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 $z^{2}k[[z]]$ 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 $z^{2}k[[z]]$ is conjugate to a subsemigroup of the semigroup of monomials.