On the amenability of semigroups of entire maps and formal power series

In this article, we investigate some relations between dynamical and algebraic properties of semigroups of entire maps with applications to semigroups of formal series. We show that two entire maps fixing the origin share the set of preperiodic points, whenever these maps generate a semigroup which contains neither free nor free abelian non-cyclic subsemigroups and one of the maps has the origin as a superattracting fixed point. We show that a subgroup of formal series generated by rational elements is amenable, whenever contains no free non-cyclic subsemigroup generated by rational elements. We prove that a left-amenable semigroup S of entire maps admits a invariant probability measure for a continuous extension of S on the Stone-Cech compactification of the complex plane. Finally, given an entire map f, we associate a semigroup S such that f admits no ergodic fixed point of the Ruelle operator, whenever every finitely generated subsemigroup of S admits a left-amenable Ruelle representation.