For a circle diffeomorphism $f$ and a complex number $\omega$ in the upper half-plane, one can define the complex rotation number of $f+\omega$ by computing the modulus of a certain torus. This construction is related to the regular rotation number of $f$ and produces a beautiful self-similar set (bubbles) related to Arnold's tongues.

No previous knowledge is required;
I will outline all main ideas and results in circle dynamics and complex dynamics related to the topic.

Partially based on the joint work with X.Buff.