非线性化芽的拟共形变形
Quasi-conformal deformations of nonlinearizable germs
摘要：
设$f(z) = e^{2πi α}z + O(z^2), α\in \mathbb{R}$是$\mathbb{C}$中全纯双同态的芽.对于$α$有理数和$f$的无限阶，拓扑共轭到$f$的芽的共形共轭类的空间由Ecalle-Voronin不变量参数化（特别是无限维的）.当$α$是无理数且$f$是非线性化的时，不知道$f$是否容许拟共形变形。证明了如果$f$有一列收敛到不动点的排斥周期轨道，则$f$嵌入到无穷维拟共形共轭芽族中，其中没有两个是共形共轭的.
英文摘要：
Let $f(z) = e^{2πi α}z + O(z^2), α\in \mathbb{R}$ be a germ of holomorphic diffeomorphism in $\mathbb{C}$. For $α$ rational and $f$ of infinite order, the space of conformal conjugacy classes of germs topologically conjugate to $f$ is parametrized by the Ecalle-Voronin invariants (and in particular is infinite-dimensional). When $α$ is irrational and $f$ is nonlinearizable it is not known whether $f$ admits quasi-conformal deformations. We show that if $f$ has a sequence of repelling periodic orbits converging to the fixed point then $f$ embeds into an infinite-dimensional family of quasi-conformally conjugate germs no two of which are conformally conjugate.
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