  We use Lerman's contact cut construction to find a sufficient condition for
Hamiltonian diffeomorphisms of compact surfaces to embed into a closed
3-manifold as Poincar\'e return maps on a global surface of section for a Reeb
flow. In particular, we show that the irrational pseudorotations of the 2-disc
constructed by Fayad-Katok embed into the Reeb flow of a dynamically convex
contact form on the 3-sphere.
