Forgive me if this is too obvious, but somebody should say it. The Koopman operator is bracket preserving with respect to the canonical Poisson bracket.

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 $\bigwedge^{2n}(M)$.