Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes

Marian Mrozek

Published 2015-05-29Version 1

Working in the algebraic setting of free chain complexes with a distinguished basis (Lefschetz complexes) we introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features with respect to the $T_0$ topology of Lefschetz complexes. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.