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Dynamical torsion for contact Anosov flows

Yann Chaubet, Nguyen Viet Dang

Abstract

We introduce a new object, the dynamical torsion, which extends the potentially ill-defined value at $0$ of the Ruelle zeta function of a contact Anosov flow twisted by an acyclic representation of the fundamental group. We show important properties of the dynamical torsion: it is invariant under deformations among contact Anosov flows, it is holomorphic in the representation and it has the same logarithmic derivative as some refined combinatorial torsion of Turaev. This shows that the ratio between this torsion and the Turaev torsion is locally constant on the space of acyclic representations. In particular, for contact Anosov flows path connected to the geodesic flow of some hyperbolic manifold among contact Anosov flows, we relate the leading term of the Laurent expansion of $ζ$ at the origin, the Reidemeister torsion and the torsions of the finite dimensional complexes of the generalized resonant states of both flows for the resonance $0$. This extends previous work of~\cite{dang2018fried} on the Fried conjecture near geodesic flows of hyperbolic $3$--manifolds, to hyperbolic manifolds of any odd dimension.

Dynamical torsion for contact Anosov flows

Abstract

We introduce a new object, the dynamical torsion, which extends the potentially ill-defined value at of the Ruelle zeta function of a contact Anosov flow twisted by an acyclic representation of the fundamental group. We show important properties of the dynamical torsion: it is invariant under deformations among contact Anosov flows, it is holomorphic in the representation and it has the same logarithmic derivative as some refined combinatorial torsion of Turaev. This shows that the ratio between this torsion and the Turaev torsion is locally constant on the space of acyclic representations. In particular, for contact Anosov flows path connected to the geodesic flow of some hyperbolic manifold among contact Anosov flows, we relate the leading term of the Laurent expansion of at the origin, the Reidemeister torsion and the torsions of the finite dimensional complexes of the generalized resonant states of both flows for the resonance . This extends previous work of~\cite{dang2018fried} on the Fried conjecture near geodesic flows of hyperbolic --manifolds, to hyperbolic manifolds of any odd dimension.

Paper Structure

This paper contains 6 sections, 8 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Refined versions of torsion
  4. Dynamical torsion
  5. Statement of the results.
  6. Methods of proof

Key Result

Theorem 1

Let $M = SZ$ be the unit tangent bundle of some closed oriented hyperbolic manifold $Z$, and denote by $X$ its geodesic vector field on $M$. Assume that $\rho : \pi_1(M) \to O(d)$ is an acyclic and unitary representation. Then $\zeta_{X,\rho}$ extends meromorphically to $\mathbb{C}$. Moreover, it is where $2r+1 = \dim M$, and $\tau_\textrm{R}(\rho)$ is the Reidemeister torsion of $(M, \rho)$.

Theorems & Definitions (9)

  • Theorem 1: Fried fried1986analytic
  • Theorem 2: Dyatlov--Zworski
  • Theorem 3: D--Rivière--Guillarmou--Shen
  • Theorem 4
  • Theorem 5
  • Remark 2.1
  • Theorem 6
  • Theorem 7
  • Corollary 8