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.
@article{SM_1972_16_1_a4,
author = {Yu. S. Ilyashenko},
title = {Cycle breaking in fiberings on analytic curves},
journal = {Sbornik. Mathematics},
pages = {60--68},
year = {1972},
volume = {16},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1972_16_1_a4/}
}
TY - JOUR
AU - Yu. S. Ilyashenko
TI - Cycle breaking in fiberings on analytic curves
JO - Sbornik. Mathematics
PY - 1972
SP - 60
EP - 68
VL - 16
IS - 1
UR - http://geodesic.mathdoc.fr/item/SM_1972_16_1_a4/
LA - en
ID - SM_1972_16_1_a4
ER -
%0 Journal Article
%A Yu. S. Ilyashenko
%T Cycle breaking in fiberings on analytic curves
%J Sbornik. Mathematics
%D 1972
%P 60-68
%V 16
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1972_16_1_a4/
%G en
%F SM_1972_16_1_a4
Yu. S. Ilyashenko. Cycle breaking in fiberings on analytic curves. Sbornik. Mathematics, Tome 16 (1972) no. 1, pp. 60-68. http://geodesic.mathdoc.fr/item/SM_1972_16_1_a4/
[1] I. G. Petrovskii, E. M. Landis, “O chisle predelnykh tsiklov uravneniya $dy/dx=P(x,y)/Q(x,y)$, gde $P$ i $Q$ – mnogochleny vtoroi stepeni”, Matem. sb., 37(79):2 (1955), 209–250 | MR | Zbl
[2] L. Khërmander, Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, Moskva, 1969
