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Transactions of the American Mathematical Society

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Siegel discs, Herman rings and the Arnold family

Author: Lukas Geyer
Journal: Trans. Amer. Math. Soc. 353 (2001), 3661-3683
MSC (2000): Primary 30D05; Secondary 58F03, 58F08
Published electronically: April 24, 2001
MathSciNet review: 1837254
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We show that the rotation number of an analytically linearizable element of the Arnold family $f_{a,b}(x)=x+a+b\sin(2\pi x)\pmod 1$, $a,b\in{\mathbb R}$, $0<b<1/(2\pi)$, satisfies the Brjuno condition. Conversely, for every Brjuno rotation number there exists an analytically linearizable element of the Arnold family. Along the way we prove the necessity of the Brjuno condition for linearizability of $P_{\lambda,d}(z)=\lambda z(1+z/d)^d$ and $E_\lambda(z)=\lambda z e^z$, $\lambda=e^{2\pi i\alpha}$, at 0. We also investigate the complex Arnold family and classify its possible Fatou components. Finally, we show that the Siegel discs of $P_{\lambda,d}$ and $E_\lambda$ are quasidiscs with a critical point on the boundary if the rotation number is of constant type.

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Additional Information

Lukas Geyer
Affiliation: Universität Dortmund, FB Mathematik, LS IX, 44221 Dortmund, Germany

Keywords: Arnold family, standard family, linearization, Herman rings, circle diffeomorphisms
Received by editor(s): December 18, 1998
Received by editor(s) in revised form: December 12, 1999
Published electronically: April 24, 2001
Additional Notes: The author wishes to thank “Studienstiftung des deutschen Volkes” and DAAD for financial support.
Article copyright: © Copyright 2001 American Mathematical Society