A remark on a paper by Floyd
A remark on a paper by Floyd
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弗洛伊德在一篇论文中的评论
DOI：
10.1007/978-1-4613-9611-6_11
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发表时间：
1988
期刊：
影响因子：
--
通讯作者：
P. Tukia
中科院分区：
文献类型：
--
作者：
P. Tukia
设G和H为Rn= Rn U {oo}的莫比乌斯变换的两个离散群，并设~:G--t H为同构。几何上可以实现到什么程度的问题一直是许多研究的主题。其中许多都是基于这样的观察：在某些情况下，总是存在一个由极限集导出的映射 f: L (G)--t L (H)。也就是说，我们有（1）！ g (x)=~(g) f (x) 对于所有 x E L (G) 和 g E G。例如，如果 G 和 H 在几何上是有限的，并且如果 ~ 将 G 的抛物线元素双射到 H 的抛物线元素上，那么就知道存在同胚！：L (G)--t L (H) 满足 (1)、d.[6、定理 3.3]（或 [2]，这也意味着这一点，我们将在下面看到）。本注释的目的是描述如果省略抛物线元素的条件会发生什么（这实际上是秩为 1 的抛物线元素的条件，因为出于代数原因保留了秩 k> 1 的抛物线元素 [6，引理 3.2]）。事实证明，总是存在这样的极限集映射，但如果不满足抛物线元素的条件，则该映射在稠密集上是不连续的（但在抛物线不动点之外是连续的）。我们举一个例子来描述这种情况，它似乎呈现出一些有趣和不寻常的特征。
Let G and H be two discrete groups of Mobius transformations of Rn= Rn U {oo} and let~: G--t H be an isomorphism. The question to what extent~ can be realized geometrically has been the subject of many studies. Many of these have been based on the observation that under certain circumstances there is always a map f: L (G)--t L (H) of the limit sets inducing~. That is, we have (1)! g (x)=~(g) f (x) for all x E L (G) and g E G. For instance, if G and H are geometrically finite, and if~ carries parabolic elements of G bijectively onto parabolic elements of H, then one knows that there is a homeomorphism!: L (G)--t L (H) satisfying (1), d.[6, Theorem 3.3](or [2] which also implies this as we will see below). The purpose of this note is to describe what happens if the condition on parabolic elements is omitted (which is actually a condition on parabolic elements of rank 1 since parabolic elements of rank k> 1 are preserved for algebraic reasons [6, Lemma 3.2]).It turns out that there always is such a map of the limit sets but if the condition on parabolic elements is not satisfied, then this map is noncontinuous at a dense set (but continuous outside parabolic fixed points). We give an example to describe this situation which seems to present some interesting and unusual features.