Complex rotation numbers and bubbles

Nataliya Goncharuk (Texas A&M University, EUA)

The notion of a complex rotation number for an analytic circle dieomorphism was suggested by V.Arnold in 1978. The relation to usual rotation numbers was later established in a sequence of works by E.Risler, V. Moldavski, Yu. Ilyashenko, X.Bu, and myself. I will give a survey of results, applications, and open questions on complex rotation numbers and the related fractal-like set – bubbles.