Complex rotation numbers

TL;DR

This paper studies the behavior of complex rotation numbers associated with circle diffeomorphisms, showing convergence properties without Diophantine conditions and introducing a new fractal set based on limit values.

Contribution

It proves that the convergence of complex rotation numbers occurs for all irrational rotation numbers without Diophantine assumptions and introduces a new fractal set from these limit values.

Findings

Convergence of complex rotation numbers for all irrational rotation numbers.

Introduction of a new fractal set based on limit values of τ.

Limit values for rational rotation numbers form analytic loops.

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.