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Finiteness theorems for limit cycles: a digest of the revised proof. (English. Russian original) Zbl 1382.34032
Izv. Math. 80, No. 1, 50-112 (2016); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 80, No. 1, 55-118 (2016).
Summary: At the end of the 1970s it became clear that an old result of H. Dulac [Bull. Soc. Math. Fr. 51, 45–188 (1923; JFM 49.0304.01)], that an individual polynomial plane vector field has only a finite number of limit cycles, is not complete. Independent complete proofs of the finiteness theorem were published in two monographs: [Yu. S. Il’yashenko, Finiteness theorems for limit cycles. Transl. from the Russian by H. H. McFaden. Providence, RI: American Mathematical Society (1991; Zbl 0743.34036)] and [J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Paris: Hermann, Éditeurs des Sciences et des Arts. (1992; Zbl 1241.34003)]. Both books are highly technical and difficult to read.
The paper under review is the first of two parts in which a simplified version of Ilyashenko’s proof is presented. In fact, the following stronger result is proved.
The monodromy transformation \(\Delta\) associated with a polycycle of an analytic vector field, expressed in the logarithmic chart \(\xi =-\ln x\) on a semi-transversal \((\mathbb R^+,0)\) to the polycycle, satisfies the following bounds:
\[ \mathrm{exp}\left(-\mathrm{exp}^{[n]}\nu \xi\right)\xi <|\Delta (\xi)-\xi| <\exp\left(-\exp^{[n]}\mu\xi\right) \]
for large \(\xi\), where \(0<\mu <\nu\) are constants and \(\exp^{[n]}=\exp \circ\cdots\circ\exp\) is the \(n\)th iteration of the exponential map.

34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
37F75 Dynamical aspects of holomorphic foliations and vector fields
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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