Periodic multivariate formal power series

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.