# The action spectrum near positive definite invariant tori

Bulletin de la Société Mathématique de France (2003)

- Volume: 131, Issue: 4, page 603-616
- ISSN: 0037-9484

## Access Full Article

top## Abstract

top## How to cite

topBernard, Patrick. "The action spectrum near positive definite invariant tori." Bulletin de la Société Mathématique de France 131.4 (2003): 603-616. <http://eudml.org/doc/272320>.

@article{Bernard2003,

abstract = {We show that the Birkhoff normal form near a positive definite KAM torus is given by the function $\alpha $ of Mather. This observation is due to Siburg [Si2], [Si1] in dimension 2. It clarifies the link between the Birkhoff invariants and the action spectrum near the torus. Our extension to high dimension is made possible by a simplification of the proof given in [Si2].},

author = {Bernard, Patrick},

journal = {Bulletin de la Société Mathématique de France},

keywords = {lagrangian systems; Aubry-Mather theory; minimizing orbits; averaged action; invariant torus; normal forms; action spectrum},

language = {eng},

number = {4},

pages = {603-616},

publisher = {Société mathématique de France},

title = {The action spectrum near positive definite invariant tori},

url = {http://eudml.org/doc/272320},

volume = {131},

year = {2003},

}

TY - JOUR

AU - Bernard, Patrick

TI - The action spectrum near positive definite invariant tori

JO - Bulletin de la Société Mathématique de France

PY - 2003

PB - Société mathématique de France

VL - 131

IS - 4

SP - 603

EP - 616

AB - We show that the Birkhoff normal form near a positive definite KAM torus is given by the function $\alpha $ of Mather. This observation is due to Siburg [Si2], [Si1] in dimension 2. It clarifies the link between the Birkhoff invariants and the action spectrum near the torus. Our extension to high dimension is made possible by a simplification of the proof given in [Si2].

LA - eng

KW - lagrangian systems; Aubry-Mather theory; minimizing orbits; averaged action; invariant torus; normal forms; action spectrum

UR - http://eudml.org/doc/272320

ER -

## References

top- [1] P. Bernard – « Homoclinic orbits to invariant sets of quasi-integrable exact maps », Ergod. Th. Dynam. Sys.20 (2000), p. 1583–1601. Zbl0992.37055MR1804946
- [2] D. Mc Duff & D. Salamon – Introduction to Symplectic Topology, Oxford Math. Monographs, 1995. Zbl0844.58029MR1373431
- [3] M. Herman – « Inégalités a priori pour des tores lagrangiens invariants par des difféomorphismes symplectiques », Publ. Math. IHES70 (1989), p. 47–101. Zbl0717.58020
- [4] V. Lazutkin – KAM Theory and Semiclassical Approximations to Eigenfunctions, Springer, 1993. Zbl0814.58001MR1239173
- [5] J. Mather – « Action minimizing invariant measures for positive definite Lagrangian systems », Math. Z.207 (1991), p. 169–207. Zbl0696.58027MR1109661
- [6] K. Siburg – « Aubry-Mather theory and the inverse spectral problem for planar convex domains », Israel J. Math.113 (1999), p. 285–304. Zbl0996.37051MR1729451
- [7] —, « Symplectic invariants of elliptic fixed points », Comment. Math. Helv.75 (2000), p. 681–700. Zbl0985.37054MR1789182

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.