Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-30T17:53:03.480Z Has data issue: false hasContentIssue false
  • English
  • Français

Simultaneous linearization of germs of commuting holomorphic diffeomorphisms

Published online by Cambridge University Press:  14 September 2011

KINGSHOOK BISWAS
KINGSHOOK BISWAS*
Affiliation:
Ramakrishna Mission Vivekananda University, Belur Math, WB-711202, India (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let α1,…,αn be irrational numbers which are linearly independent over the rationals, and f1,…,fn commuting germs of holomorphic diffeomorphisms in ℂ such that fk(0)=0,f′k(0)=e2πiαk,k=1,…,n. Moser showed that f 1,…,fn are simultaneously linearizable (i.e. conjugate by a germ of holomorphic diffeomorphism h to the rigid rotations Rαk(z)=e2πiαkz) if the vector of rotation numbers (α 1,…,αn) satisfies a Diophantine condition. Adapting Yoccoz’s renormalization to the setting of commuting germs, we show that simultaneous linearization holds in the presence of a weaker Brjuno-type condition ℬ(α1,…,αn)<+∞, where ℬ(α1,…,αn) is a multivariable analogue of the Brjuno function. If there are no periodic orbits for the action of the germs f1,…,fn in a neighbourhood of the origin, then a weaker arithmetic condition ℬ′ (α 1,…,αn)<+∞ analogous to Perez-Marco’s condition for linearization in the absence of periodic orbits is shown to suffice for linearizability. Normalizing the germs to be univalent on the unit disc, in both cases the Siegel discs are shown to contain discs of radii for some universal constant C.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1216 - 1225
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[Brj71]Brjuno, A. D.. Analytical form of differential equations. Trans. Moscow Math. Soc. 25 (1971), 131–288.Google Scholar
[Mos90]Moser, J.. On commuting circle mappings and simultaneous Diophantine approximations. Math. Z. 205 (1990), 105–121.CrossRefGoogle Scholar
[PM93]Perez-Marco, R.. Sur les dynamiques holomorphes non-linearisables et une conjecture de V.I. Arnold. Ann. Sci. Éc. Norm. Supér. 26 (1993), 565–644.CrossRefGoogle Scholar
[PM95]Perez-Marco, R.. Uncountable number of symmetries for non-linearisable holomorphic dynamics. Invent. Math. 119 (1995), 67–127.CrossRefGoogle Scholar
[PM97]Perez-Marco, R.. Fixed points and circle maps. Acta Math. 179(2) (1997), 243–294.CrossRefGoogle Scholar
[Sch80]Schmidt, W. M.. Diophantine Approximation (Lecture Notes in Mathematics, 785). Springer, New York, 1980.Google Scholar
[Sie42]Siegel, C. L.. Iteration of analytic functions. Ann. of Math. (2) 43 (1942), 807–812.CrossRefGoogle Scholar
[SMY97]Moussa, P., Marmi, S. and Yoccoz, J. C.. The Brjuno functions and their regularity properties. Comm. Math. Phys. 186 (1997), 265–293.Google Scholar
[Yoc95]Yoccoz, J. C.. Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Astérisque 231 (1995), 3–88.Google Scholar