The action spectrum near positive definite invariant tori

Patrick Bernard

Bulletin de la Société Mathématique de France (2003)

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

Abstract

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We show that the Birkhoff normal form near a positive definite KAM torus is given by the function α 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].

How to cite

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Bernard, 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

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  1. [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. [2] D. Mc Duff & D. Salamon – Introduction to Symplectic Topology, Oxford Math. Monographs, 1995. Zbl0844.58029MR1373431
  3. [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. [4] V. Lazutkin – KAM Theory and Semiclassical Approximations to Eigenfunctions, Springer, 1993. Zbl0814.58001MR1239173
  5. [5] J. Mather – « Action minimizing invariant measures for positive definite Lagrangian systems », Math. Z.207 (1991), p. 169–207. Zbl0696.58027MR1109661
  6. [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. [7] —, « Symplectic invariants of elliptic fixed points », Comment. Math. Helv.75 (2000), p. 681–700. Zbl0985.37054MR1789182

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