A Hofer-like metric on the group of symplectic diffeomorphisms

TL;DR

This paper introduces a new Hofer-like metric on the group of symplectic diffeomorphisms, leveraging Hodge decomposition, and explores its properties and implications for symplectic topology.

Contribution

It constructs a novel norm on symplectic diffeomorphisms related to the Hofer norm and analyzes the closure of Hamiltonian diffeomorphisms within this framework.

Findings

The norm bounds the Hamiltonian subgroup by the Hofer norm.

Hamiltonian diffeomorphisms form a closed subgroup under the extended norm.

Applications to $C^0$ symplectic topology and spectral distances are provided.

Abstract

Using a "Hodge decomposition" of symplectic isotopies on a compact symplectic manifold $(M,\omega)$, we construct a norm on the identity component in the group of all symplectic diffeomorphisms of $(M,\omega)$ whose restriction to the group $Ham(M,\omega)$ of hamiltonian diffeomorphisms is bounded from above by the Hofer norm. Moreover, $Ham(M,\omega)$ is closed in $Symp(M,\omega)$ equipped with the topology induced by the extended norm. We give an application to the $C^0$ symplectic topology. We also discuss extensions of Oh's spectral distance.