Journal article
Herman Rings and Arnold Disks
For (λ,a)∈ C* × C, let fλ,a be the rational map defined by fλ,a(z) = λ z2 (az+1)/(z+a). If α∈ R/Z is a Brjuno number, we let Dα be the set of parameters (λ,a) such that fλ,a has a fixed Herman ring with rotation number α (we consider that (e2iπα,0)∈ Dα). Results obtained by McMullen and Sullivan imply that, for any g∈ Dα, the connected component of Dα(C* × (C/{0,1})) that contains g is isomorphic to a punctured disk.
We show that there is a holomorphic injection Fα:D→Dα such that Fα(0) = (e2iπ α,0) and F'()=(,rα), where rα is the conformal radius at 0 of the Siegel disk of the quadratic polynomial z↦ e2iπ αz(1+z). As a consequence, we show that for a∈ (0,1/3), if fl,a has a fixed Herman ring with rotation number α and if ma is the modulus of the Herman ring, then, as a→0, we have eπ ma=(rα/a) + O(a).
We finally explain how to adapt the results to the complex standard family z↦ λ\se(a/2)(z‐1/z).
Language: | English |
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Publisher: | Oxford University Press |
Year: | 2005 |
Pages: | 689-716 |
ISSN: | 14697750 and 00246107 |
Types: | Journal article |
DOI: | 10.1112/S0024610705007015 |
ORCIDs: | Henriksen, Christian |