Given an analytic circle diffeomorphism f ∶ ℝ/ℤ → ℝ/ℤ and a complex number w, ℑw > 0, consider the quotient space of the annulus 0 < ℑz < ℑw, z ∈ ℂ/ℤ, by the action of f + w. This quotient space is a torus, and we can ask about its modulus. This modulus is called the complex rotation number of f + w.

Limit values of the complex rotation number on ℝ/ℤ form a bubbly picture in the upper half-plane: infinitely many bubbles (analytic curves) grow from rational points of the real axis. Bubbles are complex analogue to Arnold tongues. In the talk, I'll give a survey of old and new results on the shapes of bubbles, with some proofs, and list open questions. (Mostly based on the joint work with Xavier Buff)