Stability of Morse-Smale vector fields on manifolds with boundary
Stability of Morse-Smale vector fields on manifolds with boundary
复制标题
有边界流形上Morse-Smale向量场的稳定性
DOI：
10.1016/0040-9383(90)90025-f
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发表时间：
1990
期刊：
影响因子：
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通讯作者：
M. Pacifico
中科院分区：
文献类型：
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作者：
R. Labarca;M. Pacifico
设A4是一个具有边界aM的紧致流形，用%^'（M，JM）表示M上的C'向量场空间，这些向量场与dM相切，具有通常的C '拓扑。在这个空间中，很自然地将结构稳定性定义为无边界的情况，即说XE a '（M，c？M）是C '结构稳定的，如果它有一个C'邻域θ使得每个YE@都拓扑等价于X，即存在一个同胚h：M，J将X的轨道映射到Y的轨道上并保持它们的时间方向.在无边界条件下，得到了结构稳定性的充分条件（遵循Palis-Smale [lo]的猜想）Robbin [12]和罗宾逊[133]以及最近Mafib [S]完成了C1结构稳定性这些条件必要性的证明。这项工作的目标是继续[3]、[4]和[6]的研究路线，其最终目的是找到5！'（M，8M）.在这里，我们将给出一个完整的答案，这个问题的向量场的非游荡集是简单的，即，由一个有限集的轨道。如[6]中所解释的，我们的结果也可以在Z，-等变向量场的稳定性的背景下解释。为了对我们的结果作出精确的说明，让我们首先介绍一些基本的符号和定义。
LET A4 be a compact manifold with boundary aM and denote by%^‘(M, JM) the space of C’vector fields on M, that are tangent to dM endowed with the usual C’topology. In this space it is natural to define structural stability as in the boundaryless case, namely saying that XE a’(M, c? M) is C’structurally stable if it has a C’neighborhood 9 such that every YE@ is topologically equivalent to X, ie, there exists a homeomorphism h: M, J mapping orbits of X onto orbits of Y and preserving their time orientation. In the boundaryless case, satisfactory sufficient conditions for structural stability have been obtained (following a conjecture of Palis-Smale [lo]) by Robbin [12] and Robinson [133 and recently Mafib [S] completed the proof of the necessity of these conditions for C1 structural stability.The objective of this work is to continue the line of research of [3],[4] and [6], whose final aim is to find a characterization of the structurally stable elements of 5!“‘(M, 8M). Here we shall give a complete answer to this question for vector fields whose nonwandering set is simple, ie, consisting of a finite set of orbits. As explained in [6], our results can also be interpreted in the context of stability of Z,-equivariant vector fields. In order to make precise statements about our results let us first introduce some basic notations and definitions.