Topological Classification of Gradient-Like Flows with Surface Dynamics on 3-Manifolds

摘要

Recall that a flow f~t given on a closed smooth manifold M~n of dimension n is called a Morse-Smale flow if its nonwandering set consists of finitely many hyperbolic equilibrium states and closed trajectories and, moreover, the stable and unstable manifolds of different equilibrium states and closed trajectories either do not intersect or intersect transversally. A Morse-Smale flow without closed trajectories is said to be gradient-like. The Morse index of an equilibrium state p (a closed trajectory 7) is the number equal to the dimension of the corresponding unstable manifold W_p~u (W_γ~u). In what follows, we assume that n = 3 and the manifold M~3 is orientable. Given a gradient-like flow f~t on M~3, we denote the set of all of its equilibrium states by Ω_(f~t), the set of all equilibrium states with Morse index i ∈ {0,1,2,3} by Ω~i and the cardinality of the set Ω~i by |Ω~i|. The equilibrium states with Morse indices 0 and 3 are said to be nodal (respectively, sink and source), and those with indices 1 and 2 are said to be saddle. We set Ω = Ω~1 ∪ Ω~2.