Flexibility of the adjoint action of the group of Hamiltonian diffeomorphisms

Abstract

On a closed and connected symplectic manifold, the group of Hamiltonian diffeomorphisms has the structure of an infinite-dimensional Fréchet Lie group, where the Lie algebra is naturally identified with the space of smooth and zero-mean normalized functions, and the adjoint action is given by pullbacks. We show that this action is flexible: for every non-zero smooth and zero-mean normalized function $u$, any other smooth and zero-mean function $f$ can be written as a finite sum of elements in the orbit of $u$ under the adjoint action. Additionally, the number of elements in this sum is dominated by the uniform norm of $f$. This result can be interpreted as a (bounded) infinitesimal version of Banyaga's theorem on the simplicity of the group of Hamiltonian diffeomorphisms.

Revised version to take into account the referees suggestions. To appear in Annales scientifiques de l'École normale supérieure