Smale flows on the three-sphere
Smale flows on the three-sphere
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DOI：
10.1090/s0002-9947-1987-0896023-7
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发表时间：
1987
影响因子：
1.3
通讯作者：
K. D. Rezende
中科院分区：
文献类型：
--
作者：
K. D. Rezende
本文给出了S3上Smale流的一个完全分类。这种分类是通过建立一个简洁的一组属性，必须满足的（抽象）李雅普诺夫图相关联的Smale流和李雅普诺夫函数。我们证明了这些性质是必要的，即给定一个Smale流和一个李雅普诺夫函数，它的李雅普诺夫图满足这组性质。我们还证明了这些性质是充分的，即给定一个满足这组性质的抽象李雅普诺夫图L'，在拓扑等价的情况下，可以实现S3上的Smale流，它的李雅普诺夫图为图L，其中L等于Lt.在证明施加在图上的条件是必要的技术涉及到一些使用的同源性理论。几何方法被用来构造与给定图相关联的S3上的流，从而建立上述条件的充分性。本文的主要定理推广了Franks [8]对S3上的非奇异Smale流进行分类的一个结果。
In this paper, a complete classification of Smale flows on S3 is obtained. This classification is presented by means of establishing a concise set of properties that must be satisfied by an (abstract) Lyapunov graph associated to a Smale flow and a Lyapunov function. We show that these properties are necessary, that is, given a Smale flow and a Lyapunov function, its Lyapunov graph satisfies this set of properties. We also show that these properties are sufficient, that is, given an abstract Lyapunov graph L' satisfying this set of properties, it is possible to realize a Smale flow on S3 that has a graph L as its Lyapunov graph where L is equal to Lt up to topological equivalence. The techniques employed in proving that the conditions imposed on the graph are necessary involve some use of homology theory. Geometrical methods are used to construct the flow on S3 associated to the given graph and therefore establish the sufficiency of the above conditions. The main theorem in this paper generalizes a result of Franks [8] who classified nonsingular Smale flows on S3.