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