Complex Valued Analytic Torsion and Dynamical Zeta Function on Locally Symmetric Spaces

Abstract

Abstract We show that the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an arbitrary flat vector bundle has a meromorphic extension to the whole complex plane and that its leading term in the Laurent series at the zero point is related to the regularised determinant of the flat Laplacian of Cappell–Miller. When the flat vector bundle is close to an acyclic and unitary one, we show that the dynamical zeta function is regular at the zero point and that its value is equal to the complex valued analytic torsion of Cappell–Miller. This generalises the author’s previous results for unitarily flat vector bundles as well as Müller and Spilioti’s results on hyperbolic manifolds.