Topological Inference of the Conley Index

TL;DR

This paper introduces a framework for inferring the Conley index of isolated critical sets in dynamical systems from finite samples, enabling robust topological analysis and Morse index estimation.

Contribution

It develops a local index pair construction with positive reach, facilitating sampling-based inference of the Conley index and Morse index from finite data.

Findings

Constructs local index pairs with positive reach for critical sets

Provides a sampling theory for robust homology inference

Enables estimation of Morse index from finite evaluations

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.