论文标题
关于内部功能的正向迭代的注释
A note on forward iteration of inner functions
论文作者
论文摘要
全体形态动力学中的一个众所周知的问题是获得Denjoy-Wolff型结果，用于单位光盘的自图组成。在这里，我们解决了内部函数的特定情况：如果$ f_n：\ mathbb {d} \ to \ mathbb {d} $是固定原点的内部函数，我们表明$ f_n \ circ \ cdots \ circ \ cird circ f_1 $ circ f_1 $是常数或内部函数。对于Blaschke产品的特殊情况，我们证明了类似的结果，此外，将某些条件施加在收敛速度的情况下，可以保证边界扩展的$ l^1 $收敛性。我们给出一个反例，表明在没有这些额外条件的情况下，边界扩展可能在$ \ partial \ mathbb {d} $的所有点上差异。
A well-known problem in holomorphic dynamics is to obtain Denjoy--Wolff-type results for compositions of self-maps of the unit disc. Here, we tackle the particular case of inner functions: if $f_n:\mathbb{D}\to\mathbb{D}$ are inner functions fixing the origin, we show that a limit function of $f_n\circ\cdots\circ f_1$ is either constant or an inner function. For the special case of Blaschke products, we prove a similar result and show, furthermore, that imposing certain conditions on the speed of convergence guarantees $L^1$ convergence of the boundary extensions. We give a counterexample showing that, without these extra conditions, the boundary extensions may diverge at all points of $\partial\mathbb{D}$.
