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)