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 Trudy Mat. Inst. Steklova, 2006, Volume 254, Pages 196–214 (Mi tm109)

Persistence Theorems and Simultaneous Uniformization

Yu. S. Ilyashenko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Among the most intriguing problems in the theory of foliations by analytic curves is that of the persistence of complex limit cycles of a polynomial vector field, as well as related problems concerning the persistence of identity cycles and saddle connections and the global extendability of the Poincaré map. It is proved that all these persistence problems have positive solutions for any foliation admitting a simultaneous uniformization of leaves. The latter means that there exists a uniformization of leaves that analytically depends on the initial condition and satisfies certain additional assumptions, called continuity and boundedness. Thus, the results obtained are conditional, but they establish a relation between very different properties of foliations.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2006, 254, 184–200

Bibliographic databases:

UDC: 517.927.7
Received in June 2005

Citation: Yu. S. Ilyashenko, “Persistence Theorems and Simultaneous Uniformization”, Nonlinear analytic differential equations, Collected papers, Trudy Mat. Inst. Steklova, 254, Nauka, MAIK «Nauka/Inteperiodika», M., 2006, 196–214; Proc. Steklov Inst. Math., 254 (2006), 184–200

Citation in format AMSBIB
\Bibitem{Ily06} \by Yu.~S.~Ilyashenko \paper Persistence Theorems and Simultaneous Uniformization \inbook Nonlinear analytic differential equations \bookinfo Collected papers \serial Trudy Mat. Inst. Steklova \yr 2006 \vol 254 \pages 196--214 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm109} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2301006} \elib{https://elibrary.ru/item.asp?id=13517734} \transl \jour Proc. Steklov Inst. Math. \yr 2006 \vol 254 \pages 184--200 \crossref{https://doi.org/10.1134/S0081543806030096} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33749419146} 

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This publication is cited in the following articles:
1. Gavrilov L., Movasati H., Nakai I., “On the non-persistence of Hamiltonian identity cycles”, J. Differential Equations, 246:7 (2009), 2706–2723
2. Alvarez S., Hussenot N., “Singularities for analytic continuations of holonomy germs of Riccati foliations”, Ann. Inst. Fourier, 66:1 (2016), 331–376
3. A. A. Scherbakov, “Uniformizatsiya sloenii s giperbolicheskimi listami i uravnenie Beltrami s parametrami”, Funkts. analiz i ego pril., 53:3 (2019), 98–100
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