Symplectic Koopmanism

Let $(M, \omega)$ be a $2n$-dimensional symplectic manifold and let $L_2(M,|\omega^n|)$ be the Hilbert space of complex-valued functions on $M$ that are square integrable with respect to the Liouville measure.

A classical (but still wonderful) remark due to Koopman is that given a measure preserving map $\phi: M \rightarrow M$, the operator $$U_\phi : L_2(M,|\omega^n|) \longrightarrow L_2(M,|\omega^n|)$$ defined by $f \mapsto f \circ \phi$ is unitary. The operator $U_\phi$ is sometimes called the Koopman operator of the map $\phi$. Since symplectomorphisms are measure preserving, it seems natural to ask the following

Vague question. Is there anything at all particular about Koopman operators of symplectomorphisms?

• Assuming that by "uniform closure", you mean "norm closure", then operators that you describe form a closed discrete set (any tow of them have distance 2). – André Henriques Jun 27 '14 at 12:39
• Of course !! It was a dumb way to try to make precise a vague question. I'm editing the question to leave it vague. ;-) – alvarezpaiva Jun 27 '14 at 12:48
• What is the motivation for your question? – André Henriques Jun 27 '14 at 15:32
• @AndréHenriques: I'm trying to see if there is any symplectic version of ergodic theory, one that uses that the maps are not just measure preserving. – alvarezpaiva Jun 27 '14 at 17:13
• Which results/concepts in ergodic theory do you expect to be affected by the fact that the transformation is a symplectomorphism? – André Henriques Jun 27 '14 at 19:05

The Koopman operator is bracket preserving with respect to the canonical Poisson bracket on $C^{\infty}(M)$. Formally, this implies the dual operator (the Frobenius-Perron operator) is a Poisson automorphism with respect to the Lie-Poisson structure on $\bigwedge^{2n}(M)$.