Periodic multivariate formal power series

2022/06/11 by Xue Zhang, Zhang, Xue
Mathematics · #13F25 #16W60 #47J07 #65Q30 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2206.05525

openalex publication_date 2022/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A system of multivariate formal power series φ with a homogeneous decomposition φ=∑_k=0^∞φ_k 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.

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