On some completions of the space of hamiltonian maps

Vincent Humilière

Bulletin de la Société Mathématique de France (2008)

  • Volume: 136, Issue: 3, page 373-404
  • ISSN: 0037-9484

Abstract

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In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of ℝ 2 n endowed with the standard symplectic form ω 0 = d p ∧ d q . We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example, discontinuous Hamiltonians. We also prove that some dynamical aspects of Hamiltonian systems are preserved in the completions.

How to cite

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Humilière, Vincent. "On some completions of the space of hamiltonian maps." Bulletin de la Société Mathématique de France 136.3 (2008): 373-404. <http://eudml.org/doc/272505>.

@article{Humilière2008,
abstract = {In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of $\mathbb \{R\}^\{2n\}$ endowed with the standard symplectic form $\omega _0=dp\wedge dq$. We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example, discontinuous Hamiltonians. We also prove that some dynamical aspects of Hamiltonian systems are preserved in the completions.},
author = {Humilière, Vincent},
journal = {Bulletin de la Société Mathématique de France},
keywords = {symplectic topology; hamiltonian dynamics; Viterbo distance; symplectic capacity; Hamilton-Jacobi equation},
language = {eng},
number = {3},
pages = {373-404},
publisher = {Société mathématique de France},
title = {On some completions of the space of hamiltonian maps},
url = {http://eudml.org/doc/272505},
volume = {136},
year = {2008},
}

TY - JOUR
AU - Humilière, Vincent
TI - On some completions of the space of hamiltonian maps
JO - Bulletin de la Société Mathématique de France
PY - 2008
PB - Société mathématique de France
VL - 136
IS - 3
SP - 373
EP - 404
AB - In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of $\mathbb {R}^{2n}$ endowed with the standard symplectic form $\omega _0=dp\wedge dq$. We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example, discontinuous Hamiltonians. We also prove that some dynamical aspects of Hamiltonian systems are preserved in the completions.
LA - eng
KW - symplectic topology; hamiltonian dynamics; Viterbo distance; symplectic capacity; Hamilton-Jacobi equation
UR - http://eudml.org/doc/272505
ER -

References

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  8. [8] P. Libermann & C.-M. Marle – Geométrie symplectique, bases théoriques de la mécanique, tome I, Publications Mathématiques de l’Université Paris VII, 1986. Zbl0643.53001
  9. [9] Y. Oh – « The group of Hamiltonian homeomorphisms and C 0 symplectic topology I », preprint, arXiv:math.SG/0402210, 2005. Zbl1144.37033
  10. [10] A. Ottolenghi & C. Viterbo – « Solutions généralisées pour l’équation d’Hamilton-Jacobi dans le cas d’évolution. », manuscript. 
  11. [11] M. Schwarz – « On the action spectrum for closed symplectically aspherical manifolds », Pacific J. Math.193 (2000), p. 419–461. Zbl1023.57020MR1755825
  12. [12] J. C. Sikorav – « Sur les immersions Lagrangiennes admettant une phase génératrice globale », Compte-rendus de l’Académie des Sciences302 (1986), p. 119–122. Zbl0602.58019MR830282
  13. [13] D. Théret – « A complete proof of Viterbo’s uniqueness theorem on generating functions », Topology and its Applications96 (1999), p. 246–266. Zbl0952.53037MR1709692
  14. [14] C. Viterbo – « Solutions d’équations de Hamilton-Jacobi », Seminaire X-EDP, Palaiseau, 1992. Zbl0878.35025
  15. [15] —, « Symplectic topology as the geometry of generating functions », Math. Annalen292 (1992), p. 685–710. Zbl0735.58019MR1157321

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