Simultaneous linearization of holomorphic germs in presence of resonances

2008/12/18 by Jasmin Raissy, Raissy, Jasmin · 1 citation
Mathematics · #32H50 #37F50 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.0812.3579

openalex publication_date 2008/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f₁, ..., f_m be m≥ 2 germs of biholomorphisms of \Cⁿ, fixing the origin, with (\d f₁)_O diagonalizable and such that f₁ commutes with f_h for any h=2,..., m. We prove that, under certain arithmetic conditions on the eigenvalues of (\d f₁)_O and some restrictions on their resonances, f₁, ..., f_m are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under f₁,..., f_m.

Citations

• Discrete holomorphic local dynamical systems

Cited by

• Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step

Related

• On linearization of biholomorphism with non-semi-simple linear part at a fixed point
• Linearization of holomorphic germs with resonant linear part
• Simultaneous linearization of commuting germs of holomorphic diffeomorphisms
• Brjuno conditions for linearization in presence of resonances
• Dynamics of two-resonant biholomorphisms