A Ruelle dynamical zeta function for equivariant flows

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.