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Proceedings of the American Mathematical Society

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On the Blaschke circle diffeomorphisms

Author: Haifeng Chu
Journal: Proc. Amer. Math. Soc. 143 (2015), 1169-1182
MSC (2010): Primary 37F50; Secondary 37F10
Published electronically: October 22, 2014
MathSciNet review: 3293732
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Abstract: We study the analytic linearizability of a special family of analytic circle diffeomorphisms defined by

$\displaystyle B_{t,a,d}(z)=e^{2\pi it}z^{d+1}\left (\dfrac {z+a}{1+az}\right )^d$

with $ t,a\in \mathbb{R},\ d\in \mathbb{N},\ $$ \text {and}\ a>2d+1.$ Using the quasiconformal surgery procedure we prove that: If $ B_{t,a,d}$ is analytically linearizable, then the rational map $ B_{t,a,d}$ has a fixed Herman ring with Brjuno type rotation number. Conversely, for any Brjuno number $ \alpha $, we can find a rational map $ B_{t,a,d}$ with $ t,a\in \mathbb{R},\ d\in \mathbb{N},\ $$ \text {and}\ a>2d+1,$ such that $ B_{t,a,d}\vert _{S^1}$ has rotation number $ \rho (B_{t,a,d}\vert _{S^1})=\alpha $ and is analytically linearizable. These present a ``bigger family'' for the prototype of the local linearization theorem of the analytic circle diffeomorphisms.

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Additional Information

Haifeng Chu
Affiliation: School of Mathematics, Northwest University, Xi’an Shaanxi 710100, People’s Republic of China

Received by editor(s): January 27, 2013
Received by editor(s) in revised form: June 23, 2013
Published electronically: October 22, 2014
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society