Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T20:12:20.028Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal geodesics†

Published online by Cambridge University Press:  19 September 2008

Victor Bangert
Victor Bangert
Affiliation:
Mathematisches Institut der Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Motivated by the close relation between Aubry-Mather theory and minimal geodesies on a 2-torus we study the existence and properties of minimal geodesics in compact Riemannian manifolds of dimension ≥3. We prove that there exist minimal geodesics with certain rotation vectors and that there are restrictions on the rotation vectors of arbitrary minimal geodesics. A detailed analysis of the minimal geodesics of the ‘Hedlund examples’ shows that – to a certain extent – our results are optimal.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 2 , June 1990 , pp. 263 - 286
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[1]Aubry, S. & Le Daeron, P. Y.. The discrete Frenkel-Kontorova model and its extensions I: Exact results for the ground states. Physica 8D (1983), 381–422.Google Scholar
[2]Bangert, V.. Mather sets for twist maps and geodesics on tori. Dynamics Reported, Vol. 1 (eds. Kirchgraber, U. & Walther, H. O.) pp. 1–56. John Wiley and B. G. Teubner: Chichester-Stuttgart, 1988.CrossRefGoogle Scholar
[3]Bernstein, D. & Katok, A.. Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians. Invent. Math. 88 (1987), 225–241.CrossRefGoogle Scholar
[4]Byalyi, M. L. & Polterovich, L. V.. Geodesic flows on the two-dimensional torus and phase transitions ‘commensurability - noncommensurability’. Functional Anal. Appl. 20 (1986), 260–266.CrossRefGoogle Scholar
(Originally published in Funktsional. Anal. i Prilozhen. 20 (1986), 9–16.)Google Scholar
[5]Denzler, J.. Mather sets for plane Hamiltonian systems. J. Appl. Math. Phys. (ZAMP) 38 (1987), 791–812.Google Scholar
[6]Federer, H.. Real flat chains, cochains and variational problems. Indiana Univ. Math. J. 24 (1974), 351–407.CrossRefGoogle Scholar
[7]Gromov, M.. Structures métriques pour les Variétés Riemanniennes. (eds. Lafontaine, J. & Pansu, P.) CEDIC: Paris, 1981.Google Scholar
[8]Gromov, M.. Filling Riemannian manifolds. J. Differential Geom. 18 (1983), 1–147.CrossRefGoogle Scholar
[9]Hedlund, G. A.. Geodesies on a two-dimensional Riemannian manifold with periodic coefficients. Ann. of Math. 33 (1932), 719–739.CrossRefGoogle Scholar
[10]Herman, M. R.. Existence et non existence de tores invariants par des diffeomorphismes symplectiques. Preprint. Ecole Polytechnique: Palaiseau, 1988.Google Scholar
[11]Katok, A.. Minimal orbits for small perturbations of completely integrable Hamiltonian systems. Preprint. California Institute of Technology: Pasadena, 1989.Google Scholar
[12]Klingenberg, W.. Geodätischer Fluss auf Mannigfaltig-keiten vom hyperbolischen Typ. Invent. Math. 14 (1971), 63–82.CrossRefGoogle Scholar
[13]Leichtweiβ, K.. Konvexe Mengen. Springer: Berlin-Heidelberg-New York, 1980.Google Scholar
[14]Mather, J. N.. A criterion for the non-existence of invariant circles. Publ. Math. IHES 63 (1986), 153–204.Google Scholar
[15]Mather, J. N.. Destruction of invariant circles. Erg. Th. & Dynatn. Sys. 8* (1988), 199–214.Google Scholar
[16]Mather, J. N.. Minimal Measures. Comment Math. Helv. 64 (1989), 375–394.CrossRefGoogle Scholar
[17]Morse, M.. A fundamental class of geodesies on any closed surface of genus greater than one. Trans. Amer. Math. Soc. 26 (1924), 25–60.CrossRefGoogle Scholar
[18]Moser, J.. Monotone twist mappings and the calculus of variations. Erg. Th. & Dynam. Sys. 6 (1986), 325–333.Google Scholar
[19]Moser, J.. Recent developments in the theory of Hamiltonian systems. SIAM Review 28 (1986), 459–485.Google Scholar
[20]Rinow, W.. Die innere Geometrie der metrischen Räume. Grundlehren der math. Wiss. 105. Springer: Berlin-Göttingen-Heidelberg, 1961.CrossRefGoogle Scholar