Document type
ArticleVersion
Published versionPublication date
All rights reserved
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/164372
Surgery on Herman rings of the complex standard family
Journal Title
Authors
Director/Tutor
Journal ISSN
Volume Title
Related resource
Abstract
We consider the standard family (or Arnold family) of circle maps given by f_{\alpha, \beta}(x)=x + \alpha + \beta \sin(x) \pmod{2\pi}, for x,\alpha\in [0,2\pi), \beta \in (0,1) and its complexification F_{\alpha,\beta}(z)=z e^{i\alpha} \exp [\frac12\beta(z-\frac{1}{z})]. If f_{\alpha,\beta} is analytically linearizable, there is a Herman ring around the unit circle in the dynamical plane of F_{\alpha,\beta}. Given an irrational rotation number \theta, the parameters (\alpha,\beta) such that f_{\alpha, \beta} has rotation number \theta form a curve T_\theta in the parameter plane. Using quasi-conformal surgery of the simplest type, we show that if \theta is a Brjuno number, the curve T_\theta can be parametrized real-analytically by the modulus of the Herman ring, from \beta=0 up to a point (\alpha_0,\beta_0) with \beta_0 \leq 1, for which the Herman ring collapses. Using a result of Herman and a construction in I. N. Baker and P. Domínguez (Complex Variables37 (1998), 67-98) we show that for a certain set of angles \theta \in \mathcal{B} \setminus \mathcal{H}, the point \beta_0 is strictly less than 1 and, moreover, the boundary of the Herman rings with the corresponding rotation number have two connected components which are quasi-circles, and do not contain any critical point. For rotation numbers of constant type, the boundary consists of two quasi-circles, each containing one of the two critical points of F_{\alpha, \beta}.
Subject (English)
Citation
Citation
GEYER, Lukas and FAGELLA RABIONET, Núria. Surgery on Herman rings of the complex standard family. Ergodic Theory and Dynamical Systems. 2003. Vol. 23, num. 2, pags. 493-508. ISSN 0143-3857. [consulted: 14 of June of 2026]. Available at: https://hdl.handle.net/2445/164372