We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible invariant stable foliations” that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse–Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse–Smale gradient-like flow on a closed manifold $M$ are the open cells of a CW-decomposition of $M$.

中文翻译：

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Morse-Smale 梯度流诱导的稳定叶理和 CW 结构

我们证明闭合流形上的 Morse-Smale 梯度流具有“兼容不变稳定叶状结构系统”，类似于 Palis 和 Smale 在证明 Morse-Smale 微分同胚和流的结构稳定性时引入的对象，但具有更精细的规律性和几何特性。我们展示了如何使用这些不变的叶状结构，以便为众所周知但相当微妙的定理提供独立的证明，该定理指出封闭流形上莫尔斯-斯梅尔梯度流的不稳定流形$\mathrm{中号}$是 CW 分解的开单元$\mathrm{中号}$。