Three-Color Graph of the Morse Flow on a Compact Surface with Boundary
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THREE-COLOR GRAPH OF THE MORSE FLOW ON A COMPACT SURFACE WITH BOUNDARY A. O. Prishlyak1 and A. A. Prus2;3
UDC 517.91
We consider Morse flows on compact surfaces with boundary and construct a complete topological invariant of these flows (an equipped three-color graph). The vertices of this graph correspond to standard domains on the surface in the form of curvilinear triangles or quadrangles. We establish conditions under which this three-color graph specifies a Morse flow. We find the number of topologically nonequivalent Morse flows with at most five standard domains on the surfaces with boundary.
1. Introduction On each closed manifold, a vector field always generates a flow. In the case of a compact manifold with boundary, a vector field generates a flow if and only if it touches the boundary at every point of this boundary [1]. A vector field X on a manifold M is called structurally stable if, in the set of all vector fields on the manifold M; there exists a neighborhood U such that an arbitrary field Y 2 U is topologically equivalent to the field X [2]. On the closed surfaces, structurally stable vector fields are Morse–Smale fields. For manifolds of higher dimensions, parallel with Morse–Smale fields, there exist other structurally stable vector fields. An analog of the Morse–Smale fields for manifolds with boundary was described in [3, 4]. The idea of classification of the two-dimensional flows belongs to Andronov and Pontryagin [5]. A topological classification of the Morse–Smale vector fields on closed surfaces was proposed by Peixoto [6] and Oshemkov and Sharko [7]. For fields with certain restrictions on three-dimensional manifolds, the corresponding classification was developed by Umanskii [8] and Prishlyak [9]. A topological classification of m-fields on two- and threedimensional manifolds with boundary can be found in [10]. The structural stability of Morse–Smale fields on closed manifolds was studied by Peixoto [6, 11], Robinson, and Percell. Under the Palis–Smale hypothesis [12], the sufficient conditions for the structural stability on closed manifolds were obtained by Robbin [13] and Robinson [3]. Later, Mane [14] completed the proof of the necessary condition for the C 1 structural stability. The trajectory equivalence of the optimal Morse flows on closed surfaces was investigated in [15]. In the present paper, we consider Morse flows on surfaces with boundary. By analogy with the flows on closed surfaces, the Morse flows form everywhere dense sets on the surfaces with boundary [4]. Among the flows for which the set of nonwandering points consists of finitely many trajectories, only Morse flows are structurally stable. The aim of the present paper is to use three-color graphs for the investigation of the topological properties of Morse flows on compact surfaces with boundary. 1 Shevchenko
Kyiv National University, Hlushkov Ave., 4e, Kyiv, 03127, Ukraine; e-mail: [email protected]. Kyiv National University, Hlushkov Ave., 4e, Kyiv, 03127, Ukraine; e-mail: [email protected]. 3 Corresponding
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