Homeomorphisms of the Circle Without Periodic Points
Homeomorphisms of the Circle Without Periodic Points
复制标题
无周期点的圆的同胚
DOI：
10.1112/plms/s3-20.4.688
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发表时间：
1970
影响因子：
1.8
通讯作者：
N. Markley
中科院分区：
文献类型：
--
作者：
N. Markley
同胚的圆首先考虑庞加莱 * 谁使用他们获得定性结果一类微分方程的环面。他分类那些有密集的轨道表明，他们是拓扑等价于旋转通过一个角度ineradable与IT。然而，Denjoy表明，存在同胚的圆没有周期点，没有密集的轨道。这建立了存在一类同胚的循环没有周期点，这是不拓扑等价的旋转通过一个角度insiderable与TT。但从那时起，这门课基本上被忽视了。本文的目的是给出无周期点的圆的所有同胚的一个分类方案。我们的方法依赖于分析那些远离所有其他点的点。(Two点x和y是远端的，如果存在e > 0，使得对于所有整数n，d（<p（x），<p（y））> s，其中d是度量，<p是同胚。在旋转的情况下，每个点都远离每个其他点。另一方面，当不存在稠密轨道时，存在一个Cantor极小集和至少一对双渐近轨道。我们将看到这些离散流具有相当大的拓扑变化。例如，有无穷多个具有相同旋转数和正好n对双渐近轨道的拓扑不同的轨道。没有周期点的圆的同胚的分类与它们的极小集的分类相同，因为每个同胚都有唯一的极小集。这就导致了一个问题，即确定一个任意的最小集合何时来自于一个圆的同胚。在§4中，我们用动力学性质完全刻画了这些极小集。近缘关系是动态客体，在这一表征中起着关键作用。最后，我们在分类中使用的不变量包含关于同态、自同态和自同构的信息。这种性质的定理在§ 3中给出。
Homeomorphisms of the circle were first considered by Poincare* who used them to obtain qualitative results for a class of differential equations on the torus. He classified those which have a dense orbit by showing that they are topologically equivalent to a rotation through an angle incommensurable with IT. However, Denjoy showed that there exist homeomorphisms of the circle without periodic points and without dense orbits. This established the existence of a class of homeomorphisms of the circle without periodic points which are not topologically equivalent to rotations through an angle incommensurable with TT. But since then this class has been largely ignored. The purpose of this paper is to present a classification scheme for all homeomorphisms of the circle without periodic points. Our method depends upon analysing those points which are distal from all other points. (Two points x and y are distal if there exists e > 0 such that d(<p(x), <p(y)) > s for all integers n where d is a metric and <p is a homeomorphism.) In the case of a rotation every point is distal from every other point. On the other hand, when there is no dense orbit there is a Cantor minimal set and at least one pair of doubly asymptotic orbits. We will see that these discrete flows have considerable topological variety. For example, there are infinitely many topologically distinct ones with the same rotation number and exactly n pairs of doubly asymptotic orbits. The classification of homeomorphisms of the circle without periodic points is the same as the classification of their minimal sets, since each one has a unique minimal set. This leads to the problem of determining when an arbitrary minimal set comes from a homeomorphism of the circle. In §4 we characterize these minimal sets completely in terms of dynamical properties. The proximal relation is the dynamical object that plays the key role in this characterization. Finally, the invariants which we use in our classification contain information about homomorphisms, endomorphisms, and automorphisms. Theorems of this nature are presented in § 3.