Aubry-Mather theory and the inverse spectral problem for planar convex domains

Israel Journal of Mathematics

Volumn 113, Issue , 1999, Pages 285-304

  Siburg, Karl Friedrich ^a  

^a RUHR UNIVERSITY BOCHUM   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033478968     PISSN: 00212172     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02780181     Document Type: Article

References (24)

1. E. Amiran, Caustics and evolutes for convex planar domains, Journal of Differential Geometry 28 (1988), 345-357.
2. E. Amiran, A dynamical approach to symplectic and spectral invariants for billiards, Communications in Mathematical Physics 154 (1993), 99-110.
3. E. Amiran, Integrable smooth planar billiards and evolutes, New York Journal of Mathematics 3 (1997), 32-47.
4. K. Anderson and R. Melrose, The propagation of singularities along gliding rays, Inventiones Mathematicae 41 (1977), 23-95.
5. V. Bangert, Mather sets for twist maps and geodesics on tori, in Dynamics Reported (U. Kirchgraber and H. O. Walther, eds.), Vol. 1, Wiley & Sons and B.G. Teubner, 1988.
6. V. Bangert, Geodesic rays, Busemann functions and monotone twist maps, Calculus of Variations 2 (1994), 49-63.
7. M. Bialy, Convex billiards and a theorem by E. Hopf, Mathematische Zeitschrift 214 (1993), 147-154.
8. C. Golé, A new proof of the Aubry-Mather's theorem, Mathematische Zeitschrift 210 (1992), 441-448.
9. V. Guillemin and R. Melrose, The Poisson summation formula for manifolds with boundary, Advances in Mathematics 32 (1979), 204-232.
10. E. Gutkin and A. Katok, Caustics for inner and outer billiards, Communications in Mathematical Physics 173 (1995), 101-133.
11. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
12. V. Kovachev and G. Popov, Invariant tori for the billiard ball map, Transactions of the American Mathematical Society 317 (1990), 45-81.
13. V. Lazutkin, The existence of caustics for a billiard problem in a convex domain, Mathematics of the USSR-Izvestiya 7 (1973), 185-214.
14. S. Marvizi and R. Melrose, Spectral invariants of convex planar regions, Journal of Differential Geometry 17 (1982), 475-502.
15. J. N. Mather, Minimal measures, Commentarii Mathematici Helvetici 64 (1989), 375-394.
16. J. N. Mather, Differentiability of the minimal average action as a function of the rotation number, Boletim da Sociedade Brasileira de Matemática, Nova Série 21 (1990), 59-70.
17. J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Mathematische Zeitschrift 207 (1991), 169-207.
18. J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, in Transition to Chaos in Classical and Quantum Mechanics (S. Graffi, ed.), Lecture Notes in Mathematics 1589, Springer-Verlag, Berlin, 1992.
19. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, Berlin, 1989.
20. J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse (1962), 1-20.
21. G. Popov, Invariants of the length spectrum and spectral invariants of planar convex domains, Communications in Mathematical Physics 161 (1994), 335-364.
22. K. F. Siburg, Action minimizing measures and the geometry of the Hamiltonian diffeomorphism group, Duke Mathematical Journal 92 (1998), 295-319.
23. K. F. Siburg, A dynamical systems approach to Birkhoff's Theorem, L'Enseignement Mathématique 44 (1998), 291-303.
24. S. Tabachnikov, Billiards, Panoramas et Synthèses 1, SMF, 1995.