We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspace of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for \v{C}ech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive \v{C}ech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.

中文翻译：

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\v{C}ech-Delaunay 梯度流和自映射的同源性推断

我们将通过一组离散的点值对显示自身的连续自映射称为采样动态系统。通过 Delaunay 复合体上的链图捕获可用信息，我们使用持久同源性来量化重复行为的证据。我们建立了一个采样定理来恢复由自映射引起的同源性内同态的特征空间。使用由 \v{C}ech 和 Delaunay 复合物的离散莫尔斯理论产生的组合梯度流，我们构建了一个链图，将问题从自然但昂贵的 \v{C}ech 复合物转换为计算效率高的 Delaunay 三角剖分。快速链映射算法具有超越动力系统的应用。