Functional equations in formal power series

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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.