{
  "id": "2206.10198",
  "version": "v1",
  "published": "2022-06-21T08:54:44.000Z",
  "updated": "2022-06-21T08:54:44.000Z",
  "title": "Topological Inference of the Conley Index",
  "authors": [
    "Ka Man Yim",
    "Vidit Nanda"
  ],
  "comment": "33 pages, comments welcome",
  "categories": [
    "math.DS",
    "math.AT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-21T08:54:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B30",
      "37B35",
      "55N31"
    ],
    "keywords": [
      "conley index",
      "topological inference",
      "sufficiently large finite point sample",
      "specific index pair",
      "robust homology inference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}