Contractibility of a persistence map preimage

TL;DR

This paper investigates the preimage structure of the persistence map in topological data analysis, establishing conditions under which these preimages are contractible, aiding the understanding of dynamical systems from persistence diagram time series.

Contribution

It introduces a definition of persistence diagrams for points in \\mathbb{R}^N and provides conditions ensuring the contractibility of their preimages, linking topological features to dynamical system solutions.

Findings

Preimages of the persistence map are contractible under certain conditions.

Conditions for fixed points in dynamical systems based on persistence diagram time series.

A new framework for analyzing solutions of dynamical systems via topological data analysis.

Abstract

This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of solutions snapshots, what conclusions can be drawn about solutions of the original dynamical system? In this paper we provide a definition of a persistence diagram for a point in $\mathbb{R}^N$ modeled on piecewise monotone functions. We then provide conditions under which time series of persistence diagrams can be used to guarantee the existence of a fixed point of the flow on $\mathbb{R}^N$ that generates the time series. To obtain this result requires an understanding of the preimage of the persistence map. The main theorem of this paper gives conditions under which these preimages are contractible simplicial complexes.