Towards a formal tie between combinatorial and classical vector field dynamics
Tomasz Kaczyński, Marian Mrożek, Thomas Wanner · Journal of Computational Dynamics · 2016
Forman's combinatorial vector fields on simplicial complexesare a discrete analogue of classical flows generated by dynamicalsystems. Over the last decade, many notions from dynamical systemstheory have found analogues in this combinatorial setting, such asfor example discrete gradient flows and Forman's discrete Morsetheory. So far, however, there is no formal tie between the twotheories, and it is not immediately clear what the precise relationbetween the combinatorial and the classical setting is. The goal ofthe present paper is to establish such a formal tie on the levelof the induced dynamics. Following Forman's paper [6], wework with possibly non-gradient combinatorial vector fields onfinite simplicial complexes, and construct a flow-like uppersemi-continuous acyclic-valued mapping on the underlying topologicalspace whose dynamics is equivalent to the dynamics of Forman'scombinatorial vector field on the level of isolated invariantsets and isolating blocks.