A new fractal set of “complexified Arnold’s tongues” will be discribed.
It occurs in the following way. Consider an analytic diffeomorphism $f$ of
a unit circle $S^1$ into itself. Let $f_\lambda = \lambda f$ , and $|\lambda| \leq 1$. If $|\lambda| = 1$, then $f_\lambda$ is
still an analytic diffeomorphism of a circle into itself; let $\rho(\lambda)$ be its rotation
number. If $|\lambda| < 1$, then an elliptic curve occurs as a factor space of an
action of $f_\lambda$ . Let $\mu(\lambda)$ be the “multiplicative modulus” of this elliptic curve.
A “moduli map” of unit discs $\lambda \to \mu(\lambda)$ occurs. The problem is to describe
its limit values.
It appears that the boundary values of the moduli map form a fractal set:
a union of $S^1$ and a countable number of “bubbles” adjacent to all the roots
of unity from inside $S^1$ . Relations of these limit values and rotation numbers
$\rho(\lambda)$ will be described.
These results are motivated by problems stated by Arnold and Yoccoz,
and are due to Risler, Moldavskis, Buff, Goncharuk and the speaker. Some
open problems will be stated.