Topological inference of the Conley index

Journal article

The Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference. This allows us to estimate the Conley index, and as a direct consequence, we are also able to estimate the Morse index of any critical point of a Morse function using finitely many local evaluations.

Authors: Yim, KM; Nanda, V

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer Nature
• Journal: Journal of Dynamics and Differential Equations
• Volume: 37
• Issue: 2
• Pages: 1565–1597
• Publication date: 2023-09-23
• Acceptance date: 2023-08-28
• DOI: 10.1007/s10884-023-10310-1
• EISSN: 1572-9222
• ISSN: 1040-7294
• Language: English
• Keywords: manifold learning; homology inference; Conley index theory