Mathematische Zeitschrift
1432-1823
0025-5874
Springer Nature - SN SciGraph project
http://link.springer.com/10.1007%2FPL00004638
1998-08
2019-04-10T23:28
1998-08-01
en
research_article
articles
https://scigraph.springernature.com/explorer/license/
629-681
false
Combinatorial vector fields and dynamical systems
In this paper we introduce the notion of a combinatorial dynamical system on any CW complex. Earlier in [Fo3] and [Fo4], we presented the idea of a combinatorial vector field (see also [Fo1] for the one-dimensional case), and studied the corresponding Morse Theory. Equivalently, we studied the homological properties of gradient vector fields (these terms were defined precisely in [Fo3], see also Sect. 2 of this paper). In this paper we broaden our investigation and consider general combinatorial vector fields. We first study the homological properties of such vector fields, generalizing the Morse Inequalities of [Fo3]. We then introduce various zeta functions which keep track of the closed orbits of the corresponding flow, and prove that these zeta functions, initially defined only on a half plane, can be analytically continued to meromorphic functions on the entire complex plane. Lastly, we review the notion of Reidemeister Torsion of a CW complex (introduced in [Re], [Fr]) and show that the torsion is equal to the value at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $z=0$\end{document} of one of the zeta functions introduced earlier. Much of this paper can be viewed as a combinatorial analogue of the work on smooth dynamical systems presented in [P-P], [Fra], [Fri1, 2] and elsewhere.
Forman
Robin
10.1007/pl00004638
doi
readcube_id
84edebdb7b03363fd343f860ecd4e121669454b7d335b361f1475c3813013e21
Pure Mathematics
dimensions_id
pub.1027904863
228
4
Mathematical Sciences
Rice University
Department of Mathematics, Rice University, Houston, TX 77251, USA (e-mail: forman@math.rice.edu), US