Periodic multivariate formal power series

A system of multivariate formal power series $\phi$ with a homogeneous decomposition $\phi ={\sum }_{k=0}^{\mathrm{\infty }}{\phi }_k$ is invertible under composition if ${\phi }_0=0$ and $\mathrm{d}\mathrm{e}\mathrm{t}({\phi }_1)\mathrm{\ne }0.$ All invertible series over a field $K$ form a formal transformation group $G_{\mathrm{\infty }}(n,K)\mathrm{.}$ We prove that every periodic series $\phi \in G_{\mathrm{\infty }}(n,K)$ with ${\phi }_1$ diagonalizable is conjugate to ${\phi }_1\mathrm{.}$ This classifies all periodic series in $G_{\mathrm{\infty }}(n,\mathbb{C})\mathrm{.}$ A constraint for a periodic series is obtained when its first term is a multiple of identity.