Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field plays a fundamental role, relating the geometric structure of phase space to a combinatorial object consisting of critical points and separatrices. Such concepts led Forman to a satisfactory theory of discrete vector fields, in close analogy to the continuous case. In this paper, we introduce discrete line fields. Again, our definition is rich enough to provide the counterparts of the basic results in the theory of continuous line fields: a Euler-Poincar\'e formula, a Morse--Smale decomposition and a topologically consistent cancellation of critical elements, which allows for topological simplification of the original discrete line field.

中文翻译：

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表面上的离散线场

矢量场和线场，它们在切线上没有方向的对应物，是动力系统理论中熟悉的对象。在他们研究中使用的技术中，（通用）场的 Morse--Smale 分解起着重要作用，它将相空间的几何结构与由临界点和分离点组成的组合对象相关联。这些概念导致 Forman 提出了一个令人满意的离散向量场理论，与连续情况非常相似。在本文中，我们引入了离散线场。同样，我们的定义足够丰富，可以提供连续线场理论中基本结果的对应物：Euler-Poincar\'e 公式、Morse--Smale 分解和关键元素的拓扑一致抵消，