Access the full text.
Sign up today, get DeepDyve free for 14 days.
Art Duval (1994)
A combinatorial decomposition of simplicial complexesIsrael Journal of Mathematics, 87
R. Stanley (1993)
A combinatorial decomposition of acyclic simplicial complexesDiscret. Math., 120
J. Franks (1982)
Homology and Dynamical Systems
C. Conley (1978)
Isolated Invariant Sets and the Morse Index, 38
R. Forman (1993)
Determinants of Laplacians on graphsTopology, 32
Albert Lundell, S. Weingram (1969)
The Topology of CW Complexes
D. Ray, I. Singer (1971)
R-Torsion and the Laplacian on Riemannian manifoldsAdvances in Mathematics, 7
R. Forman (1998)
Morse Theory for Cell ComplexesAdvances in Mathematics, 134
W. Parry, M. Pollicott (1990)
Zeta functions and the periodic orbit structure of hyperbolic dynamics
W. Franz (1935)
Über die Torsion einer Überdeckung.Journal für die reine und angewandte Mathematik (Crelles Journal), 1935
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 $z=0$ 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.
Mathematische Zeitschrift – Springer Journals
Published: Aug 1, 1998
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.