We provide an algorithm for constructing a Kirby diagram of a 4-dimensional open book given a Heegaard diagram of the page. As an application, we show that any open book with trivial monodromy is diffeomorphic to an open book constructed with a punctured handlebody as page and a composition of torus twists and sphere twists as monodromy.

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 Poincaré 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 and Katok embed into the Reeb flow of a dynamically convex contact form on the $3$ -sphere.