Cycle breaking in fiberings on analytic curves

Yu. S. Ilyashenko

Sbornik. Mathematics, Tome 16 (1972) no. 1, pp. 60-68

Résumé

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 $-1\operatorname{Re}\zeta1$ and $D_1\subset D$ the strip $-1\operatorname{Re}\zeta0$. 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.