Connection matrices in combinatorial topological dynamics

TL;DR

This paper extends the concept of connection matrices within Conley's topological dynamics framework to combinatorial vector fields, providing a new tool for analyzing dynamical systems with enhanced flexibility.

Contribution

It introduces a generalized notion of connection matrices for poset filtered chain complexes and establishes their properties in combinatorial vector fields.

Findings

Connection matrices are unique for gradient combinatorial vector fields.

The new framework generalizes classical results to combinatorial settings.

Provides a method to analyze connecting orbits in combinatorial dynamical systems.

Abstract

Connection matrices are one of the central tools in Conley's approach to the study of dynamical systems, as they provide information on the existence of connecting orbits in Morse decompositions. They may be considered a generalisation of the boundary operator in the Morse complex in Morse theory. Their computability has recently been addressed by Harker, Mischaikow, and Spendlove in the context of lattice filtered chain complexes. In the current paper, we extend the recently introduced Conley theory for combinatorial vector and multivector fields on Lefschetz complexes by transferring the concept of connection matrix to this setting. This is accomplished by the notion of connection matrix for arbitrary poset filtered chain complexes, as well as an associated equivalence, which allows for changes in the underlying posets. We show that for the special case of gradient combinatorial vector fields in the sense of Forman, connection matrices are necessarily unique. Thus, the classical results of Reineck have a natural analogue in the combinatorial setting.