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Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles
Author(s):
Pavao
Mardesic;
Mariana
Saavedra
Journal:
Proc. Amer. Math. Soc.
135
(2007),
3273-3282.
MSC (2000):
Primary 34C07;
Secondary 34C25, 34M35
Posted:
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  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:
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.
Posted:
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
Copyright of article:
Copyright
2007,
American Mathematical Society
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