Persistent Homology of Morse Decompositions in Combinatorial Dynamics

TL;DR

This paper explores the use of persistent homology to analyze Morse decompositions in combinatorial dynamical systems on simplicial complexes, aiming to validate dynamical reconstructions from sampled data.

Contribution

It introduces a framework extending classical persistence theory to finite topological spaces derived from sampled dynamics, focusing on Morse decompositions.

Findings

Homological persistence of Morse decompositions can validate reconstructed dynamics.

Experimental results demonstrate the effectiveness of the approach on numerical examples.

Abstract

We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics directly from the sample. We study the homological persistence of {\em Morse decompositions} of such systems, an important descriptor of the dynamics, as a tool for validating the reconstruction. Our framework can be viewed as a step toward extending the classical persistence theory to "vector cloud" data. We present experimental results on two numerical examples.