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Pseudo‐Abelian integrals along Darboux cycles
Author(s) -
Bobieński Marcin,
Mardes̆ić Pavao
Publication year - 2008
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn015
Subject(s) - mathematics , integrable system , abelian group , meromorphic function , bounded function , iterated function , pure mathematics , darboux integral , hamiltonian system , mathematical analysis , geometry , curvature
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 a j of polynomials in log h with meromorphic functions of h 1 / a jas coefficients.