2023/03/01 by Peter Hochs, Hemanth Saratchandran, Hochs, Peter +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2303.00312
openalex publication_date 2023/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For proper group actions on smooth manifolds, with compact quotients, we define an equivariant version of the Ruelle dynamical ζ-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 ζ-function. We ask the question in what situations an equivariant generalisation of Fried's conjecture holds, relating the equivariant Ruelle ζ-function to equivariant analytic torsion. We compute the equivariant Ruelle ζ-function in several examples, including examples where the classical Ruelle ζ-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.