Leafwise cohomological expression of dynamical zeta functions on foliated dynamical systems

TL;DR

This paper presents a leafwise cohomological expression for dynamical zeta functions on 3-dimensional Riemannian foliated dynamical systems, linking geometric structures with dynamical zeta functions in the context of arithmetic topology.

Contribution

It provides a novel leafwise cohomological formulation of dynamical zeta functions on RFDS^3, advancing the understanding of their geometric and analytic properties.

Findings

Leafwise cohomological expression derived for dynamical zeta functions.

Connections established between foliated dynamical systems and arithmetic topology.

Framework supports conjectural links to arithmetic schemes as suggested by Deninger.

Abstract

A Riemmanian foliated dynamical system of 3-dimension $(\mathrm{RFDS}^{3})$ is a closed Riemannian 3-manifold with additional structures: foliation, dynamical system. In the context of arithmetic topology, it is a geometric/analytic analogue of an arithmetic scheme with a conjectural dynamical system suggested by C. Deninger. In this paper, we show leafwise cohomological expression of dynamical zeta function on a Riemannian foliated dynamical system.