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 S1 into itself. Let fλ = λf , and |λ| ≤ 1. If |λ| = 1, then fλ is still an analytic diffeomorphism of a circle into itself; let ρ(λ) be its rotation number. If |λ| < 1, then an elliptic curve occurs as a factor space of an action of fλ . Let μ(λ) be the "multiplicative modulus" of this elliptic curve. A "moduli map" of unit discs λ → μ(λ) 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 S1 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 ρ(λ) 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.