Combinatorial vector fields and dynamical systems
This work follows a sequence of papers by the same author [Topology 32, 35-46 (1993; Zbl 0780.05041); Geometry, Topology and Physics for Raoul Bott, Conf. Proc. Lect. Notes Geom. Topol. 4, 112-125 (1995; Zbl 0867.57018); Adv. Math. 134, 90-145 (1998; Zbl 0896.57023); `Witten-Morse theory for cell complexes' (preprint)]. After introducing the notion of a combinatorial dynamical system, he studies its homological properties and generalizes the Morse inequalities of a previous paper. He then introduces zeta-functions which keep track of the closed orbits of the corresponding flow and shows that they can be continued to meromorphic functions on the complex plane. Then Reidemeister torsion is proved to be equal to the value at zero of one of these zeta-functions. The paper may be viewed as a combinatorial analogue of the work on smooth dynamical systems [see for example \textit{W. Parry} and \textit{M. Pollicott}, Zeta-functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990; Zbl 0726.58003)]. Finally, note that the combinatorial theory presented here has implications for the theory of smooth dynamical systems.
scientific article; zbMATH DE number 1229588