Combinatorial Vector Field Dynamics

Forman's discrete Morse theory is an analogy of the classical Morse theory with informal ties numerically explored by computational geometry and visualization communities. In his 1998 paper on "Combinatorial vector fields and dynamical systems", Forman extends his discrete theory to non-gradient combinatorial vector fields with the aim of investigating periodic trajectories. However, his concept of V-paths des not suit analysis of asymptotic dynamics. We extend V-paths so to fill that gap and continue with dynamical attributes such as isolated invariant sets, index pairs, and Morse decomposition. The ultimate goal is to establish a formal tie between continuous and combinatorial vector fields on the level of dynamical systems. This is a joint work with M. Mrozek and Th. Wanner.