Complex valued analytic torsion and dynamical zeta function on locally symmetric spaces

TL;DR

This paper establishes a deep connection between the Ruelle dynamical zeta function and analytic torsion on locally symmetric spaces, extending previous results to more general flat vector bundles and providing meromorphic continuation and regularity results.

Contribution

It proves the meromorphic extension of the Ruelle zeta function for twisted bundles and relates its behavior at zero to the regularised determinant and analytic torsion, generalizing prior work.

Findings

Ruelle zeta function has a meromorphic extension to the complex plane.

At zero, the zeta function's leading term relates to the regularised determinant of the flat Laplacian.

When the bundle is close to acyclic and unitary, the zeta function is regular at zero and equals the analytic torsion.

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 author's previous results for unitarily flat vector bundles as well as M\"uller and Spilioti's results on hyperbolic manifolds.