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 $\mathrm{rot}(f+{\omega }_0)$ is Diophantine and if $\omega$ tends to ${\omega }_0$ non tangentially to the real axis, then $\tau$ tends to $\mathrm{rot}(f+{\omega }_0)$. We show that the Diophantine and non tangential assumptions are unnecessary: if $\mathrm{rot}(f+{\omega }_0)$ is irrational then $\tau$ tends to $\mathrm{rot}(f+{\omega }_0)$ 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 $\mathrm{rot}(f+{\omega }_0)$, these limits do not necessarily coincide with $\mathrm{rot}(f+{\omega }_0)$ and form a countable number of analytic loops in the upper half-plane.

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