AI 中文总结
本文推广了自共形系统指数分离的可检验条件，证明前 Schwarz 导数分离蕴含强指数分离，并在低维及 Jordan 域上建立典型性，解决维数下降猜想。
AI 中文摘要
在实直线上，Bárány、Kolossváry 和 Troscheit [5] 通过解析函数空间上的对偶迭代函数系统，使解析自共形系统的指数分离变得可检验。他们的条件所分离的量是前 Schwarz 导数，这一观察使我们能够将该构造推广到任意维数的共形迭代函数系统，其中前 Schwarz 导数为 $T_f=\nabla\log\\|Df\\|$。前 Schwarz 余循环的点态分离蕴含强指数分离条件，而在维数一和二中，Schwarz 余循环的分离在模 Möbius 变换的意义下蕴含相同条件。每个假设都等价于一个均匀间隙条件，并且这些条件是 $\mathcal{C}^2$-开的。在直线上它们也是稠密的，因此两个分离条件在 $\mathcal{C}^2$-意义下都是典型的；这加强了 Bárány、Kolossváry 和 Troscheit 的典型性定理，他们的开稠密集仅承载普通条件。在平面中，它们在具有单连通延拓域的 Jordan 域上是稠密的，并且两个分离条件在 $\mathcal{C}^2$-意义下也是典型的。在维数至少为三时，每个共形映射都是 Möbius 变换，因此没有系统满足模 Möbius 变换的条件，前 Schwarz 假设变为极点分离条件，当生成元强烈收缩且远离相似变换时是稠密的，但一般情况下不稠密。在 Jordan 域上，平面典型性和 Feng 与 Rapaport [13] 的一个定理解决了具有单射生成元的 $\mathcal{C}^2$-典型平面系统的维数下降猜想。
英文摘要
On the real line, Bárány, Kolossváry, and Troscheit [5] made exponential separation of analytic self-conformal systems checkable through a dual iterated function system on a space of analytic functions. The quantity their condition separates is the pre-Schwarzian derivative, and this observation lets us carry the construction to conformal iterated function systems in every dimension, where the pre-Schwarzian is $T_f=\nabla\log\|Df\|$. A pointwise separation of the pre-Schwarzian cocycle implies the strong exponential separation condition, and in dimensions one and two separation of the Schwarzian cocycle implies the same condition modulo Möbius maps. Each hypothesis is equivalent to a uniform gap condition, and these are $\mathcal{C}^2$-open. On the line they are also dense, so both separation conditions are $\mathcal{C}^2$-generic there; this sharpens the genericity theorem of Bárány, Kolossváry, and Troscheit, whose open and dense set carries only the plain condition. In the plane they are dense on Jordan domains with simply connected extension domains, and both separation conditions are $\mathcal{C}^2$-generic there too. In dimensions at least three every conformal map is Möbius, so no system satisfies the condition modulo Möbius maps, and the pre-Schwarzian hypothesis becomes a pole-separation condition, dense when the generators contract strongly enough and stay away from the similarities, but not dense in general. On Jordan domains, the planar genericity and a theorem of Feng and Rapaport [13] settle the dimension drop conjecture for $\mathcal{C}^2$-generic planar systems with injective generators.
Comments115 pages
