Creating semiflows on simplicial complexes from combinatorial vector fields

Abstract

Combinatorial vector fields on simplicial complexes introduced by Robin Forman constitute a combinatorial analogue of classical flows. They have found numerous and varied applications in recent years. Yet, their formal relationship to classical dynamical systems has been less clear. In this paper we prove that for every combinatorial vector field on a finite simplicial complex X one can construct a semiflow on the underlying polytope X which exhibits the same dynamics. 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.