Dynamical Systems Seminar
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 bubbles, with some proofs, and list open questions.
(Mostly based on the joint work with X. Buff)