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Journal of the London Mathematical Society 2005 72(3):689-716; doi:10.1112/S0024610705007015
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© The London Mathematical Society

Herman Rings and Arnold Disks

Xavier Buff, NúRia Fagella, Lukas Geyer and Christian Henriksen

Laboratoire Emile Picard, Université Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex, France buff{at}picard.ups-tlse.fr
Departimento de Matematica Aplicada i Analisi, Universitat de Barcelona Gran via 585, 08007 Barcelona, Spain e-mail: fagella{at}maia.ub.es
Department of Mathematics, Montana State University PO Box 172400, Bozeman, MT 59717-2400, USA geyer{at}math.montana.edu
Department of Mathematics, Technical University of Denmark Matematiktorvet, Building 303, DK-2800 Kgs. Lyngby, Denmark christian.henriksen{at}mat.dtu.dk

Received 19 May 2004. Revision received 8 March 2005.

For ({lambda},a)isin C* x C, let f{lambda},a be the rational map defined by f{lambda},a(z) = {lambda} z2 (az+1)/(z+a). If {alpha}isin R/Z is a Brjuno number, we let D{alpha} be the set of parameters ({lambda},a) such that f{lambda},a has a fixed Herman ring with rotation number {alpha} (we consider that (e2i{pi}{alpha},0)isin D{alpha}). Results obtained by McMullen and Sullivan imply that, for any gisin D{alpha}, the connected component of D{alpha}(C* x (C/{0,1})) that contains g is isomorphic to a punctured disk.

We show that there is a holomorphic injection F{alpha}:D->D{alpha} such that F{alpha}(0) = (e2i{pi} {alpha},0) and Formula, where r{alpha} is the conformal radius at 0 of the Siegel disk of the quadratic polynomial z↦ e2i{pi} {alpha}z(1+z).

As a consequence, we show that for aisin (0,1/3), if fl,a has a fixed Herman ring with rotation number {alpha} and if ma is the modulus of the Herman ring, then, as a->0, we have e{pi} ma=(r{alpha}/a) + O(a).

We finally explain how to adapt the results to the complex standard family z↦ {lambda} e(a/2)(z-1/z).


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