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 Mat. Sb. (N.S.), 1972, Volume 87(129), Number 1, Pages 58–66 (Mi msb3035)

Cycle breaking in fiberings on analytic curves

Yu. S. Ilyashenko

Abstract: In this paper various fiberings are constructed on analytic curves in which the topological reconstruction of the fibers proceeds on a nonanalytic set.
Let $D\subset C^1$ be the strip $-1<\operatorname{Re}\zeta<1$ and $D_1\subset D$ the strip $-1<\operatorname{Re}\zeta<0$. Analytic mappings $F\colon C^5\to C^4$ and $f\colon D\to C^5$ are constructed such that 1) for each $\zeta\in D_1$ the fiber $\chi_\zeta$ of the mapping $F$ which passes through the point $f(\zeta)$ has nontrivial fundamental group; 2) for each $\zeta\in{D\setminus D_1}$ the fiber $\chi_\zeta$ is simply connected.
Next it is shown that the generalization of the Petrovskii–Landis Hypothesis on the conservation of cycles for the equations $\dot z=V(z)$, $z\in C^n$, with analytic right-hand side $V(z)$, is valid. Indeed, in $C^2$ we construct a family $\alpha_\zeta$ of equations of the indicated form, analytic in $\zeta$ and such that 1) for each $\zeta\in D_1\setminus N$ ($N$ is a countable set) on one of the solutions of the equations $\alpha_\zeta$ there is a limit cycle $l(\zeta)$; 2) the cycle $l(\zeta)$ changes continuously as $\zeta$ runs over $D_1\setminus N$ and is broken as $\zeta$ approaches a point on the straight line $\operatorname{Re}\zeta=0$. Some related examples are also constructed.
Bibliography: 6 titles.

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English version:
Mathematics of the USSR-Sbornik, 1972, 16:1, 60–68

Bibliographic databases:

UDC: 516.2+517.9
MSC: Primary 32C15, 32E10; Secondary 32L05

Citation: Yu. S. Ilyashenko, “Cycle breaking in fiberings on analytic curves”, Mat. Sb. (N.S.), 87(129):1 (1972), 58–66; Math. USSR-Sb., 16:1 (1972), 60–68

Citation in format AMSBIB
\Bibitem{Ily72} \by Yu.~S.~Ilyashenko \paper Cycle breaking in fiberings on analytic curves \jour Mat. Sb. (N.S.) \yr 1972 \vol 87(129) \issue 1 \pages 58--66 \mathnet{http://mi.mathnet.ru/msb3035} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=307251} \zmath{https://zbmath.org/?q=an:0234.34034|0263.34026} \transl \jour Math. USSR-Sb. \yr 1972 \vol 16 \issue 1 \pages 60--68 \crossref{https://doi.org/10.1070/SM1972v016n01ABEH001349} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. S. Ilyashenko, “Fiberings into analytic curves”, Math. USSR-Sb., 17:4 (1972), 551–569
2. D. V. Anosov, “On generic properties of closed geodesics”, Math. USSR-Izv., 21:1 (1983), 1–29
3. Ivashkovich S., “Vanishing Cycles in Holomorphic Foliations by Curves and Foliated Shells”, Geom Funct Anal, 21:1 (2011), 70–140
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