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A Ruelle dynamical zeta function for equivariant flows

Peter Hochs, Hemanth Saratchandran

TL;DR

This work extends the classical Ruelle dynamical ζ-function to flows with proper cocompact group actions by introducing an equivariant ζ-function $R^g_{\varphi,\nabla^F}(\sigma)$ defined via a distributional flat $g$-trace and a $g$-delocalised fixed-point framework. It relies on an equivariant Guillemin trace formula to express the ζ-function as a trace over $g$-periodic data, and it analyzes its basic properties, including cutoff-independence, subgroup restrictions, and connections to the classical ζ-function. The paper also formulates an equivariant Fried conjecture relating $R^g_{\varphi,\nabla^F}(0)$ to the equivariant analytic torsion $T_g(\nabla^F)$ under vanishing $L^2$-kernel assumptions, and verifies the conjecture in several 1D and Euclidean/discrete-model examples while presenting counterexamples where the $L^2$-kernel is nontrivial. Overall, the results illuminate how equivariant zeta-type invariants interact with analytic torsion and suggest concrete conditions under which an equivariant Fried-type relation may hold, guiding future investigations into suspension flows and other equivariant dynamical systems.

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.

A Ruelle dynamical zeta function for equivariant flows

TL;DR

This work extends the classical Ruelle dynamical ζ-function to flows with proper cocompact group actions by introducing an equivariant ζ-function defined via a distributional flat -trace and a -delocalised fixed-point framework. It relies on an equivariant Guillemin trace formula to express the ζ-function as a trace over -periodic data, and it analyzes its basic properties, including cutoff-independence, subgroup restrictions, and connections to the classical ζ-function. The paper also formulates an equivariant Fried conjecture relating to the equivariant analytic torsion under vanishing -kernel assumptions, and verifies the conjecture in several 1D and Euclidean/discrete-model examples while presenting counterexamples where the -kernel is nontrivial. Overall, the results illuminate how equivariant zeta-type invariants interact with analytic torsion and suggest concrete conditions under which an equivariant Fried-type relation may hold, guiding future investigations into suspension flows and other equivariant dynamical systems.

Abstract

For proper group actions on smooth manifolds, with compact quotients, we define an equivariant version of the Ruelle dynamical -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 -function. We ask the question in what situations an equivariant generalisation of Fried's conjecture holds, relating the equivariant Ruelle -function to equivariant analytic torsion. We compute the equivariant Ruelle -function in several examples, including examples where the classical Ruelle -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.
Paper Structure (24 sections, 41 theorems, 198 equations)

This paper contains 24 sections, 41 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. The equivariant Ruelle dynamical $\zeta$-function
  3. Definition of the equivariant Ruelle dynamical $\zeta$-function
  4. An equivariant Fried conjecture
  5. Properties of the equivariant Ruelle dynamical $\zeta$-function
  6. An equivariant Guillemin trace formula
  7. An expression for the equivariant Ruellle dynamical $\zeta$-function
  8. Alternative expressions
  9. Restriction to subgroups
  10. Relations with the classical Ruelle dynamical $\zeta$-function
  11. The classical decomposition of the Ruelle dynamical $\zeta$-function for geodesic flow
  12. The line and the circle
  13. The real line
  14. The circle
  15. Geodesic flow on $\mathbb{R}^n$
  16. ...and 9 more sections

Key Result

Lemma 2.5

Suppose that $\varphi$ is $g$-nondegenerate and let $l \in L_{g}(\varphi)$. Then the set $\Gamma^{g}_l(\varphi)$ is discrete, in the sense that it is countable, and the images of the curves in $\Gamma^{g}_l(\varphi)$ together form the closed subset $M^{g^{-1}\varphi_l} \subset M$ of points fixed by

Theorems & Definitions (111)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 101 more