Periodic multivariate formal power series

Abstract

A system of multivariate formal power series φ with a homogeneous decomposition φ=∑k=0∞φk is invertible under composition if φ0=0 and det(φ1)≠0. All invertible series over a field K form a formal transformation group G∞(n,K). We prove that every periodic series φ∈G∞(n,K) with φ1 diagonalizable is conjugate to φ1. This classifies all periodic series in G∞(n,C). A constraint for a periodic series is obtained when its first term is a multiple of identity.