论文标题
环形图的分析线性化
Analytic linearization of conformal maps of the annulus
论文作者
论文摘要
我们考虑在$ \ mathbb r/\ mathbb z $ in $ \ mathbb c/\ mathbb z $中定义的圆形图。 E. Risler证明，在此类地图的通用分析家族中，包含Brjuno旋转$ f_0（z）= z+α$的$f_ζ$，所有与此旋转都偶联的地图形成了codimension-1分析性亚法，接近$ f_0 $。 在本文中，我们获得了Risler的结果，作为以下构造的推论。我们在$ \ mathbb r/\ mathbb z $的附近介绍了一个重新归一化的操作员。我们证明该操作员是双曲线的，一个不稳定的方向与翻译相对应。我们进一步使用了全体形状运动的论点和Yoccoz的定理，以表明其稳定的叶面由与旋转偶联的差异性组成。
We consider holomorphic maps defined in an annulus around $\mathbb R/\mathbb Z$ in $\mathbb C/\mathbb Z$. E. Risler proved that in a generic analytic family of such maps $f_ζ$ that contains a Brjuno rotation $f_0(z)=z+α$, all maps that are conjugate to this rotation form a codimension-1 analytic submanifold near $f_0$. In this paper, we obtain the Risler's result as a corollary of the following construction. We introduce a renormalization operator on the space of univalent maps in a neighborhood of $\mathbb R/\mathbb Z$. We prove that this operator is hyperbolic, with one unstable direction corresponding to translations. We further use a holomorphic motions argument and Yoccoz's theorem to show that its stable foliation consists of diffeomorphisms that are conjugate to rotations.
