On the adjoint action of the group of symplectic diffeomorphisms

We study the action of Hamiltonian diffeomorphisms of a compact symplectic manifold ($X,\omega$) on $C^\infty(X)$ and on functions $C^\infty(X)\to \mathbb R$. We describe various properties of invariant convex functions on $C^\infty(X)$. Among other things we show that continuous convex functions $C^\infty(X)\to \mathbb R$ that are invariant under the action are automatically invariant under so called strict rearrangements and they are continuous in the sup norm topology of $C^\infty(X)$; but this is not generally true if the convexity condition is dropped.