Periodic multivariate formal power series

Abstract

A system of multivariate formal power series ϕ with a homogeneous decomposition ϕ = ∞ k=0 ϕ k is invertible under composition if ϕ 0 = 0 and det(ϕ 1 ) = 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 ϕ 1 diagonalizable is conjugate to ϕ 1 . 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.