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<p>Up to topological equivalence, a continuous flow in a surface is determined by its separatrix configuration. This claim is widely known as Markus–Neumann theorem, but it is false with the definition of separatrix originally considered. Recently, Espín Buendía and Jiménez López delivered counterexamples and proposed an updated definition of separatrix in order that the claim becomes true. The main objective of this paper is to provide a detailed and self-contained proof of Markus–Neumann theorem taking into account this new definition of separatrices, with a slight generalization.</p>

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Revisiting Markus–Neumann theorem

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Francisco Braun,
  • Rodrigo Thomaz

摘要

Up to topological equivalence, a continuous flow in a surface is determined by its separatrix configuration. This claim is widely known as Markus–Neumann theorem, but it is false with the definition of separatrix originally considered. Recently, Espín Buendía and Jiménez López delivered counterexamples and proposed an updated definition of separatrix in order that the claim becomes true. The main objective of this paper is to provide a detailed and self-contained proof of Markus–Neumann theorem taking into account this new definition of separatrices, with a slight generalization.