2024/05/31 by J. J. Sánchez-Gabites, Sánchez-Gabites, J. J.
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Mathematical Dynamics and Fractals #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2405.20945
Consider a flow in ℝ3 and let K be the biggest invariant subset of some compact region of interest N ⊆ ℝ3. The set K is often not computable, but the way the flow crosses the boundary of N can provide indirect information about it. For example, classical tools such as Ważewski's principle or the Poincaré-Hopf theorem can be used to detect whether K is nonempty or contains rest points, respectively. We present a criterion that can establish whether K has a nontrivial homology by looking at the subset of the boundary of N along which the flow is tangent to N. We prove that the criterion is as sharp as possible with the information it uses as an input. We also show that it is algorithmically checkable.