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Surgery on Herman rings of the complex standard family

Published online by Cambridge University Press:  22 September 2003


NÚRIA FAGELLA
Affiliation:
Universitat de Barcelona, Departamento de Matemàtica, Aplicada i Anàlisi, Gran Via 585, Barcelona, Spain (e-mail: fagella@maia.ub.es)
LUKAS GEYER
Affiliation:
Universität Dortmund, Fachbereich Mathematik Vogelpothsweg 87, D-44221 Dortmund, Germany (e-mail: geyer@math.uni-dortmund.de)

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}.


Type
Research Article
Copyright
2003 Cambridge University Press

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