Čech-Delaunay gradient flow and homology inference for self-maps (Q2221301)

The goal of the paper under review is to leverage persistent homology tools to study dynamical systems. More specifically, the authors are concerned with computational aspects of estimating the homology of a self-map from a finite sample. Let \(M\) be a compact subset of \(\mathbb{R}^n\) and \(f\colon M\to M\) a continuous self-map with finite Lipschitz constant. A \textit{sample} is a finite set \(X\subseteq M\) along with a self map \(g\colon X\to X\) and a real number \(\rho\) such that \(\|f(x)-g(x)\|\leq \rho\) for all \(x\in X\). The authors improve upon their previous work [\textit{H. Edelsbrunner} et al., Found. Comput. Math. 15, No. 5, 1213--1244 (2015; Zbl 1330.55009)] by working with Delaunay complexes as opposed to the Čech complex. They define chain maps between Delaunay complexes, using the Čech complex as an intermediary. The authors give inference results that depend only on the self-map and the domain. They then analyze their algorithms and present the results of their computational experiments. These are compared with the results of previous algorithms developed by the authors. The paper concludes with a discussion and open questions.