  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 [BuffGoncharuk2015] arXiv:1308.3510 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 $\alpha$.
