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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles


Authors: Pavao Mardesic and Mariana Saavedra
Journal: Proc. Amer. Math. Soc. 135 (2007), 3273-3282
MSC (2000): Primary 34C07; Secondary 34C25, 34M35
Published electronically: June 22, 2007
MathSciNet review: 2322759
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Abstract: We call Poincaré time the time associated to the Poincaré (or first return) map of a vector field. In this paper we prove the non-accumulation of isolated critical points of the Poincaré time $ T$ on hyperbolic polycycles of polynomial vector fields. The result is obtained by proving that the Poincaré time of a hyperbolic polycycle either has an unbounded principal part or is an almost regular function. The result relies heavily on the proof of Il'yashenko's theorem on non-accumulation of limit cycles on hyperbolic polycycles.


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Additional Information

Pavao Mardesic
Affiliation: Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S. Université de Bourgogne, B.P. 47 870 21078 Dijon Cedex, France
Email: mardesic@u-bourgogne.fr

Mariana Saavedra
Affiliation: Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Chile
Email: mariansa@udec.cl

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09026-0
PII: S 0002-9939(07)09026-0
Keywords: Critical period, finiteness, non-accumulation, quasi-analyticity, Dulac problem
Received by editor(s): July 5, 2006
Published electronically: June 22, 2007
Additional Notes: This work was partially supported by Fondecyt Projects 1061006 and 7060107, Escuela de Graduados de la Universidad de Concepción and Proyecto Fundación Andes C13955/12
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society