Given a complex number $\omega$, $\Im\omega>0$, consider the annulus $A\subset\mathbb C/\mathbb Z$ bounded by $\mathbb R/\mathbb Z$ and $\mathbb R/\mathbb Z+\omega$. Take an analytic circle diffeomorphism $f$, and consider the quotient space $A / (f+\omega)$.

This quotient is a complex torus. Its modulus is called the complex rotation number of $f+\omega$. As $\omega \to \mathbb R$, the annulus $A$ degenerates; the boundary values of the complex rotation number form a fractal set called “bubbles”. Bubbles is a complex analogue of Arnold tongues.

I will outline what is known about bubbles (the results of E.Risler, V.Moldavskii, Yu.Ilyashenko, X.Buff, and myself), then focus on recent results: self-similarity of bubbles near rational points, and the question on continuous dependence of bubbles on the diffeomorphism $f$.