Reduction of symplectic homeomorphisms

TL;DR

This paper investigates how symplectic homeomorphisms that preserve coisotropic submanifolds behave under reduction, demonstrating that in specific cases they maintain spectral invariants using Floer theory.

Contribution

It shows that reduced symplectic homeomorphisms preserve spectral invariants on tori, introducing a new class of spectral invariants via Lagrangian Floer theory.

Findings

Reduced homeomorphisms preserve spectral invariants on tori.

Introduction of a new class of spectral invariants with a non-standard triangle inequality.

Spectral capacity is preserved under reduction in the studied setting.

Abstract

In a previous article, 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.