Please use this identifier to cite or link to this item: http://hdl.handle.net/2445/164372
 Title: Surgery on Herman rings of the complex standard family Author: Geyer, LukasFagella Rabionet, Núria Keywords: Sistemes dinàmics de baixa dimensióSistemes dinàmics complexosLow-dimensional dynamical systemsComplex dynamical systems Issue Date: Apr-2003 Publisher: Cambridge University Press 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}. Note: Reproducció del document publicat a: https://doi.org/10.1017/S0143385702001323 It is part of: Ergodic Theory and Dynamical Systems, 2003, vol. 23, num. 2, p. 493-508 URI: http://hdl.handle.net/2445/164372 Related resource: https://doi.org/10.1017/S0143385702001323 ISSN: 0143-3857 Appears in Collections: Articles publicats en revistes (Matemàtiques i Informàtica)

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