Right amenability in semigroups of formal power series

Abstract

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