Topological Inference of the Conley Index

Ka Man Yim; Vidit Nanda

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arXiv · 2206.10198

Topological Inference of the Conley Index

Ka Man Yim·Vidit Nanda

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Abstract

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.

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BibTeX RIS

Ka Man Yim, Vidit Nanda. 2022-06-21. Topological Inference of the Conley Index. https://arxiv.org/abs/2206.10198

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