American Institute of Mathematical Sciences

Advanced
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

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 & Continuous Dynamical Systems - A, 2011, 29 (3) : 839-844. doi: 10.3934/dcds.2011.29.839
References:
[1]

I. N. Baker, Fractional iteration near a fixed point of multiplier 1,, J. Austral. Math. Soc, 4 (1964), 143.  doi: 10.1017/S144678870002334X.  Google Scholar

[2]

A. D. Brjuno, Analytical form of differential equations,, Transactions Moscow Math. Soc., 25 (1971), 131.   Google Scholar

[3]

J. Ecalle, Théorie itérative: introduction à la théorie des invariantes holomorphes,, J. Math pures et appliqués, 54 (1975), 183.   Google Scholar

[4]

J. Moser, On commuting circle mappings and simultaneous Diophantine approximations,, Math. Zeitschrift, 205 (1990), 105.  doi: 10.1007/BF02571227.  Google Scholar

[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 ().   Google Scholar

[6]

R. Perez-Marco, Uncountable number of symmetries for non-linearisable holomorphic dynamics,, Inventiones Mathematicae, 119 (1995), 67.  doi: 10.1007/BF01245175.  Google Scholar

[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.   Google Scholar

[8]

J. C. Yoccoz, Petits diviseurs en dimension 1,, Asterisque, 231 (1995), 3.   Google Scholar

show all references

References:
[1]

I. N. Baker, Fractional iteration near a fixed point of multiplier 1,, J. Austral. Math. Soc, 4 (1964), 143.  doi: 10.1017/S144678870002334X.  Google Scholar

[2]

A. D. Brjuno, Analytical form of differential equations,, Transactions Moscow Math. Soc., 25 (1971), 131.   Google Scholar

[3]

J. Ecalle, Théorie itérative: introduction à la théorie des invariantes holomorphes,, J. Math pures et appliqués, 54 (1975), 183.   Google Scholar

[4]

J. Moser, On commuting circle mappings and simultaneous Diophantine approximations,, Math. Zeitschrift, 205 (1990), 105.  doi: 10.1007/BF02571227.  Google Scholar

[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 ().   Google Scholar

[6]

R. Perez-Marco, Uncountable number of symmetries for non-linearisable holomorphic dynamics,, Inventiones Mathematicae, 119 (1995), 67.  doi: 10.1007/BF01245175.  Google Scholar

[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.   Google Scholar

[8]

J. C. Yoccoz, Petits diviseurs en dimension 1,, Asterisque, 231 (1995), 3.   Google Scholar

[1]

Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773
[2]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[3]

Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333
[4]

Percy Fernández-Sánchez, Jorge Mozo-Fernández, Hernán Neciosup. Dicritical nilpotent holomorphic foliations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3223-3237. doi: 10.3934/dcds.2018140
[5]

John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047
[6]

Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537
[7]

Victor Kozyakin. Polynomial reformulation of the Kuo criteria for v- sufficiency of map-germs. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 587-602. doi: 10.3934/dcdsb.2010.14.587
[8]

João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837
[9]

David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317
[10]

Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847
[11]

Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1
[12]

Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443
[13]

Lisa C. Jeffrey and Frances C. Kirwan. Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface. Electronic Research Announcements, 1995, 1: 57-71.

[14]

Kazuhiro Sakai. The oe-property of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 581-591. doi: 10.3934/dcds.1998.4.581
[15]

Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281
[16]

Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615
[17]

Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204
[18]

Masoumeh Gharaei, Ale Jan Homburg. Random interval diffeomorphisms. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 241-272. doi: 10.3934/dcdss.2017012
[19]

Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075
[20]

Sheldon Newhouse. Distortion estimates for planar diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 345-412. doi: 10.3934/dcds.2008.22.345

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences