# A Ruelle dynamical zeta function for equivariant flows

## TL;DR

This paper introduces an equivariant Ruelle dynamical zeta function for flows on manifolds with group actions, extending classical concepts and exploring its relation to equivariant analytic torsion with concrete examples.

## Contribution

It defines a new equivariant Ruelle zeta function for proper group actions, generalizing Guillemin's trace formula and analyzing its properties and applications.

## Key findings

- The equivariant Ruelle zeta function is computed in several examples.
- The equivariant Fried conjecture holds when the Laplacian kernel vanishes.
- The construction extends classical dynamical zeta functions to equivariant settings.

## Abstract

For proper group actions on smooth manifolds, with compact quotients, we define an equivariant version of the Ruelle dynamical $\zeta$-function for equivariant flows satisfying a nondegeneracy condition. The construction is based on an equivariant generalisation of Guillemin's trace formula, obtained in a companion paper. This formula implies several properties of the equivariant Ruelle $\zeta$-function. We ask the question in what situations an equivariant generalisation of Fried's conjecture holds, relating the equivariant Ruelle $\zeta$-function to equivariant analytic torsion. We compute the equivariant Ruelle $\zeta$-function in several examples, including examples where the classical Ruelle $\zeta$-function is not defined. The equivariant Fried conjecture holds in the examples where the condition of the conjecture (vanishing of the kernel of the Laplacian) is satisfied.