Functional equations in formal power series

Let k be an algebraically closed field of characteristic zero, and k[[z]] the ring of formal power series over k. In this paper, we study equations in the semigroup z²k[[z]] with the semigroup operation being composition. We prove a number of general results about such equations and provide some applications. In particular, we answer a question of Horwitz and Rubel about decompositions of “even” formal power series. We also show that every right amenable subsemigroup of z²k[[z]] is conjugate to a subsemigroup of the semigroup of monomials.