Right amenability in semigroups of formal power series

TL;DR

This paper characterizes right amenable finitely generated subsemigroups of formal power series under composition over an algebraically closed field, linking their structure to conjugation by invertible series and roots of unity.

Contribution

It provides new characterizations of right amenability in semigroups of formal power series, connecting algebraic structure with conjugation and roots of unity.

Findings

A subsemigroup is right amenable iff it can be conjugated to a semigroup generated by monomials with roots of unity.

Characterization involves the existence of an invertible element conjugating generators to monomials.

Results apply to finitely generated subsemigroups of formal power series under composition.

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 $\omega_i,$ $1\leq i \leq k.$