Abstract
We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial vector field give rise to isomorphic objects in the multivalued map case.
Highlights
In [19] we proved that for any combinatorial vector field on the collection of simplices of a simplicial complex, one can construct an acyclic-valued and upper semicontinuous map on the underlying geometric realization whose dynamics on the level of invariant sets exhibits the same complexity
This is what one would expect because the multivalued map we construct is modeled on a combinatorial analogue of a classical vector field giving rise to flow-type dynamics
In order to formulate the definition of the Morse decomposition of an isolated invariant set, we need the concepts of α- and ω-limit sets
Summary
In the years since Forman [14,15] introduced combinatorial vector fields on simplicial complexes, they have found numerous applications in such areas as visualization and mesh compression [21], graph braid groups [13], homology computation [17,25], Communicated by Shmuel Weinberger.
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