Complex Rotation Numbers and Renormalization

The complex rotation number (suggested by V.Arnold) is an invariant related to the dynamics of an analytic circle diffeomorphism $f$. Complex rotation numbers give rise to a nice fractal set "bubbles" analogous to classical Arnold's tongues. 
I will give a survey on complex rotation numbers and list open problems. Also, I will explain how the renormalization operator makes the "bubbles" self-similar and controls their sizes.

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