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Proceedings of the London Mathematical Society Advance Access published online on April 22, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn015
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© 2008 London Mathematical Society

Pseudo-Abelian integrals along Darboux cycles

Marcin Bobienski

Institute of Mathematics
Warsaw University
ul. Banacha 2
02-097 Warsaw
Poland
mbobi@mimuw.edu.pl

Pavao Mardesic

Université de Bourgogne
Institut de Mathématiques de Bourgogne
UMR 5584
du CNRS
BP 47870
21078 Dijon Cedex
France

Received 16 May 2007. Revision received 30 January 2008.

We study polynomial perturbations of integrable, non-Hamiltonian system with first integral of Darboux-type with positive exponents. We assume that the unperturbed system admits a period annulus. The linear part of the Poincaré return map is given by pseudo-Abelian integrals. In this paper we investigate analytic properties of these integrals. We prove that iterated variations of these integrals vanish identically. Using this relation we prove that the number of zeros of these integrals is locally uniformly bounded under generic hypothesis. This is a generic analog of the Varchenko-Khovanskii theorem for pseudo-Abelian integrals. Finally, under some arithmetic properties of exponents, the pseudo-Abelian integrals are a sum over exponents aj of polynomials in log h with meromorphic functions of h1/aj as coefficients.

2000 Mathematics Subject Classification 34C07, 34C08.

This research was supported by Conseil Regional de Bourgogne 2006 (No 05514AA010S4115) and KBN Grant No 2 P03A 015 29.


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