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

Mardešić, Pavao; Saavedra, Mariana

doi:10.1090/S0002-9939-07-09026-0

Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles
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by Pavao Mardešić and Mariana Saavedra

Proc. Amer. Math. Soc. 135 (2007), 3273-3282

DOI: https://doi.org/10.1090/S0002-9939-07-09026-0

Published electronically: June 22, 2007

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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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