1. Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow
- Author
-
Alberto Abbondandolo and Pietro Majer
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Closed manifold ,stable foliation ,CW-decomposition ,gradient flow ,Morse-Smale flow ,Structure (category theory) ,Dynamical Systems (math.DS) ,Morse code ,Mathematics::Geometric Topology ,law.invention ,Flow (mathematics) ,law ,FOS: Mathematics ,Mathematics::Differential Geometry ,Geometry and Topology ,Mathematics - Dynamical Systems ,Invariant (mathematics) ,Balanced flow ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
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 $M$., 57 pages, 1 figure. The list of references has been expanded and the discussion about the history of the problem and future perspectives has been improved thanks to the suggestions of some readers
- Published
- 2021