# On some completions of the space of hamiltonian maps

Bulletin de la Société Mathématique de France (2008)

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

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topHumilière, 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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- [9] Y. Oh – « The group of Hamiltonian homeomorphisms and ${C}^{0}$ symplectic topology I », preprint, arXiv:math.SG/0402210, 2005. Zbl1144.37033
- [10] A. Ottolenghi & C. Viterbo – « Solutions généralisées pour l’équation d’Hamilton-Jacobi dans le cas d’évolution. », manuscript.
- [11] M. Schwarz – « On the action spectrum for closed symplectically aspherical manifolds », Pacific J. Math.193 (2000), p. 419–461. Zbl1023.57020MR1755825
- [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
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- [15] —, « Symplectic topology as the geometry of generating functions », Math. Annalen292 (1992), p. 685–710. Zbl0735.58019MR1157321

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