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 Fr\'echet 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.