On the adjoint action of the group of symplectic diffeomorphisms

TL;DR

This paper investigates the properties of invariant convex functions under the action of Hamiltonian diffeomorphisms on a symplectic manifold, revealing invariance and continuity characteristics related to strict rearrangements.

Contribution

It characterizes invariant convex functions under symplectic diffeomorphisms and explores their invariance under strict rearrangements and continuity properties.

Findings

Invariant convex functions are invariant under strict rearrangements.

Such functions are continuous in the sup norm topology.

Dropping convexity loses these invariance and continuity properties.

Abstract

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.