Dynamical Resistance Day

Title: Complex rotation numbers (video)

Given a complex number w, Im w>0, and an analytic circle diffeomorphism f, consider a complex torus which is a quotient space of a strip in C/Z by the action of f+w. Its modulus is called the complex rotation number of f+w. As w tends to the real line, limit values of the complex rotation number of f+w form a fractal set “bubbles”.

“Bubbles” are closely related to rotation numbers of circle diffeomorphisms and to Arnold tongues. I will talk about geometrical structure of “bubbles”, their shapes and self-similarity, and some open questions.