Creating Semiflows on Simplicial Complexes from Combinatorial Vector Fields

TL;DR

This paper constructs semiflows on simplicial complexes from combinatorial vector fields, establishing a direct link to classical dynamical systems and enabling analysis via Conley-Morse graphs.

Contribution

It introduces a method to build semiflows on simplicial complexes that replicate the dynamics of combinatorial vector fields using purely combinatorial data.

Findings

Constructed a semiflow matching combinatorial vector field dynamics.

Established equivalence in Conley-Morse graph sense.

Used tiling of space to create isolating blocks from combinatorial info.

Abstract

Combinatorial vector fields on simplicial complexes as introduced by Robin Forman have found numerous and varied applications in recent years. Yet, their relationship to classical dynamical systems has been less clear. In recent work it was shown that for every combinatorial vector field on a finite simplicial complex one can construct a multivalued discrete-time dynamical system on the underlying polytope X which exhibits the same dynamics as the combinatorial flow in the sense of Conley index theory. However, Forman's original description of combinatorial flows appears to have been motivated more directly by the concept of flows, i.e., continuous-time dynamical systems. In this paper, it is shown that one can construct a semiflow on X which exhibits the same dynamics as the underlying combinatorial vector field. The equivalence of the dynamical behavior is established in the sense of Conley-Morse graphs and uses a tiling of the topological space X which makes it possible to directly construct isolating blocks for all involved isolated invariant sets based purely on the combinatorial information.