Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Consequences of contractible geodesics
on surfaces

Authors: J. Denvir and R. S. Mackay
Journal: Trans. Amer. Math. Soc. 350 (1998), 4553-4568
MSC (1991): Primary 58F17
MathSciNet review: 1615951
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The geodesic flow of any Riemannian metric on a geodesically convex surface of negative Euler characteristic is shown to be semi-equivalent to that of any hyperbolic metric on a homeomorphic surface for which the boundary (if any) is geodesic. This has interesting corollaries. For example, it implies chaotic dynamics for geodesic flows on a torus with a simple contractible closed geodesic, and for geodesic flows on a sphere with three simple closed geodesics bounding disjoint discs.

References [Enhancements On Off] (What's this?)

  • [Ar] V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
    V. I. Arnol′d, Mathematical methods of classical mechanics, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR 0690288
  • [AM] Ashcroft, N.W. and Mermin, N.D., Solid State Physics, W. B. Saunders, (1976).
  • [Ba] V. Bangert, Mather sets for twist maps and geodesics on tori, Dynamics reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl., vol. 1, Wiley, Chichester, 1988, pp. 1–56. MR 945963
  • [Be] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777
  • [Bi] Birkhoff, G.D., Dynamical Systems with Two Degrees of Freedom, Trans. AMS, 18, (1917), 219.
  • [BS] Joan S. Birman and Caroline Series, Dehn’s algorithm revisited, with applications to simple curves on surfaces, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 451–478. MR 895628
  • [Bol1] Sergei V. Bolotin, Homoclinic orbits of geodesic flows on surfaces, Russian J. Math. Phys. 1 (1993), no. 3, 275–288. MR 1259485
  • [Bol2] Sergey Bolotin, Variational criteria for nonintegrability and chaos in Hamiltonian systems, Hamiltonian mechanics (Toruń, 1993) NATO Adv. Sci. Inst. Ser. B Phys., vol. 331, Plenum, New York, 1994, pp. 173–179. MR 1316675
  • [Boy] Philip Boyland, Topological methods in surface dynamics, Topology Appl. 58 (1994), no. 3, 223–298. MR 1288300,
  • [Bu] B. Buffoni, Periodic and homoclinic orbits for Lorentz-Lagrangian systems via variational methods, Nonlinear Anal. 26 (1996), no. 3, 443–462. MR 1359225,
  • [C] C. Carathéodory, Calculus of variations and partial differential equations of the first order. Part II: Calculus of variations, Translated from the German by Robert B. Dean, Julius J. Brandstatter, translating editor, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1967. MR 0232264
  • [CDP] M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994
  • [Gr] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829,
  • [GH] É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648
  • [Ha] Michael Handel, Global shadowing of pseudo-Anosov homeomorphisms, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 373–377. MR 805836,
  • [He] Hedlund, G.A., Geodesics on a Two-Dimensional Riemannian Manifold with Periodic Coefficients, Ann. Math., 33 (1932) 719-739.
  • [K] Wilhelm Klingenberg, Lectures on closed geodesics, Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Vol. 230. MR 0478069
  • [LY] Li, T-Y., and Yorke, J.A., Period three implies chaos, Amer. Math. Monthly, 82 (1975) 985-992.
  • [M1] Morse, H.M., A Fundamental Class of Geodesics on any Closed Surface of Genus Greater than One, Trans. AMS, 26, (1924), 25-60.
  • [M2] Morse, H.M., Instability and Transitivity, Journ. de Math, XIV, (1935), 49-71.
  • [Pe] L. A. Bunimovich, I. P. Cornfeld, R. L. Dobrushin, M. V. Jakobson, N. B. Maslova, Ya. B. Pesin, Ya. G. Sinaĭ, Yu. M. Sukhov, and A. M. Vershik, Dynamical systems. II, Encyclopaedia of Mathematical Sciences, vol. 2, Springer-Verlag, Berlin, 1989. Ergodic theory with applications to dynamical systems and statistical mechanics; Edited and with a preface by Sinaĭ; Translated from the Russian. MR 1024068
  • [Po] L. V. Polterovich, Geodesics on a two-dimensional torus with two rotation numbers, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 4, 774–787, 895 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 1, 101–114. MR 966984
  • [S1] Caroline Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 601–625. MR 873435,
  • [S2] C. Series, Some geometrical models of chaotic dynamics, Proc. Roy. Soc. London Ser. A 413 (1987), no. 1844, 171–182. MR 909276
  • [Sh] Sharkovskii, AN, Co-existence of the cycles of a continuous mapping of the line into itself, Ukrain Math Zh 16:1 (1964) 61-71 (Russian).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F17

Retrieve articles in all journals with MSC (1991): 58F17

Additional Information

J. Denvir
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
Address at time of publication: Department of Mathematics, Truman State University, Kirksville, Missouri 63501

R. S. Mackay
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
Address at time of publication: DAMTP, Silver Street, University of Cambridge, CB3 9EW, U.K.

Keywords: Geodesics, semiconjugacy, surface, dynamics
Received by editor(s): September 7, 1995
Received by editor(s) in revised form: September 13, 1996
Article copyright: © Copyright 1998 American Mathematical Society