Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix.These matrices, a generalization of Morse boundary operators from classical Morse theory, capture the connections made by the flows among the critical structures - such as attractors, repellers, and orbits - in a vector field. Recently, in the context of combinatorial dynamics, an efficient persistence-like algorithm to compute connection matrices has been proposed in~\cite{DLMS24}. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of~\cite{DLMS24} and bringing the theory of persistence closer to combinatorial dynamical systems. We supplement this main result with an observation: the concept of persistence as defined for scalar fields naturally adapts to Morse decompositions whose Morse sets are filtered with a Lyapunov function. We conclude by presenting preliminary experimental results.

翻译：Morse分解将向量场中的流划分为等价结构。给定此类分解，可通过所谓的连接矩阵进一步定义其流结构的摘要。这些矩阵作为经典Morse理论中Morse边界算子的推广，能够捕捉向量场中临界结构（如吸引子、排斥子与轨道）之间通过流建立的连接关系。最近在组合动力学背景下，文献~\cite{DLMS24}提出了一种类持续性高效算法以计算连接矩阵。本文证明：实际上，采用穷举归约的经典持续性算法即可获取连接矩阵，这不仅简化了文献~\cite{DLMS24}的算法，还将持续性理论与组合动力系统更紧密地联系起来。我们通过一项观察补充此主要结论：为标量场定义的持续性概念可自然适用于其Morse集通过Lyapunov函数进行过滤的Morse分解。最后，我们展示了初步实验结果。