A dynamical systems analogue of Lichtenbaum's conjectures on special values of Hasse-Weil zeta functions

Abstract

In recent years Lichtenbaum has conjectured a description for the special values of Hasse--Weil zeta functions in terms of ``Weil-\'etale cohomology''. In earlier papers we studied a class of foliated dynamical systems which had some similarities with arithmetic schemes. Assuming that certain zeta regularized determinants exist we now prove an analogue of Lichtenbaum's conjectures for a particularly simple class of such dynamical systems: Using results of \'Alvarez L\'opez and Kordyukov we express the leading coefficient $\zeta^*_R (0)$ of the Ruelle zeta function $\zeta_R (s)$ at $s = 0$ in terms of analytic torsion. We then apply the Cheeger--M\"uller theorem to replace the analytic torsion by the Reidemeister torsion with respect to harmonic bases. For our dynamical systems the latter can be expressed by the same recipe as the one in Lichtenbaum's conjectures.