Symplectic invariants of elliptic fixed points

Commentarii Mathematici Helvetici

Volumn 75, Issue 4, 2000, Pages 681-700

  Siburg, Karl Friedrich ^a  

^a RUHR UNIVERSITY BOCHUM   (Germany)

Author keywords

Aubry mather theory; Birkhoff normal form; Elliptic fixed point; Geodesic flow; Length spectrum; Minimal action; Period spectrum; Reeb flow

Indexed keywords

EID: 0040736056     PISSN: 00102571     EISSN: None     Source Type: Journal    
DOI: 10.1007/s000140050145     Document Type: Article

References (28)

1. S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase, Physica 7D (1983), 240-258.
2. V. Bangert, Mather sets for twist maps and geodesics on tori, in: Kirchgraber, Walther (eds.): Dynamics Reported 1, 1-56, J. Wiley & Sons and B.G. Teubner 1988.
3. Y. Colin de Verdière: Spectre du Laplacien et longueurs des géodésiques périodiques I, Comp. Math. 27 (1973), 83-106.
4. Y. Colin de Verdière, Sur les longueurs des trajectoires périodiques d'un billard, in: Dazord, Desolneux (eds.): Géométrie symplectique et de contact, Sem. Sud-Rhod. Géom. (1984), 122-139.
5. J.P. Françoise, V. Guillemin, On the period spectrum of a symplectic mapping, J. Funct. Anal. 100 (1991), 317-358.
6. V. Guillemin, R. Melrose, The Poisson summation formula for manifolds with boundary, Adv. Math. 32 (1979), 204-232.
7. C. Golé, A new proof of the Aubry-Mather's theorem, Math. Z. 210 (1992), 441-448.
8. H. Ito, Convergence of Birkhoff normal forms for integrable systems, Comment. Math. Helv. 64 (1989), 412-461.
9. R. Iturriaga, H. Sánchez-Morgado: A minimax selector for a class of Hamiltonians on cotangent bundles, preprint.
10. A. Katok, Minimal orbits for small perturbations of completely integrable Hamiltonian systems, in: McGehee, Meyer (eds.): Twist Mappings and Their Applications, IMA Volumes in Mathematics and its Applications 44, Springer-Verlag, 1992.
11. A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
12. V. Lazutkin, KAM Theory and Semiclassical Approximations to Eigenfunctions, Springer-Verlag, 1993.
13. J.N. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology 21 (1982), 457-467.
14. J.N. Mather, Differentiability of the minimal average action as a function of the rotation number, Bol. Soc. Bras. Mat., Nova Ser. 21 (1990), 59-70.
15. J.N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), 169-207.
16. J.N. Mather, G. Forni, Action minimizing orbits in Hamiltonian systems, in: Graffi (ed.): Transition to Chaos in Classical and Quantum Mechanics, Springer LNM 1589 (1992), 92-186.
17. J. Moser, Proof of a generalized form of a fixed point theorem due to G.D. Birkhoff, in: do Carmo, Palis (eds.): Geometry and Topology, Springer LNM 597 (1977), 464-494.
18. J. Moser, Stable and random motion in dynamical systems, Princeton University Press, 1977.
19. J. Moser, Monotone twist mappings and the calculus of variations, Ergod. Th. & Dynam. Sys. 6 (1986), 401-413.
20. G. Popov, Length spectrum invariants of Riemannian manifolds, Math. Z. 213 (1993), 311-351.
21. R.T. Rockafellar, R.J.-B. Wets, Variational Analysis, Springer-Verlag, 1998.
22. K.F. Siburg, Action minimizing measures and the geometry of the Hamiltonian diffeomorphism group, Duke Math. J. 92 (1998), 295-319.
23. K.F. Siburg, A dynamical systems approach to Birkhoff's Theorem, Enseign. Math. 44 (1998), 291-303.
24. K.F. Siburg, Aubry-Mather theory and the inverse spectral problem for planar convex domains, Israel J. Math. 113 (1999), 285-304.
25. K.F. Siburg, The minimal action in geometry and dynamics, preprint.
26. C.L. Siegel, J. Moser, Lectures on Celestial Mechanics, Springer-Verlag, 1971.
27. S. Zelditch, Normal forms and inverse spectral theory, Journées Équations aux dérivées partielles, 2-5 June 1998, GDR 1151 (CNRS).
28. S. Zelditch: The inverse spectral problem for surfaces of revolution, J. Diff. Geom. 49 (1998), 207-264.