Reduction of symplectic homeomorphisms

Humilière, Vincent; Leclercq, Rémi; Seyfaddini, Sobhan

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.

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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