Conley\u2013Morse\u2013Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes

Marian Mrozek

doi:10.1007/s10208-016-9330-z

Abstract

We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. 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 combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.

Highlights

In the late 1990s of the twentieth century Forman [10] introduced the concept of a combinatorial vector field and presented a version of Morse theory for acyclic combinatorial vector fields
Kaczynski et al [16] defined the concept of an isolated invariant set and the Conley index in the case of a combinatorial vector field on the collection of simplices of a simplicial complex and observed that such a combinatorial field has a counterpart on the polytope of the simplicial complex in the form of a multivalued, upper semicontinuous, acyclic valued and homotopic to identity map
The presented theory shows that combinatorialization of dynamics, started by Forman’s paper [11], may be extended to cover such concepts as isolated invariant set, Conley index, attractor, repeller, attractor–repeller pair and Morse decomposition

Summary

Introduction

In the late 1990s of the twentieth century Forman [10] introduced the concept of a combinatorial vector field and presented a version of Morse theory for acyclic combinatorial vector fields. Conley theory [9] is a generalization of Morse theory to the setting of nonnecessarily gradient or gradient-like flows on locally compact metric spaces. In this theory the concepts of a non-degenerate critical point and its Morse index are replaced by the more general concept of an isolated invariant set and its Conley index. The Conley theory reduces to the Morse theory in the case of a flow on a smooth manifold defined by a smooth gradient vector field with non-degenerate critical points. Kaczynski et al [16] defined the concept of an isolated invariant set and the Conley index in the case of a combinatorial vector field on the collection of simplices of a simplicial complex and observed that such a combinatorial field has a counterpart on the polytope of the simplicial complex in the form of a multivalued, upper semicontinuous, acyclic valued and homotopic to identity map

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