Abstract: The Tits alternative states that any finitely generated
linear group is either virtually solvable or contains a non-abelian
free group. This implies that every finitely generated linear group has
either polynomial or exponential growth. We will show that a similar
result holds for finitely generated semigroups of rational functions
in characteristic 0. This is joint work with Jason Bell, Keping
Huang, and Wayne Peng.