Rotation Numbers and Moduli of Elliptic Curves

Given a circle diffeomorphism $f$, we can construct a map taking each real number $a$ to the rotation number of the diffeomorphism $f+a$. In 1978, V. I. Arnold suggested a complex analog To this map. Given a complex number $z$ with $\operatorname{Im}z>0$, Arnold used the map $f+z$ to construct an elliptic curve. The moduli map takes every number $z$ to the modulus $\mu(z)$ of this elliptic curve.
In this article, we investigate the limit behaviour of the map $\mu$ in neighborhoods of the real intervals on which the rotation number of the diffeomorphism $f+a$ is rational. We show that the map $\mu$ extends analytically to any interior point of such an interval, excluding some finite set of exceptional points. Near exceptional points and the endpoints of the interval, the values of the function $\mu$ tend to the rotation number of the map $f+a$.
The union of the images of such intervals under the map $\mu$ is a fractal set in the upper half-plane. This fractal set is a complex analog to Arnold tongues.