Let f be a germ of a holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in ${\mathbb C}$ (i.e. $f(0) = 0, f'(0) = e^{2\pi i \alpha }, \alpha \in {\mathbb R} - {\mathbb Q}$ ). Pérez-Marco [Fixed points and circle maps. Acta Math.179(2) (1997), 243–294] showed the existence of a unique continuous monotone one-parameter family of non-trivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families $(g_t)$ of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps $(g_t)$ is the orbit of a locally defined semigroup $(\Phi _t)$ on the space of analytic circle maps, which we show has a well-defined infinitesimal generator X. The explicit form of X is obtained by using the Loewner equation associated to the family of hulls $(K_t)$ . We show that the Loewner measures $(\mu _t)$ driving the equation are 2-conformal measures on the circle for the circle maps $(g_t)$ .

中文翻译：

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刺猬的 Loewner 进化和圆图的 2-保角测量

让F是全纯微分同胚的胚芽，在原点处有一个非理性无差异不动点${\mathbb C}$（IE$f(0) = 0, f'(0) = e^{2\pi i \alpha }, \alpha \in {\mathbb R} - {\mathbb Q}$）。Pérez-Marco [固定点和圆形地图。数学学报。179(2) (1997), 243-294] 表明存在一个独特的连续单调单参数族的非平凡不变全连续体，包含称为 Siegel compacta 的不动点，并给出了细菌和族之间的对应关系$(g_t)$通过将这些紧致的补码共形映射到单位圆盘的补码而获得的圆图。圆形地图家族$(g_t)$是一个局部定义的半群的轨道$(\披_t)$在解析圆图的空间上，我们展示了一个定义明确的无穷小生成器X. 的显式形式X通过使用与船体族相关的 Loewner 方程获得$(K_t)$. 我们证明了 Loewner 措施$(\亩_t)$驱动方程的是圆图的圆上的 2-conformal 度量$(g_t)$.