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 $-1<\operatorname{Re}\zeta<1$ 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.

Bibliographie

[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

[3] R. Ganning, X. Rossi, Analiticheskie funktsii mnogikh kompleksnykh peremennykh, Mir, Moskva, 1969 | MR

[4] O. Forster, K. J. Ramspott, “Analytisch Modulgarben und Endromisbtindel”, Invent. Math., 2:2 (1966), 145–170 | DOI | MR | Zbl

[5] L. S. Pontryagin, Obyknovennye differentsialnye uravneniya, Fizmatgiz, Moskva, 1961

[6] J. Wermer, “The hull on a curve in $C^n$”, Ann. Math., 68:3 (1958), 550–501 | DOI | MR