循环群的动力强迫
Dynamical forcing of circular groups
摘要：
本文介绍并研究了动力强迫的概念。基本上，我们开发了一个工具包的技术，以产生的非线性提出的群体，只能作用于循环与某些规定的动力学性质。作为应用，我们证明了可由群强制的旋转数集X是一个无限生成的Q-模，包含可数无穷多个代数独立的超越数.我们还证明了，当G在所有可数群上变化时，群G的元素g在所有作用下的旋转数的集合所构成的圆的子集的集合，恰好是包含0的圆的闭子集的集合，并且在交换x和-x的对合下是不变的。作为另一个应用，我们构造了一个有限生成群，它忠实地作用在圆上，但不允许任何忠实的C^1作用，从而否定地回答了约翰·弗兰克斯的一个问题。
英文摘要：
In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties. As an application, we show that the set X of rotation numbers which can be forced by finitely presented groups is an infinitely generated Q-module, containing countably infinitely many algebraically independent transcendental numbers. We also show that the set of subsets of the circle which are the set of rotation numbers of an element g of a group G under all actions of G on a circle, as G varies over all countable groups, are exactly the set of closed subsets of the circle which contain 0, and are invariant under the involution which interchanges x and -x. As another application, we construct a finitely generated group which acts faithfully on the circle, but which does not admit any faithful C^1 action, thus answering in the negative a question of John Franks.
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