A classic result of Jacques Tits states any finitely generated linear group contains either a nonabelian free group or a solvable group of finite index. This is often called the “Tits alternative” for finitely generated linear groups. As a consequence (via work of Milnor and Wolf), one has that every finitely generated linear group either has polynomial growth or contains a nonabelian free group. Tits’ proof relies heavily on techniques from number theory. We ask if a variant of the Tits alternative holds for finitely generated semigroups of finite self-maps of varieties and show that it does in the case of P^1 in characteristic 0. This represents joint work with Jason Bell, Keping Huang, and Wayne Peng.