Does anyone know any undecidable problems in the dynamics of functions (not necessarily monoid homomorphisms or anything) from \Sigma^* to \Sigma^* where \Sigma is a finite set? In particular, I'm wondering if there is an algorithm that takes as input a (computable) function f:Sigma^* \rightarrow \Sigma^*, a w \in \Sigma^*, and a language L and decides if there is an n such that f^n(w) is an element of L. Does anyone have a reference where questions similar to this are addressed? Thanks.

EDIT: I had to take the dollar signs out of the question. The site was having trouble with the Tex.