  We prove an analog of the Tits alternative for rational functions. In
particular, we show that if $S$ is a finitely generated semigroup of rational
functions over the complex numbers, then either $S$ has polynomially bounded
growth or $S$ contains a nonabelian free semigroup. We also show that if f and
g are polarizable maps over any field that do not have the same set of
preperiodic points, then the semigroup generated by f and g contains a
nonabelian free semigroup.
