On the Blaschke circle diffeomorphisms

Chu, Haifeng

doi:10.1090/S0002-9939-2014-12359-8

On the Blaschke circle diffeomorphisms
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Proc. Amer. Math. Soc. 143 (2015), 1169-1182 Request permission

Abstract:

We study the analytic linearizability of a special family of analytic circle diffeomorphisms defined by \[ B_{t,a,d}(z)=e^{2\pi it}z^{d+1}\left (\dfrac {z+a}{1+az}\right )^d\] 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 $B_{t,a,d}|_{S^1}$ has rotation number $\rho (B_{t,a,d}|_{S^1})=\alpha$ 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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