On donne une généralisation à la dimension supérieure des résultats obtenus par Birkhoff et Mather sur l'existence d'orbites errant dans les zones d'instabilité des applications de l'anneau déviant la verticale. Notre généralisation s'inspire fortement de celle proposée par Mather. Elle présente cependant l'avantage de contenir effectivement l'essentiel des résultats de Birkhoff et Mather sur les difféomorphismes de l'anneau.
We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps.
@article{AIF_2002__52_5_1533_0,
author = {Bernard, Patrick},
title = {Connecting orbits of time dependent Lagrangian systems},
journal = {Annales de l'Institut Fourier},
volume = {52},
year = {2002},
pages = {1533-1568},
doi = {10.5802/aif.1924},
mrnumber = {1935556},
zbl = {1008.37035},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2002__52_5_1533_0}
}
Bernard, Patrick. Connecting orbits of time dependent Lagrangian systems. Annales de l'Institut Fourier, Tome 52 (2002) pp. 1533-1568. doi : 10.5802/aif.1924. http://gdmltest.u-ga.fr/item/AIF_2002__52_5_1533_0/
[1] One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rat. Mach. Anal, Tome 90 (1985), pp. 325-388 | Article | MR 801585 | Zbl 0585.49002
[2] Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris, Série I, Tome 327 (1998), pp. 267-270 | MR 1650261 | Zbl 1052.37514
[3] Book (In preparation)
[4] Failure of convergence of the Lax-Oleinik semi-group in the time periodic case, Bull. Soc. Math. France, Tome 128 (2000), pp. 473-483 | Numdam | MR 1792479 | Zbl 0989.37035
[5] Lagrangian flows: The dynamics of globally minimizing orbits ; Lagrangian flows: the dynamics of globally minimizing orbits. II, Bol. Soc. Bras. Mat, Tome 28 (1997), p. 141-153 ; 155-196 | MR 1479499 | Zbl 0892.58065
[6] Aubry set and Mather's action functional, Preprint (2001)
[7] Destruction of invariant circles, Erg. The. and Dyn. Syst, Tome 8 (1988), pp. 199-214 | Article | MR 967638 | Zbl 0688.58024
[8] Differentiability of the minimial average action as a function of the rotation number, Bol. Soc. Bras. Math, Tome 21 (1990), pp. 59-70 | Article | MR 1139556 | Zbl 0766.58033
[9] Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc, Tome 4 (1991), pp. 207-263 | Article | MR 1080112 | Zbl 0737.58029
[10] Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z, Tome 207 (1991), pp. 169-207 | Article | MR 1109661 | Zbl 0696.58027
[11] Variational construction of connecting orbits, Ann. Inst. Fourier (1993) | Numdam | MR 1275203 | Zbl 0803.58019
[12] Action minimizing orbits in Hamiltonian systems, Transition to chaos in classical and quantum mechanics, Springer (Lect. Notes in Math.) Tome 1589 (1994) | Zbl 0822.70011
[13] Monotone twist Mappings and the Calculs of Variations, Ergodic Theory and Dyn. Syst, Tome 6 (1986), pp. 401-413 | MR 863203 | Zbl 0619.49020
[14] Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations, J. Math. Pures Appl, Tome 80 (2001), pp. 85-104 | Article | MR 1810510 | Zbl 0979.35033