The tiered Aubry set for autonomous Lagrangian functions

Marie-Claude Arnaud[1]

  • [1] Université d’Avignon et des Pays de Vaucluse Laboratoire d’Analyse non linéaire et Géométrie (EA 2151) 84 018 Avignon (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 5, page 1733-1759
  • ISSN: 0373-0956

Abstract

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Let L : T M → ℝ be a Tonelli Lagrangian function (with M compact and connected and dim M ≥ 2 ). The tiered Aubry set (resp. Mañé set) 𝒜 T ( L ) (resp. 𝒩 T ( L ) ) is the union of the Aubry sets (resp. Mañé sets) 𝒜 ( L + λ ) (resp. 𝒩 ( L + λ ) ) for λ closed 1-form. Then1.the set 𝒩 T ( L ) is closed, connected and if dim H 1 ( M ) ≥ 2 , its intersection with any energy level is connected and chain transitive;2.for L generic in the Mañé sense, the sets 𝒜 T ( L ) ¯ and 𝒩 T ( L ) ¯ have no interior;3.if the interior of 𝒜 T ( L ) ¯ is non empty, it contains a dense subset of periodic points.We then give an example of an explicit Tonelli Lagrangian function satisfying 2 and an example proving that when M = 𝕋 2 , the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different.

How to cite

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Arnaud, Marie-Claude. "The tiered Aubry set for autonomous Lagrangian functions." Annales de l’institut Fourier 58.5 (2008): 1733-1759. <http://eudml.org/doc/10361>.

@article{Arnaud2008,
abstract = {Let $L\colon TM \rightarrow \{\mathbb\{R\}\}$ be a Tonelli Lagrangian function (with $M$ compact and connected and $\dim M\ge 2$). The tiered Aubry set (resp. Mañé set) $\mathcal\{A\}^T(L)$ (resp. $\mathcal\{N\}^T(L)$) is the union of the Aubry sets (resp. Mañé sets) $\mathcal\{A\}(L+\lambda )$ (resp. $\mathcal\{N\}(L+\lambda )$) for $\lambda $ closed 1-form. Then1.the set is closed, connected and if , its intersection with any energy level is connected and chain transitive;2.for generic in the Mañé sense, the sets and have no interior;3.if the interior of is non empty, it contains a dense subset of periodic points.We then give an example of an explicit Tonelli Lagrangian function satisfying 2 and an example proving that when $M=\mathbb\{T\}^2$, the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different.},
affiliation = {Université d’Avignon et des Pays de Vaucluse Laboratoire d’Analyse non linéaire et Géométrie (EA 2151) 84 018 Avignon (France)},
author = {Arnaud, Marie-Claude},
journal = {Annales de l’institut Fourier},
keywords = {Lagrangian dynamics; Hamiltonian dynamics; Aubry-Mather theory; Mañé set; Tonelli Lagrangian function; Aubry set},
language = {eng},
number = {5},
pages = {1733-1759},
publisher = {Association des Annales de l’institut Fourier},
title = {The tiered Aubry set for autonomous Lagrangian functions},
url = {http://eudml.org/doc/10361},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Arnaud, Marie-Claude
TI - The tiered Aubry set for autonomous Lagrangian functions
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 5
SP - 1733
EP - 1759
AB - Let $L\colon TM \rightarrow {\mathbb{R}}$ be a Tonelli Lagrangian function (with $M$ compact and connected and $\dim M\ge 2$). The tiered Aubry set (resp. Mañé set) $\mathcal{A}^T(L)$ (resp. $\mathcal{N}^T(L)$) is the union of the Aubry sets (resp. Mañé sets) $\mathcal{A}(L+\lambda )$ (resp. $\mathcal{N}(L+\lambda )$) for $\lambda $ closed 1-form. Then1.the set is closed, connected and if , its intersection with any energy level is connected and chain transitive;2.for generic in the Mañé sense, the sets and have no interior;3.if the interior of is non empty, it contains a dense subset of periodic points.We then give an example of an explicit Tonelli Lagrangian function satisfying 2 and an example proving that when $M=\mathbb{T}^2$, the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different.
LA - eng
KW - Lagrangian dynamics; Hamiltonian dynamics; Aubry-Mather theory; Mañé set; Tonelli Lagrangian function; Aubry set
UR - http://eudml.org/doc/10361
ER -

References

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