Herman Rings and Arnold Disks |
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Authors: | Buff, Xavier Fagella, NuRia Geyer, Lukas Henriksen, Christian |
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Affiliation: | 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 |
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Abstract: | For (,a) C* x C, let f,a be the rational map defined by f,a(z)= z2 (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f,a has a fixed Hermanring with rotation number (we consider that (e2i,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e2i ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e2i z(1+z). As a consequence, we show that for a (0,1/3), if fl,a has afixed Herman ring with rotation number and if ma is the modulusof the Herman ring, then, as a0, we have e ma=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e(a/2)(z-1/z). |
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