Periodic multivariate formal power series

TL;DR

This paper classifies periodic multivariate formal power series with invertibility under composition, showing they are conjugate to their linear part when diagonalizable, and provides constraints when the first term is scalar multiple of identity.

Contribution

It proves that all periodic series with diagonalizable linear part are conjugate to their linear component, offering a complete classification of such series over complex numbers.

Findings

Periodic series with diagonalizable linear part are conjugate to their linear component.

A classification of all periodic series in the formal transformation group over complex numbers.

Constraints are derived when the first term is a scalar multiple of the identity.

Abstract

A system of multivariate formal power series $\varphi$ with a homogeneous decomposition $\varphi=\sum_{k=0}^\infty\varphi_k$ is invertible under composition if $\varphi_0=0$ and $\mathrm{det}(\varphi_1)\ne 0.$ All invertible series over a field $K$ form a formal transformation group $G_\infty(n,K).$ We prove that every periodic series $\varphi\in G_\infty(n,K)$ with $\varphi_1$ diagonalizable is conjugate to $\varphi_1.$ This classifies all periodic series in $G_\infty(n,\mathbb{C}).$ A constraint for a periodic series is obtained when its first term is a multiple of identity.