Morse-Smale flow, Milnor metric, and dynamical zeta function

TL;DR

This paper develops a new metric called the Milnor metric on the cohomology of manifolds with flat bundles, linking it to dynamical zeta functions and generalizing classical notions in dynamical systems.

Contribution

It introduces the Milnor metric using Morse-Smale flows, extending the understanding of dynamical zeta functions and connecting it with Ray-Singer metrics.

Findings

Defined the Milnor metric via fixed points and closed orbits

Established a formula relating Milnor and Ray-Singer metrics

Extended the concept of the zeta function's value at zero

Abstract

We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem.