Renormalization and scaling of bubbles
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914565893128192 |
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| author | Goncharuk, Nataliya Gorbovickis, Igors |
| author_facet | Goncharuk, Nataliya Gorbovickis, Igors |
| contents | 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 $α$ has order $d^{ξ(α)} \cdot q^{-2}$, where $ξ(α)>0$, and $d$ is the distance between $α$ and $p/q$.
Proofs are based on a renormalization technique. In particular, $ξ(α)$ is related to the unstable and the top stable eigenvalues of the renormalization operator at the rotation by $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11308 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Renormalization and scaling of bubbles Goncharuk, Nataliya Gorbovickis, Igors Dynamical Systems 37E45, 37E20, 37E10 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 $α$ has order $d^{ξ(α)} \cdot q^{-2}$, where $ξ(α)>0$, and $d$ is the distance between $α$ and $p/q$. Proofs are based on a renormalization technique. In particular, $ξ(α)$ is related to the unstable and the top stable eigenvalues of the renormalization operator at the rotation by $α$. |
| title | Renormalization and scaling of bubbles |
| topic | Dynamical Systems 37E45, 37E20, 37E10 |
| url | https://arxiv.org/abs/2312.11308 |