Creating semiflows on simplicial complexes from combinatorial vector fields

71. Marian Mrozek, Thomas Wanner:
    Creating semiflows on simplicial complexes from combinatorial vector fields
    Journal of Differential Equations 304, pp. 375-434, 2021.

Abstract

Combinatorial vector fields on simplicial complexes as introduced by Robin Forman have found numerous and varied applications in recent years. Yet, their relationship to classical dynamical systems has been less clear. In recent work it was shown that for every combinatorial vector field on a finite simplicial complex one can construct a multivalued discrete-time dynamical system on the underlying polytope X which exhibits the same dynamics as the combinatorial flow in the sense of Conley index theory. However, Forman’s original description of combinatorial flows appears to have been motivated more directly by the concept of flows, i.e., continuous-time dynamical systems. In this paper, it is shown that one can construct a semiflow on the polytope which exhibits the same dynamics as the underlying combinatorial vector field. The equivalence of the dynamical behavior is established in the sense of Conley-Morse graphs and uses a tiling of the topological space X which makes it possible to directly construct isolating blocks for all involved isolated invariant sets based purely on the combinatorial information.

Links

The preprint version of the paper can be downloaded from https://arxiv.org/abs/2005.11647, while the published version of the paper can be found at https://doi.org/10.1016/j.jde.2021.10.001.

Bibtex

@article{mrozek:wanner:21a,
   author = {Marian Mrozek and Thomas Wanner},
   title = {Creating semiflows on simplicial complexes from combinatorial vector fields},
   journal = {Journal of Differential Equations},
   volume = {304},
   year = {2021},
   pages = {375--434},
   doi = {10.1016/j.jde.2021.10.001}
   }