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.

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The author’s research was partially supported by NSF grant DMS 1764167.

Received 8 January 2021

Received revised 2 September 2021

Accepted 29 September 2021

Published 13 May 2022