We will present a recent proof of Beaumont that shows that if f and g are two polarizable self-maps of a variety that do not share the same set of preperiodic points, then they generate a free semi-group under composition. The proof builds on work of Bell, Huang, Peng, and the speaker, and uses a ping-pong lemma similar to the one used by Tits in his proof of the Tits alternative for finitely generated linear groups. We will also propose some conjectured generalizations of this result that relate to a Tits alternative for finite self-maps of varieties.