Regularized determinant formulas for the zeta functions of 3-dimensional Riemannian foliated dynamical systems

TL;DR

This paper establishes a regularized determinant formula for zeta functions associated with 3D Riemannian foliated dynamical systems, confirming Deninger's conjecture using a novel connection with dynamical spectral functions.

Contribution

It provides the first proof of Deninger's conjectured formula relating zeta functions to infinitesimal operators on leafwise cohomologies in this setting.

Findings

Proves a regularized determinant formula for the zeta functions.

Relates dynamical spectral functions to zeta functions via Lefschetz trace formula.

Confirms Deninger's conjecture in the context of 3D Riemannian foliated systems.

Abstract

We prove a regularized determinant formula for the zeta functions of certain 3-dimensional Riemannian foliated dynamical systems, in terms of the infinitesimal operator induced by the flow acting on the reduced leafwise cohomologies. It is the formula conjectured by Deninger. The proof is based on relating the dynamical spectral $\xi$-functions, analogues of the $\xi$-function in analytic number theory, with the zeta function, by applying the distributional dynamical Lefschetz trace formula.