Journal of Dynamical and Control Systems

, Volume 19, Issue 2, pp 195–235 | Cite as

Integrating factors for groups of formal complex diffeomorphisms

  • M. Martelo
  • B. Scárdua


We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf of a foliation. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, …). Such system of equivalences also characterizes the existence of suitable integrating factors, i.e., invariant vector fields and one-forms associated to the group. In this paper we search the basic lines of such dictionary for the case of several complex variables groups. For abelian, metabelian, solvable or nilpotent groups we investigate the existence of suitable formal vector fields and closed diferential forms which exhibit an invariance property under the group action. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimension.

Key words and phrases

Formal diffeomorphism metabelian group dicritical map solvability length 

2000 Mathematics Subject Classification

Primary 37F75 32S65 Secondary 34M35 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Instituto de Matemática e Estatística - UFFRio de JaneiroBrasil
  2. 2.Instituto de Matemática-UFRJRio de JaneiroBrazil

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