Linking combinatorial and classical dynamics: Conley index and Morse decompositions

TL;DR

This paper establishes a rigorous connection between combinatorial and classical dynamical systems by demonstrating that combinatorial systems induce multivalued dynamical systems with equivalent invariant sets and decompositions.

Contribution

It proves that combinatorial dynamical systems on simplicial complexes correspond directly to multivalued systems with matching invariant structures and decompositions.

Findings

Isolated invariant sets correspond between systems

Conley indices are preserved across the systems

Morse decompositions are in one-to-one correspondence

Abstract

We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions, and Conley-Morse graphs of the two dynamical systems are in one-to-one correspondence.