Renormalization and scaling of bubbles
Institutional Repository Document
Overview
Research
View All
Overview
abstract
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 $alpha$.
