Connection matrices in combinatorial topological dynamics

78. Marian Mrozek, Thomas Wanner:
    Connection matrices in combinatorial topological dynamics
    Preprint, submitted for publication, 115 pages, 2023.

Abstract

Connection matrices are one of the central tools in Conley’s approach to the study of dynamical systems, as they provide information on the existence of connecting orbits in Morse decompositions. They may be considered as a generalization of the Morse complex boundary operator in Morse theory. Their computability has recently been addressed by Harker et al. (2021) in the context of lattice filtered chain complexes. In the current paper, we extend the newly introduced Conley theory for combinatorial vector and multivector fields on Lefschetz complexes due to Lipinski, Kubica, Mrozek, and Wanner (2023) by transferring the concept of connection matrix to this setting. This is accomplished by connection matrices for arbitrary poset filtered chain complexes, as well as an associated equivalence, which allows for changes in the underlying posets. We show that for the special case of gradient combinatorial vector fields in the sense of Forman (1998), connection matrices are necessarily unique. Thus, the classical results of Reineck (1990, 1995) have a natural analogue in the combinatorial setting.

Links

The preprint version of the paper can be downloaded from https://arxiv.org/abs/2103.04269.

Bibtex

@article{mrozek:wanner:p23a,
   author = {Marian Mrozek and Thomas Wanner},
   title = {Connection matrices in combinatorial topological dynamics},
   journal = {Submitted for publication},
   year = 2023
   }