Complex rotation numbers

Abstract

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 $\omega \in \mathbb C/\mathbb Z$ be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus $\{z \in \mathbb C/\mathbb Z \mid 0< \Im(z)< \Im({\omega})\}$ via the map $f+{\omega}$. This complex torus is isomorphic to $\mathbb C/(\mathbb Z+{\tau} \mathbb Z)$ for some appropriate ${\tau} \in \mathbb C/\mathbb Z$. According to Moldavskis (2001), if the ordinary rotation number $\operatorname{rot} (f+\omega_0)$ is Diophantine and if ${\omega}$ tends to $\omega_0$ non tangentially to the real axis, then ${\tau}$ tends to $\operatorname{rot} (f+\omega_0)$. We show that the Diophantine and non tangential assumptions are unnecessary: if $\operatorname{rot} (f+\omega_0)$ is irrational then ${\tau}$ tends to $\operatorname{rot} (f+\omega_0)$ as ${\omega}$ tends to $\omega_0$. This, together with results of N.Goncharuk (2012), motivates us to introduce a new fractal set, given by the limit values of ${\tau}$ as ${\omega}$ tends to the real axis. For the rational values of $\operatorname{rot} (f+\omega_0)$, these limits do not necessarily coincide with $\operatorname{rot} (f+\omega_0)$ and form a countable number of analytic loops in the upper half-plane.