Right amenability in semigroups of formal power series

Abstract

Let $k$ be an algebraically closed field of characteristic zero, and $k[[z]]$ the ring of formal power series over $k$. We provide several characterizations of right amenable finitely generated subsemigroups of $z^2k[[z]]$ with the semigroup operation $\circ $ being composition. In particular, we show that a subsemigroup $S=\langle Q_1,Q_2,\dots, Q_k\rangle$ of $z^2k[[z]]$ is right amenable if and only if there exists an invertible element $\beta$ of $zk[[z]]$ such that $\beta^{-1}\circ Q_i \circ \beta =\omega_i z^{d_i},$ $1\leq i \leq k,$ for some integers $d_i$, $1\leq i \leq k,$ and roots of unity $\omega_i,$ $1\leq i \leq k.$