arXiv subjects
Publications and source records attributed to Nataliya Goncharuk.
In this article we prove in a new way that a generic polynomial vector field in $\mathbb C^2$ possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.
We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let $f: \mathbb R/\mathbb Z \to \mathbb R/\mathbb Z$ be an orientation preserving circle diffeomorphism and let $ω\in \mathbb C/\mathbb Z$ be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus $\{z \in \mathbb C/\mathbb Z \mid 0< \Im(z)< \Im(ω)\}$ via the map $f+ω$. This complex torus is isomorphic to $\mathbb C/(\mathbb Z+τ \mathbb Z)$ for some appropriate $τ \in \mathbb C/\mathbb Z$. According to Moldavskis (2001), if the ordinary rotation number $\operatorname{rot} (f+ω_0)$ is Diophantine and if $ω$ tends to $ω_0$ non tangentially to the real axis, then $τ$ tends to $\operatorname{rot} (f+ω_0)$. We show that the Diophantine and non tangential assumptions are unnecessary: if $\operatorname{rot} (f+ω_0)$ is irrational then $τ$ tends to $\operatorname{rot} (f+ω_0)$ as $ω$ tends to $ω_0$. This, together with results of N.Goncharuk (2012), motivates us to introduce a new fractal set, given by the limit values of $τ$ as $ω$ tends to the real axis. For the rational values of $\operatorname{rot} (f+ω_0)$, these limits do not necessarily coincide with $\operatorname{rot} (f+ω_0)$ and form a countable number of analytic loops in the upper half-plane.