Reduction of symplectic homeomorphisms
[Réduction des homéomorphismes symplectiques]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 3, pp. 633-668.

Nous avons démontré dans [9], qu'un homéomorphisme symplectique qui laisse invariante une sous-variété coïsotrope C, préserve également son feuilletage caractéristique. Il induit donc un homéomorphisme sur la réduction symplectique de C.

Dans cet article, nous démontrons que l'homéomorphisme ainsi obtenu exhibe certaines propriétés symplectiques. En particulier, dans le cas où la variété symplectique ambiante est un tore et la sous-variété coïsotrope est un sous-tore standard, nous démontrons que l'homéomorphisme réduit préserve les invariants spectraux et donc aussi la capacité spectrale.

Pour démontrer notre résultat principal, nous construisons, à l'aide de l'homologie de Floer lagrangienne, une nouvelle famille d'invariants spectraux qui satisfont un nouveau type d'inégalité triangulaire.

In [9], we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C.

In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity.

To prove our main result, we use Lagrangian Floer theory to construct a new class of spectral invariants which satisfy a non-standard triangle inequality.

Publié le :
DOI : 10.24033/asens.2292
Classification : 53D40; 37J05.
Keywords: Symplectic manifolds, symplectic reduction, $C^0$--symplectic topology, spectral invariants.
Mot clés : Variétés symplectiques, réduction symplectique, topologie symplectique $C^0$, invariants spectraux. 
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     title = {Reduction of symplectic homeomorphisms},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {633--668},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Humilière, Vincent; Leclercq, Rémi; Seyfaddini, Sobhan. Reduction of symplectic homeomorphisms. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 3, pp. 633-668. doi : 10.24033/asens.2292. http://www.numdam.org/articles/10.24033/asens.2292/

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