Renormalization and scaling of bubbles

The paper explores scaling properties of bubbles -- a complex analogue of Arnold tongues, associated to a one-dimensional family of analytic circle diffeomorphisms. Bubbles are smooth loops in the upper half-plane attached at all rational points of the real line. Results of a paper by X.~Buff and N.~Goncharuk (2015) show that the size of a $p/q$-bubble has order at most $q^{-2}$. In the current paper we improve this estimate by showing that the size of a $p/q$-bubble near a bounded-type irrational number $\alpha$ has order $d^{\xi(\alpha)} \cdot q^{-2}$, where $\xi(\alpha)>0$, and $d$ is the distance between $\alpha$ and $p/q$. Proofs are based on a renormalization technique. In particular, $\xi(\alpha)$ is related to the unstable and the top stable eigenvalues of the renormalization operator at the rotation by $\alpha$.