A Ruelle dynamical zeta function for equivariant flows

Peter Hochs; Hemanth Saratchandran

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arXiv · 2303.00312

A Ruelle dynamical zeta function for equivariant flows

Peter Hochs·Hemanth Saratchandran

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Abstract

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.

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BibTeX RIS

Peter Hochs, Hemanth Saratchandran. 2025-02-12. A Ruelle dynamical zeta function for equivariant flows. https://arxiv.org/abs/2303.00312

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