Maximal abelian torsion subgroups of Diff(<b> C</b>,0)

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July 2011, 29(3): 839-844. doi: 10.3934/dcds.2011.29.839

Maximal abelian torsion subgroups of Diff( C,0)

Kingshook Biswas ^1,

1.	Ramakrishna Mission Vivekananda University, Belur Math, WB-711202

Received July 2009 Revised August 2010 Published November 2010

Abstract
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In the study of the local dynamics of a germ of diffeomorphism fixing the origin in $\mathbb C$, an important problem is to determine the centralizer of the germ in the group Diff$(\mathbb C,0)$ of germs of diffeomorphisms fixing the origin. When the germ is not of finite order, then the centralizer is abelian, and hence a maximal abelian subgroup of Diff$(\mathbb C,0)$. Conversely any maximal abelian subgroup which contains an element of infinite order is equal to the centralizer of that element. A natural question is whether every maximal abelian subgroup contains an element of infinite order, or whether there exist maximal abelian torsion subgroups; we show that such subgroups do indeed exist, and moreover that any infinite subgroup of the rationals modulo the integers $\mathbb{Q/Z}$ can be embedded into Diff$(\mathbb C,0)$ as such a subgroup.

Keywords: Germs of holomorphic diffeomorphisms, centralizers..

Mathematics Subject Classification: Primary: 37F5.

Citation: Kingshook Biswas. Maximal abelian torsion subgroups of Diff( C,0). Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 839-844. doi: 10.3934/dcds.2011.29.839

References:

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References:

[1]	I. N. Baker, Fractional iteration near a fixed point of multiplier 1, J. Austral. Math. Soc, 4 (1964), 143-148. doi: 10.1017/S144678870002334X.
[2]	A. D. Brjuno, Analytical form of differential equations, Transactions Moscow Math. Soc., 25 (1971), 131-288.
[3]	J. Ecalle, Théorie itérative: introduction à la théorie des invariantes holomorphes, J. Math pures et appliqués, 54 (1975), 183-258.
[4]	J. Moser, On commuting circle mappings and simultaneous Diophantine approximations, Math. Zeitschrift, 205 (1990), 105-121. doi: 10.1007/BF02571227.
[5]	R. Perez-Marco, Solution complète au problème de Siegel de linéarisation d'une application holomorphe au voisinage d'un point fixe (d'après J.C.Yoccoz), Seminar Bourbaki, 34 Exp. No. 753.
[6]	R. Perez-Marco, Uncountable number of symmetries for non-linearisable holomorphic dynamics, Inventiones Mathematicae, 119 (1995), 67-127. doi: 10.1007/BF01245175.
[7]	S. M. Voronin, Analytic classification of germs of conformal mappings (C,0) $\to$ (C,0) with identity linear part, Funktsional Anal. i Prilozhen, 16 (1982), 1-17.
[8]	J. C. Yoccoz, Petits diviseurs en dimension 1, Asterisque, 231 (1995), 3-88.

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