Persistence of Morse decompositions over grid resolution for maps and time series

TL;DR

This paper investigates how Morse decompositions of discretized maps and time series persist across different grid resolutions, analyzing their stability using various notions of persistence.

Contribution

It introduces a framework for studying the persistence of Morse decompositions over grid resolutions, combining graph, homology, and mixing perspectives.

Findings

Morse decompositions exhibit measurable persistence across resolutions

Persistence methods reveal stability of global and local dynamical structures

The approach applies to maps and time series in compact metric spaces

Abstract

We can approximate a continuous self-map $f$ of a compact metric space by discretizing the space into a grid. Through either the map itself or a time series, $f$ induces a multivalued grid map $\mathcal F$. The dynamical properties of $\mathcal F$ depend on the resolution of the grid, and we study the persistence of these properties as we change the resolution. In particular, we look at the persistence of Morse decompositions, at both the global (Morse graph) and local (individual Morse set) levels, using several notions of persistence -- graph structure, persistent homology, and mixing properties.