A Poincaré-Dulac renormalization theorem for attracting rigid germs in ℂ^d

2011/03/14 by Matteo Ruggiero, Ruggiero, Matteo
Mathematics · Physics and Astronomy · #37F25 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS #msc:37F25

paper · pdf · doi:10.48550/arxiv.1103.2804

15 pages, 0 figures, the paper has been withdrawn by the author since all results have been generalized by another author's paper

openalex publication_date 2011/03/14 · arxiv created 2011/09/30 · arxiv updated 2011/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Studying the dynamics of attracting rigid germs f:(ℂ^d, 0) → (ℂ^d, 0) in dimension d ≥ 3, a new phenomenon arise: principal resonances. The resonances of the classic Poincaré-Dulac theory are given by (multiplicative) relations between the eigenvalues of df₀; principal resonances arise as (multiplicative) relations between the non-null eigenvalues of df₀, and the "leading term" for the superattracting part of f. We shall prove that for attracting rigid germs there are only finitely-many principal resonances, and a Poincaré-Dulac renormalization theorem in this case. We shall conclude with some considerations on the classification of a special class of attracting rigid germs in any dimension, and we specialize the result to the 3-dimensional case.

Related

• Formal Poincare'-Dulac renormalization for holomorphic germs
• Contracting rigid germs in higher dimensions
• Résonances près de seuils d'opérateurs magnétiques de Pauli et de Dirac
• Propagation des singularités et résonances
• Renormalization for a Class of Dynamical Systems: some Local and Global Properties