Brouwer, L. E. J.. Über die periodischen Transformationen der Kugel. Math. Ann. 80 (1919), 39–41.Google Scholar
Brown, M.. Fixed points for orientation preserving homeomorphisms of the plane which interchange two points. Pacific J. Math. 143(1) (1990), 37–41.CrossRefGoogle Scholar
Carter, P. H.. An improvement of the Poincaré–Birkhoff fixed point theorem. Trans. Amer. Math. Soc. 269(1) (1982), 285–299.Google Scholar
Constantin, A. and Kolev, B.. The theorem of Kérékjártó on periodic homeomorphisms of the disk and the sphere. Enseign. Math. 40 (1994), 193–204.Google Scholar
Collier, B., Kerman, E., Reiniger, B., Turmunkh, B. and Zimmer, A.. A symplectic proof of a theorem of Franks. Compos. Math. 148(6) (2012), 1969–1984.Google Scholar
Conley, C. C.. On some new long periodic solutions of the plane restricted three body problem. Comm. Pure Appl. Math. 16 (1963), 449–467.CrossRefGoogle Scholar
De Vogelaere, R.. Surface de section dans le probléme de Störmer. Acad. Roy. Belg. Bull. Cl. Sci. (5) 40 (1954), 705–714.Google Scholar
Devaney, R. L.. Reversible diffeomorphisms and flows. Trans. Amer. Math. Soc. 218 (1976), 89–113.CrossRefGoogle Scholar
Ding, W.-Y.. A generalization of the Poincaré–Birkhoff fixed point theorem. Proc. Amer. Math. Soc. 88(2) (1983), 341–346.Google Scholar
Franks, J. and Handel, M.. Periodic points of Hamiltonian surface diffeomorphisms. Geom. Topol. 7 (2003), 713–756.Google Scholar
Frauenfelder, U. and Kang, J.. Real holomorphic curves and invariant global surfaces of section. Proc. Lond. Math. Soc. 112 (2016), 477–511.Google Scholar
Franks, J.. Generalizations of the Poincaré–Birkhoff theorem. Ann. of Math. (2) 128(1) (1988), 139–151.Google Scholar
Franks, J.. Geodesics on S 2 and periodic points of annulus homeomorphisms. Invent. Math. 108(1) (1992), 403–418.Google Scholar
Franks, J.. Area preserving homeomorphisms of open surfaces of genus zero. New York J. Math. 2 (1996), 1–19.Google Scholar
Guillou, L.. A simple proof of P. Carter’s theorem. Proc. Amer. Math. Soc. 125(5) (1997), 1555–1559.Google Scholar
Hingston, N.. On the growth of the number of closed geodesics on the two-sphere. Int. Math. Res. Not. (9) (1993), 253–262.Google Scholar
Hryniewicz, U., Momin, A. and Salamão, P. S.. A Poincaré–Birkhoff theorem for tight Reeb flows on S 3 . Invent. Math. 199(2) (2015), 333–422.CrossRefGoogle Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. The dynamics on a strictly convex energy surface in ℝ4 . Ann. of Math. (2) 148 (1998), 197–289.Google Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. Finite energy foliations of tight three-spheres and Hamiltonian dynamics. Ann. of Math. (2) 157 (2003), 125–257.Google Scholar
Kang, J.. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete Contin. Dyn. Syst. 34(12) (2014), 5229–5245.Google Scholar
Kerman, E.. On primes and period growth for Hamiltonian diffeomorphisms. J. Mod. Dyn. 6(1) (2012), 41–58.Google Scholar
von Kérékjartó, B.. Uber die periodischen Transformationen der Kreisscheibe und der Kugelfläche. Math. Ann. 80 (1919–1920), 36–38.CrossRefGoogle Scholar
Ku, H.-T. and Ku, M.-C.. The Lefschetz fixed point theorem for involutions. Proceedings of the Conference on Transformation Groups. Springer, New Orleans, 1967, pp. 341–342.Google Scholar
Kirillov, A. and Starkov, V.. Some extensions of the Poincaré–Birkhoff theorem. J. Fixed Point Theory Appl. 13 (2013), 611–625.CrossRefGoogle Scholar
Le Calvez, P. and Wang, J.. Some remarks on the Poincaré–Birkhoff theorem. Proc. Amer. Math. Soc. 138(2) (2010), 703–715.Google Scholar
Le Calvez, P.. Periodic orbits of Hamiltonian homeomorphisms of surfaces. Duke Math. J. 133(1) (2006), 125–184.Google Scholar
McGehee, R. P.. Some homoclinic orbits for the restricted three-body problem. PhD Thesis, The University of Wisconsin-Madison, 1969.Google Scholar
McDuff, D. and Salamon, D.. Introduction to Symplectic Topology. Oxford University Press, Oxford, 1998.Google Scholar
Neumann, W.. Generalizations of the Poincaré–Birkhoff fixed point theorem. Bull. Aust. Math. Soc. 17(3) (1977), 375–389.Google Scholar
Poincaré, H.. Les méthodes nouvelles de la méchanique céleste. Gauthiers-Villars, Paris, 1899.Google Scholar
Pelayo, Á. and Rezakhanlou, F.. Poincaré–Birkhoff theorems in random dynamics. Preprint, 2013, arXiv:1306.0821.Google Scholar
Störmer, C.. Sur les trajectoires des corpuscules électrisés dans l’espace sous l’action du magnétisme terrestre avec application aux aurores boréales. Arch. Sei. Phys. Nat. 24 (1907), 113–158.Google Scholar
Yong, L. and Zhenghua, L.. A constructive proof of the Poincaré–Birkhoff theorem. Trans. Amer. Math. Soc. 347(6) (1982), 2111–2126.CrossRefGoogle Scholar