2009/11/14 by Kingshook Biswas, Biswas, Kingshook
Mathematics · Physics and Astronomy · #37F50 #Advanced Algebra and Geometry #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.0911.2766
openalex publication_date 2009/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f1,...,fN be commuting germs of holomorphic diffeomorphisms in C fixing the origin with irrational rationally independent rotation numbers alpha1,...,alphaN. We adapt Yoccoz' renormalization of germs to this setting to show that a Brjuno-type condition on simultaneous Diophantine approximability of the rotation numbers is sufficient for simultaneous linearizability of f1,...,fN. This generalizes a result of Moser's. In the absence of periodic orbits we show that a weaker arithmetic condition analogous to that of Perez-Marco's for the case of a single germ is sufficent for linearizability. We also obtain lower bounds for the conformal radii of the Siegel disks in both cases in terms of the arithmetic functions defining the arithmetic conditions.