Yann Chaubet
email: yann.chaubet@math.u-psud.fr

Viet Nguyen

Dang
email: dang@math.univ-lyon1.fr

And Nguyen

Viet Dang

DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS

We introduce a new object, called dynamical torsion, which extends the potentially ill-defined value at 0 of the Ruelle zeta function ζ of a contact Anosov flow twisted by an acyclic representation of the fundamental group. The dynamical torsion depends analytically on the representation and is invariant under deformations among contact Anosov flows. Moreover, we show that the ratio between this torsion and the refined combinatorial torsion of Turaev, for an appropriate choice of Euler structure, is locally constant on the space of acyclic representations. In particular, for contact Anosov flows path connected to a geodesic flow of a 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 [DGRS18] on the Fried conjecture near geodesic flows of hyperbolic 3-manifolds, to hyperbolic manifolds of any odd dimensions.

Introduction

Let M be a closed odd dimensional manifold and (E, ∇) be a flat vector bundle over M . The parallel transport of the connection ∇ induces a conjugacy class of representation ρ ∈ Hom(π 1 (M ), GL(C d )). Moreover, ∇ defines a differential on the complex Ω • (M, E) of E-valued differential forms on M and thus cohomology groups H • (M, ∇) = H • (M, ρ) (note that we use the notation ∇ also for the twisted differential induced by ∇ whereas it can be denoted by d ∇ in other references). We will say that ∇ (or ρ) is acyclic if those cohomology groups are trivial. If ρ is unitary (or equivalently, if there exists a hermitian structure on E preserved by ∇) and acyclic, Reidemeister [START_REF] Reidemeister | Homotopieringe und linsenräume[END_REF] introduced a combinatorial invariant τ R (ρ) of the pair (M, ρ), the so-called Franz-Reidemeister torsion (or R-torsion), which is a positive number. This allowed him to classify lens spaces in dimension 3; this result was then extended in higher dimension by Franz [START_REF] Franz | Über die torsion einer Überdeckung[END_REF] and De Rham [dR36].

Later, Ray-Singer [START_REF] Ray | R-torsion and the Laplacian on Riemannian Manifolds[END_REF] introduced another invariant τ RS (ρ), the analytic torsion, defined via the derivative at 0 of the spectral zeta function of the Laplacian given by the Hermitian metric on E and some Riemannian metric on M . They conjectured the equality of the analytic and Reidemeister torsions. This conjecture was proved independently by Cheeger [START_REF] Cheeger | Analytic torsion and the heat equation[END_REF] and Müller [START_REF] Müller | Analytic torsion and R-torsion of riemannian manifolds[END_REF], assuming only that ρ is unitary (both R-torsion and analytic torsion have a natural extension if ρ is unitary and not acyclic). The Cheeger-Müller theorem was extended to unimodular flat vector bundles by Müller [START_REF] Müller | Analytic torsion and R-torsion for unimodular representations[END_REF] and to arbitrary flat vector bundles by Bismut-Zhang [START_REF] Bismut | An extension of a theorem by Cheeger and Müller[END_REF].

To remove indeterminacies arising in the definition of the combinatorial torsion, Turaev [START_REF] Turaev | Reidemeister torsion in knot theory[END_REF][START_REF] Turaev | Euler structures, nonsingular vector fields, and torsions of Reidemeister type[END_REF][START_REF] Turaev | Torsion invariants of Spin c -structures on 3-manifolds[END_REF] introduced in the acyclic case a refined version of the combinatorial 1 R-torsion, the refined combinatorial torsion. It is a complex number τ e,o (ρ) which depends on additional combinatorial data, namely an Euler structure e and a cohomological orientation o of M . We refer the reader to subsection 8.2 for precise definitions. Later, Farber-Turaev [START_REF] Farber | Poincaré-Reidemeister metric, Euler structures, and torsion[END_REF] extended this object to non-acyclic representations. In this case, τ e,o (ρ) is an element of the determinant line of cohomology det H • (M, ρ).

Motivated by the work of Turaev, but from the analytic side, Braverman-Kappeler [BK07c, BK + 08, BK07b] introduced a refined version of the Ray-Singer analytic torsion called refined analytic torsion τ an (ρ). It is complex valued in the acyclic case. Their construction heavily relies on the existence of a chirality operator Γ g , that is,

Γ g : Ω • (M, E) → Ω n-• (M, E), Γ 2 g = Id,
which is a renormalized version of the Hodge star operator associated to some metric g. They showed that the ratio ρ → τ an (ρ) τ e,o (ρ) is a holomorphic function on the representation variety given by an explicit local expression, up to a local constant of modulus one. This result is an extension of the Cheeger-Müller theorem. Simultaneously, Burghelea-Haller [START_REF] Burghelea | Complex-valued Ray-Singer torsion[END_REF] introduced a complex valued analytic torsion, which is closely related to the refined analytic torsion [START_REF] Braverman | Comparison of the refined analytic and the Burghelea-Haller torsions[END_REF] when it is defined; see [H + 07] for comparison theorems.

In the context of hyperbolic dynamical systems, Fried [START_REF] Fried | Lefschetz formulas for flows[END_REF] was interested in the link between the R-torsion and the Ruelle zeta function of an Anosov flow X which is defined by

ζ X,ρ (s) = γ∈G # X det 1 -ε γ ρ([γ])e -s (γ) , Re(s) 0,
where G # X is the set of primitive closed orbits of X, (γ) is the period of γ and ε γ = 1 if the stable bundle of γ is orientable and ε γ = -1 otherwise. Using Selberg's trace formula, Fried could relate the behavior of ζ X,ρ (s) near s = 0 with τ R , as follows.

Theorem 1 (Fried [Fri86]). 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 ρ : π 1 (M ) → O(d) is an acyclic and unitary representation. Then ζ X,ρ extends meromorphically to C. Moreover, it is holomorphic near s = 0 and |ζ X,ρ (0)| (-1) r = τ R (ρ),

(1.1)

where 2r + 1 = dim M , and τ R (ρ) is the Reidemeister torsion of (M, ρ).

In [START_REF] Fried | Lefschetz formulas for flows[END_REF], Fried conjectured that the same holds true for negatively curved locally symmetric spaces. This was proved by Moscovici-Stanton [START_REF] Moscovici | R-torsion and zeta functions for locally symmetric manifolds[END_REF], Shen [START_REF] Shen | Analytic torsion, dynamical zeta functions, and the Fried conjecture[END_REF].

For analytic Anosov flows, the meromorphic continuation of ζ X,ρ was proved by Rugh [START_REF] Henrik | Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems[END_REF] in dimension 3 and by Fried [START_REF] Fried | Meromorphic zeta functions for analytic flows[END_REF] in higher dimensions. Then Sanchez-Morgado [START_REF] Sánchez-Morgado | Lefschetz formulae for Anosov flows on 3-manifolds[END_REF][START_REF] Sánchez-Morgado | R-torsion and zeta functions for analytic Anosov flows on 3-manifolds[END_REF] proved in dimension 3 that if ρ is acyclic, unitary, and satisfies that ρ([γ]) -ε j γ is invertible for every j ∈ {0, 1} for some closed orbit γ, then (1.1) is true.

For general smooth Anosov flows, the meromorphic continuation of ζ X,ρ was proved by Giuletti-Liverani-Pollicott [START_REF] Giulietti | Anosov flows and dynamical zeta functions[END_REF] and alternatively by Dyatlov-Zworski [START_REF] Dyatlov | Dynamical zeta functions for Anosov flows via microlocal analysis[END_REF], the Axiom A case was treated by Dyatlov-Guillarmou in [START_REF] Dyatlov | Afterword: Dynamical zeta functions for Axiom A flows[END_REF]. These results are important byproducts of new functional methods to study hyperbolic flows. All these works rely on the construction of spaces of anisotropic distributions adapted to the dynamics, initiated by Kitaev [START_REF] Yu | Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness[END_REF], Blank-Keller-Liverani [START_REF] Blank | Ruelle-Perron-Frobenius spectrum for Anosov maps[END_REF], Baladi [START_REF] Baladi | Anisotropic Sobolev spaces and dynamical transfer operators: C ∞ foliations[END_REF][START_REF] Baladi | Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps-A Functional Approach[END_REF], Baladi-Tsujii [START_REF] Baladi | Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms (espaces anisotropes de types Hölder et Sobolev)[END_REF], Gouëzel-Liverani [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF], Liverani [START_REF] Liverani | Fredholm determinants, Anosov maps and Ruelle resonances[END_REF], Butterley-Liverani [START_REF] Butterley | Smooth Anosov flows: correlation spectra and stability[END_REF][START_REF] Butterley | Robustly invariant sets in fiber contracting bundle flows[END_REF], and many others where we refer to the recent book [START_REF] Baladi | Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps-A Functional Approach[END_REF] for precise references. These spaces allow to define a suitable notion of spectrum for the operator L ∇ X = ∇ι X + ι X ∇, where ι is the interior product, acting on Ω • (M, E). This spectrum is the set of so-called Pollicott-Ruelle resonances Res(L ∇ X ), which forms a discrete subset of C and contains all zeros and poles of ζ X,ρ . Faure-Roy-Sjöstrand [START_REF] Faure | Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances[END_REF], Faure-Sjöstrand [START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov flows[END_REF] initiated the use of microlocal methods to describe these anisotropic spaces of distributions giving a purely microlocal approach to study Ruelle resonances. Combining this point of view with propagation estimates originating from scattering theory [START_REF] Vasy | Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)[END_REF][START_REF] Melrose | Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces[END_REF] allowed the authors of [START_REF] Dyatlov | Dynamical zeta functions for Anosov flows via microlocal analysis[END_REF] to control the wavefront set of the resolvent L ∇ X + s -1 yielding another proof of the analytic continuation of the zeta function. Using this microlocal approach, one could try to address questions relating topology and dynamical zeta functions. Quoting the commentary from Zworski [START_REF] Zworski | Commentary on "Differentiable Dynamical Systems" by Stephen Smale[END_REF] on Smale's seminal paper [START_REF] Smale | Differentiable dynamical systems[END_REF], equation (1.1) "would link dynamical, spectral and topological quantities. [. . . ] In the case of smooth manifolds of variable negative curvature, equation (1.1) remains completely open". However in [START_REF] Dyatlov | Ruelle zeta function at zero for surfaces[END_REF], the authors were able to prove the following.

Theorem 2 (Dyatlov-Zworski). Suppose (Σ, g) is a negatively curved orientable Riemannian surface. Let X denote the associated geodesic vector field on the unitary cotangent bundle M = S * Σ. Then for some c = 0, we have as s → 0

ζ X,1 (s) = cs |χ(Σ)| (1 + O(s)) , (1.2)
where 1 is the trivial representation π 1 (S * Σ) → C * and χ(Σ) is the Euler characteristic of Σ. In particular, the length spectrum (γ), γ ∈ G # X determines the genus.

This result was generalized in a recent preprint of Cekić-Paternain [CePa] to volume preserving Anosov flows in dimension 3.

In the same spirit and using similar microlocal methods, Guillarmou-Rivière-Shen and the second author [START_REF] Nguyen | The Fried conjecture in small dimensions[END_REF] showed Theorem 3 (D-Rivière-Guillarmou-Shen). The map

X → ζ X,ρ (0)
is locally constant on the open set of smooth vector fields which are Anosov and for which 0 is not a Ruelle resonance, that is, 0 / ∈ Res(L ∇ X ). By an approximation argument, equation (1.1) holds true for M of dimension 3, if X preserves a smooth volume form, and b 1 (M ) = 0 or under the same assumption used in [START_REF] Sánchez-Morgado | R-torsion and zeta functions for analytic Anosov flows on 3-manifolds[END_REF], still assuming ρ unitary and acyclic.

If 0 ∈ Res(L ∇ X ), the results of [START_REF] Nguyen | The Fried conjecture in small dimensions[END_REF] no longer apply. In the case where X induces a contact flow, which means X = X ϑ is the Reeb vector field of some contact form ϑ on M , we overcome this difficulty by introducing a dynamical torsion τ ϑ (ρ) which is defined for any acyclic ρ and which coincides with ζ X,ρ (0) ±1 if 0 / ∈ Res(L ∇ X ). To do so, in the spirit of [START_REF] Braverman | Refined analytic torsion as an element of the determinant line[END_REF], we use a chirality operator associated to the contact form ϑ, Γ ϑ : Ω • (M, E) → Ω n-• (M, E), Γ 2 ϑ = Id, cf. §5, analogous to the Hodge star operator associated to a metric. Let us briefly describe the construction of the dynamical torsion. Let C • ⊂ D • (M, E) be the finite dimensional space of Pollicott-Ruelle generalized resonant states of L ∇ X for the resonance 0, that is,

C • = u ∈ D • (M, E), WF(u) ⊂ E * u , ∃N ∈ N, (L ∇ X ) N u = 0
, where WF is the Hörmander wavefront set, E * u ⊂ T * M is the dual of the stable bundle of X, cf. §4, and D (M, E) denotes the space of E-valued currents. Then ∇ induces a differential on the complex C • and a result from [START_REF] Viet | Topology of Pollicott-Ruelle resonant states[END_REF] implies that the complex (C • , ∇) is acyclic if we assume that ∇ is. Because Γ ϑ commutes with L ∇ X , it induces a chirality operator on C • . Therefore we can compute the torsion τ (C • , Γ ϑ ) of the finite dimensional complex (C • , ∇) with respect to Γ ϑ , as described in [START_REF] Braverman | Refined analytic torsion as an element of the determinant line[END_REF] (see §2). Then we set τ ϑ (ρ) (-1) q = τ ϑ (∇) (-1)

q = ±τ (C • , Γ ϑ ) (-1) q lim s→0 s -m(X,ρ) ζ X,ρ (s) ∈ C \ 0,
where the sign ± will be given later, m(X, ρ) is the order of ζ X,ρ (s) at s = 0 and q is the dimension of the unstable bundle of X. Note that the order m(X, ρ) ∈ Z is a priori not stable under perturbations of (X, ρ) whereas τ ϑ (ρ) has interesting invariance properties as we will see below.

We denote by Rep ac (M, d) the set of acyclic representations π 1 (M ) → GL(C d ) and by A ⊂ C ∞ (M, T M ) the space of contact forms on M whose Reeb vector field induces an Anosov flow. This is an open subset of the space of contact forms. For any ϑ ∈ A, we denote by X ϑ its Reeb vector field. The fact that the dynamical torsion τ ϑ (ρ) is defined even if 0 ∈ Res(L ∇ X ϑ ) allows us to compare it with Turaev's refined torsion. The main result of this paper is the following Theorem 4. Let (M, ϑ) be a contact manifold such that the Reeb vector field of ϑ induces an Anosov flow. Then the following holds true.

(1) There exists a canonical Euler structure e ϑ associated to ϑ such that for any cohomological orientation o, the map

ρ ∈ Rep ac (M, d) -→ τ ϑ (ρ) τ e ϑ ,o (
(η, ρ) ∈ A × Rep ac (M, d) -→ τ η (ρ) τ ϑ (ρ) det ρ, cs(X η , X ϑ )
is locally constant on A×Rep ac (M, d), where for any nonvanishing vector fields Y, X, cs(Y, X) ∈ H 1 (M, Z) is the Chern-Simons class of the pair (Y, X).

The Chern-Simons class cs(Y, X) ∈ H 1 (M, Z) measures the obstruction to find a homotopy among non singular vector fields connecting Y and X. In particular, if ϑ and η are connected by some path in A, then cs(Y η , X ϑ ) = 0, and thus τ ϑ (ρ) = τ η (ρ) for any acyclic ρ. We refer the reader to subsection 8.1 for the definition of Chern-Simons classes. This theorem generalizes some results from [START_REF] Nguyen | The Fried conjecture in small dimensions[END_REF] in the case of contact flows.

Because the dynamical torsion is constructed with the help of the dynamical zeta function ζ X,ρ , we deduce from the above theorem some informations about the behavior of ζ X,ρ (s) near s = 0.

Corollary 5. Let M be a closed odd dimensional manifold. Then for every connected open subsets U ⊂ Rep 0 (M, d) and V ⊂ A, there exists a constant C such that for every Anosov contact form ϑ ∈ V and every representation ρ ∈ U,

ζ X ϑ ,ρ (s) (-1) q = Cs (-1) q m(ρ,X ϑ ) τ e X ϑ ,o (ρ) τ (C • (ϑ, ρ) , Γ ϑ ) (1 + O(s)) , (1.3)
where X ϑ is the Reeb vector field of ϑ, (E ρ , ∇ ρ ) is the flat vector bundle over M induced by

ρ, C • (ϑ, ρ) ⊂ D • (M, E ρ )
is the space of generalized resonant states for the resonance 0 of L ∇ρ X ϑ and m(X ϑ , ρ) is the vanishing order of ζ X ϑ ,ρ (s) at s = 0.

Combining these results with Fried's Theorem 1 and some classical properties of Turaev's refined torsion, we can state a result in the spirit of Theorem 2 and Fried's conjecture. Corollary 6. Let (Z, g) be a compact hyperbolic manifold of odd dimension and denote by ϑ the contact form on S * Z whose Reeb flow is the geodesic flow. Let ρ be the lift to π 1 (S * Z) of some acyclic unitary representation π 1 (Z) → GL(C d ). Suppose that η is a contact form path connected to ϑ in the space of contact forms whose Reeb flow are Anosov. Then

ζ Xη,ρ (s) (-1) q = |s| (-1) q m(Xη,ρ) τ R (ρ) R-torsion τ (C • (ϑ, ρ), Γ ϑ ) τ (C • (η, ρ), Γ η ) (1 + O(s)) , (1.4)
where (E ρ , ∇ ρ ) is the flat vector bundle on M induced by ρ and

C • (η, ρ) (resp. C • (ϑ, ρ)) is the finite dimensional space of generalized eigenvectors of L ∇ρ Xη (resp. L ∇ρ X ϑ ) for the resonance 0.
This last corollary is non trivial for dim(Z) = 3 since hyperbolic 3-manifolds always admit unitary, acyclic connections by the result of [FN14, section 3].

Let us briefly sketch the proof of Theorem 4, which relies essentially on two variational arguments : we compute the variation of τ ϑ (∇) when we perturb the contact form ϑ and the connection ∇. As we do so, the space C • (ϑ, ∇) of Pollicott-Ruelle resonant states of L ∇ X ϑ for the resonance 0 may radically change. Therefore, it is convenient to consider the space C • [0,λ] (ϑ, ∇) instead, which consists of the generalized resonant states for L ∇ X ϑ for resonances s such that |s| ≤ λ, where λ ∈ (0, 1) is chosen so that {|s| = λ} ∩ Res(L ∇ X ϑ ) = ∅. Then using [BK07c, Proposition 5.6] and multiplicativity of torsion, one can show that

τ ϑ (∇) = ±τ C • [0,λ] (ϑ, ∇), Γ ϑ ζ (λ,∞) X ϑ ,ρ (0) (-1) q , (1.5) where ζ (λ,∞)
X ϑ ,ρ is a renormalized version of ζ X ϑ ,ρ (we remove all the poles and zeros of ζ X ϑ ,ρ within {s ∈ C, |s| ≤ λ}), see §5. Thus we can work with the space C • [0,λ] (ϑ, ∇), which behaves nicely under perturbations of X thanks to Bonthonneau's construction of uniform Anisotropic Sobolev spaces for families of Anosov flows [START_REF] Guedes | Perturbation of Ruelle resonances and Faure-Sjöstrand anisotropic space[END_REF], and also under perturbations of ∇. Now consider a smooth family of contact forms (ϑ t ) t for |t| < ε such that their Reeb vector fields (X t ) t induce Anosov flows. Then Theorem 7 says that for any acyclic ∇, the map t → τ ϑt (∇) is differentiable and its derivative vanishes. This follows from a result of [START_REF] Braverman | Refined analytic torsion as an element of the determinant line[END_REF] which allows to compute the variation of the torsion of a finite dimensional complex when the chirality operator is perturbed, and on a variation formula of the map t → ζ Xt,ρ (s) for Re(s) big enough obtained in [START_REF] Nguyen | The Fried conjecture in small dimensions[END_REF].

Next, consider a smooth family of flat connections z → ∇(z), where z is a complex number varying in a small neighborhood of the origin and write ∇(z) = ∇ + zα + o(z) where α ∈ Ω 1 (M, End(E)). Then we show in §7, in the same spirit as the variation formula obtained for the map t → τ ϑt (∇), that z → τ ϑ (∇(z)) is complex differentiable and its logarithmic derivative reads

∂ z | z=0 log τ ϑ (∇(z)) = -tr s αKe -εL ∇ X ϑ ,
where ε > 0 is small enough, tr s is the super flat trace, cf. §3.4, and K : Ω

• (M, E) → D • (M, E) is a cochain contraction, that is, it satisfies ∇K + K∇ = Id Ω • (M,E)
. On the other hand, we can compute, using the formalism of [START_REF] Viet | Spectral analysis of Morse-Smale flows II: resonances and resonant states[END_REF],

∂ z | z=0 log τ e ϑ ,o (∇(z)) = -tr s α Ke -εL ∇ -X - e tr α,
where e ϑ is an Euler structure canonically associated to ϑ, K is another cochain contraction, X is a Morse-Smale gradient vector field and e ∈ C 1 (M, Z) is a singular one-chain representing the Euler structure e ϑ , cf. §8. Now using the fact that K and K are cochain contractions, one can see that α Ke

-εL ∇ X ϑ -Ke -εL ∇ X = αR ε + [∇, αG ε ],
where R ε is an operator of degree -1 whose kernel is, roughly speaking, the union of graphs of the maps e -εXu , where (X u ) u is a non-degenerate family of vector fields interpolating X ϑ and X, cf. §8.3, and G ε is some operator of degree -2. Therefore we obtain by cyclicity of the flat trace

∂ z | z=0 log τ ϑ (∇(z)) τ e ϑ ,o (∇(z)) = tr s αR ε - e tr α = 0, (1.6)
where the last equality comes from differential topology arguments. Using the analytical structure of the representation variety, we may deduce from (1.6) the first point of Theorem 4. The second point then follows from the invariance of the dynamical torsion under small perturbations of the flow, the fact that τ e,o (ρ) = τ e ,o (ρ) det ρ, h for any other Euler structure e , where h ∈ H 1 (M, Z) satisfies e = e + h (we have that H 1 (M, Z) acts freely and transitively on the set of Euler structures, cf. §8), and the fact that, in our notations, e η -e ϑ = cs(X ϑ , X η ) for any other contact form η.

Related works. Some analogs of our dynamical torsion were introduced by Burghelea-Haller [START_REF] Burghelea | Torsion, as a function on the space of representations[END_REF] for vector fields which admit a Lyapunov closed 1-form generalizing previous works by Hutchings [H02], Hutchings-Lee [START_REF] Hutchings | Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds[END_REF][START_REF] Hutchings | Circle-valued Morse theory and Reidemeister torsion[END_REF] dealing with Morse-Novikov flows.

In that case, the dynamical torsion depends on a choice of Euler structure and is a partially defined function on Rep ac (M, d); if d = 1, it is shown in [START_REF] Burghelea | Dynamics, Laplace transform and spectral geometry[END_REF] that it extends to a rational map on the Zariski closure of Rep ac (M, 1) which coincides, up to sign, with Turaev's refined combinatorial torsion (for the same choice of Euler structure). This follows from previous works of Hutchings-Lee [START_REF] Hutchings | Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds[END_REF][START_REF] Hutchings | Circle-valued Morse theory and Reidemeister torsion[END_REF] who introduced some topological invariant involving circle-valued Morse functions. In both works, the considered object has the form Dynamical zeta function(0) × Correction term where the correction term is the torsion of some finite dimensional complex whose chains are generated by the critical points of the vector field. The chosen Euler structure gives a distinguished basis of the complex and thus a well defined torsion. This is one of the main differences with our work since in the Anosov case, there are no such choices of distinguished currents in C • . However, the chirality operator allows us to overcome this problem as described above.

We also would like to mention some interesting related works of Rumin-Seshadri [START_REF] Rumin | Analytic torsions on contact manifolds[END_REF] where they relate some dynamical zeta function involving the Reeb flow and some analytic contact torsion on 3-dimensional Seifert CR manifolds.

Plan of the paper. The paper is organized as follows. In §2, we give some preliminaries about torsion of finite dimensional complexes computed with respect to a chirality operator. In §3, we present our geometrical setting and conventions. In §4, we introduce Pollicott-Ruelle resonances. In §5, we compute the refined torsion of a space of generalized eigenvectors for nonzero resonances and we define the dynamical torsion. In §6, we prove that our torsion is unsensitive to small perturbations of the dynamics. In §7, we compute the variation of our torsion with respect to the connection. In §8, we introduce Euler structures which are some topological tools used to fix ambiguities of the refined torsion. In §9, we introduce the refined combinatorial torsion of Turaev using Morse theory and we compute its variation with respect to the connection. We finally compare it to the dynamical torsion in §10.

Finally, N.V.D acknowledges the incredible patience and love of his wife and daughter, who created the right atmosphere at home which made this possible.

Torsion of finite dimensional complexes

We recall the definition of the refined torsion of a finite dimensional acyclic complex computed with respect to a chirality operator, following [START_REF] Braverman | Refined analytic torsion as an element of the determinant line[END_REF]. Then we compute the variation of the torsion of such a complex when the differential is perturbed.

2.1. The determinant line of a complex. For a non zero complex vector space V , the determinant line of V is the line defined by det(V ) = dim V V . We declare the determinant line of the trivial vector space {0} to be C. If L is a 1-dimensional vector space, we will denote by L -1 its dual line. Any basis (v 1 , . .

. , v n ) of V defines a nonzero element v 1 ∧ • • • ∧ v n ∈ det(V ).
Thus elements of the determinant line of det(V ) should be thought of as equivalence classes of oriented basis of V .

Let

(C • , ∂) : 0 ∂ -→ C 0 ∂ -→ C 1 ∂ -→ • • • ∂ -→ C n ∂ -→ 0
be a finite dimensional complex, i.e. dim C j < ∞ for all j = 0, . . . , n. We define the determinant line of the complex

C • by det(C • ) = n j=0 det(C j ) (-1) j .
Let H • (∂) be the cohomology of (C • , ∂), that is

H • (∂) = n j=0 H j (∂), H j (∂) = ker(∂ : C j → C j+1 ) ran(∂ : C j-1 → C j ) .
We will say that the complex

(C • , ∂) is acyclic if H • (∂) = 0. In that case, det H • (∂) is canonically isomorphic to C.
It remains to define the fusion homomorphism that we will later need to define the torsion of a finite dimensional based complex [FT00, §2.3]. For any finite dimensional vector spaces V 1 , . . . , V r , we have a fusion isomorphism

µ V 1 ,...,Vr : det(V 1 ) ⊗ • • • ⊗ det(V r ) → det(V 1 ⊕ • • • ⊕ V r ) defined by µ V 1 ,...,Vr v 1 1 ∧ • • • ∧ v m 1 1 ⊗ • • • ⊗ v 1 r ∧ • • • ∧ v mr r = v 1 1 ∧ • • • ∧ v m 1 1 ∧ • • • ∧ v 1 r ∧ • • • ∧ v mr r ,
where m j = dim V j for j ∈ {1, . . . , r}.

Torsion of finite dimensional acyclic complexes.

In the present paper, we want to think of torsion of finite dimensional acyclic complexes as a map ϕ C • from the determinant line of the complex to C. We have a canonical isomorphism

ϕ C • : det(C • ) ∼ -→ C, (2.1)
defined as follows. Fix a decomposition C j = B j ⊕ A j , j = 0, . . . , n, with B j = ker(∂) ∩ C j and B j = ∂(A j-1 ) = ∂(C j-1 ) for every j. Then ∂| A j : A j → B j+1 is an isomorphism for every j.

Fix non zero elements c j ∈ det C j and a j ∈ det A j for any j. Let ∂(a j ) ∈ det B j+1 denote the image of a j under the isomorphism det A j → det B j+1 induced by the isomorphism ∂| A j : A j → B j+1 . Then for each j = 0, . . . , n, there exists a unique λ j ∈ C such that

c j = λ j µ B j ,A j ∂(a j-1 ) ⊗ a j ,
where µ B j ,A j is the fusion isomorphism defined in §2.1. Then define the isomorphism ϕ C • by

ϕ C • : c 0 ⊗ c -1 1 ⊗ • • • ⊗ c (-1) n n → (-1) N (C • ) n j=0 λ (-1) j j ∈ C,
where

N (C • ) = 1 2 n j=0 dim A j dim A j + (-1) j+1 .
One easily shows that ϕ C • is independent of the choices of a j [Tur01, Lemma 1.3]. The number τ (C • , c) = ϕ C • (c) is called the refined torsion of (C • , ∂) with respect to the element c.

The torsion will depend on the choices of c j ∈ det C j . Here the sign convention (that is, the choice of the prefactor (-1) N (C • ) in the definition of ϕ C • ) follows Braverman-Kappeler [BK07c, §2] and is consistent with Nicolaescu [START_REF] Nicolaescu | The Reidemeister Torsion of 3-manifolds[END_REF]§1]. This prefactor was introduced by Turaev and differs from [START_REF] Turaev | Reidemeister torsion in knot theory[END_REF]. See [START_REF] Nicolaescu | The Reidemeister Torsion of 3-manifolds[END_REF] for the motivation for the choice of sign.

2.3. Torsion with respect to a chirality operator. We saw above that torsion depends on the choice of an element of the determinant line. A way to fix the value of the torsion without choosing an explicit basis is to use a chirality operator as in [START_REF] Braverman | Refined analytic torsion as an element of the determinant line[END_REF]. Take n = 2r+1 an odd integer and consider a complex (C • , ∂) of length n. We will call a chirality operator an operator Γ :

C • → C • such that Γ 2 = Id C • , and Γ(C j ) = C n-j , j = 0, . . . , n.
Γ induces isomorphisms det(C j ) → det(C n-j ) that we will still denote by Γ. If ∈ L is a non zero element of a complex line, we will denote by -1 ∈ L -1 the unique element such that -1 ( ) = 1. Fix non zero elements c j ∈ det(C j ) for j ∈ {0, . . . , r} and define

c Γ = (-1) m(C • ) c 0 ⊗ c -1 1 ⊗ • • • ⊗ c (-1) r r ⊗ (Γc r ) (-1) r+1 ⊗ (Γc r-1 ) (-1) r ⊗ • • • ⊗ (Γc 0 ) -1 , where m(C • ) = 1 2 r j=0 dim C j dim C j + (-1) r+j .
Definition 2.1. The element c Γ is independent of the choices of c j for j ∈ {0, . . . , r}; the refined torsion of (C • , ∂) with respect to Γ is the element

τ (C • , Γ) = τ (C • , c Γ ).
We also have the following result which is [BK07c, Lemma 4.7] in the acyclic case about the multiplicativity of torsion.

Proposition 2.2. Let (C • , ∂) and ( C• , ∂) be two acyclic complexes of same length endowed with two chirality operators Γ and Γ. Then

τ (C • ⊕ C• , Γ ⊕ Γ) = τ (C • , Γ)τ ( C• , Γ).
2.4. Computation of the torsion with the contact signature operator. Let

B = Γ∂ + ∂Γ : C • → C • .
B is called the signature operator. Let B + = Γ∂ and B -= ∂Γ. Denote C j ± = C j ∩ ker(B ∓ ), j = 0, . . . , n.

We have that

B ± preserves C • ± . Note that B + (C j + ) ⊂ C n-j-1 + , so that B + (C j + ⊕ C n-j-1 + ) ⊂ C j + ⊕ C n-j-1 + . Note that if B is invertible on C • , B + is invertible on C • + .
If B is invertible, we can compute the refined torsion of (C • , ∂) using the following

Proposition 2.3. [BK07c, Proposition 5.6] Assume that B is invertible. Then (C • , ∂) is acyclic so that det(H • (∂)) is canonically isomorphic to C. Moreover, τ (C • , Γ) = (-1) r dim C r + det Γ∂| C r + (-1) r r-1 j=0 det Γ∂| C j + ⊕C n-j-1 + (-1) j .
2.5. Super traces and determinants. Let V • = p j=0 V j is a graded finite dimensional vector space and A : V • → V • be a degree preserving linear map. We define the super trace and the super determinant of A by

tr s,V • A = p j=0 (-1) j tr V j A, det s,V • A = p j=0 (det V j A) (-1) j .
We also define the graded trace and the graded determinant of A by

tr gr,V • A = p j=0 (-1) j j tr V j A, det gr,V • A = p j=0 (det V j A) (-1) j j .
2.6. Analytic families of differentials. The goal of the present subsection is to give a variation formula for the torsion of a finite dimensional complex when we vary the differential. This formula plays a crucial role in the variation formula of the dynamical torsion, when the representation is perturbed. Indeed, we split the dynamical torsion as the product of the torsion τ (C • (ϑ, ρ), Γ ϑ ) of some finite dimensional space of Ruelle resonant states and a renormalized value at s = 0 of the dynamical zeta function ζ X,ρ (s). Then the following formula allows us to deal with the variation of τ (C • (ϑ, ρ), Γ ϑ ).

Let (C • , ∂) be an acyclic finite dimensional complex of finite odd length n. If S : C • : C • is a linear operator, we will say that it is of degree s if S(C k ) ⊂ C k+s for any k. If S and T are two operators on C • of degrees s et t respectively then the supercommutator of S and T by [S, T ] = ST -(-1) st T S.

Cyclicity of the usual trace gives tr s,C• [S, T ] = 0 for any S, T .

Let U be a neighborhood of the origin in the complex plane and ∂(z), z ∈ U , be a family of acyclic differentials on C • which is complex differentiable at z = 0, that is,

∂(z) = ∂ + za + o(z) (2.2)
for some operator a :

C • → C • of degree 1. Note that ∂(z) • ∂(z) = 0 implies that the supercommutator [∂, a] = ∂a + a∂ = 0. (2.3)
We will denote by C • (z) the complex (C • , ∂(z)). Finally let k : C • → C • be a cochain contraction, that is a linear map of degree 1 such that

∂k + k∂ = Id C • .
(2.4)

The existence of such map is ensured by the acyclicity of (C • , ∂).

Lemma 2.4. In the above notations, for any chirality operator Γ on C • , the map z → τ (C • (z), Γ) is complex differentiable at z = 0 and

d dz z=0 log τ (C • (z), Γ) = -tr s,C • (ak).
Note that this implies in particular that tr s,C • (ak) does not depend on the chosen cochain contraction k. This is expected since if k is another cochain contraction,

[∂, akk ] = ∂akk + akk ∂ = a(k -k ),
where by (2.3) and the supertrace of a supercommutator vanishes.

Proof. First note that for non zero elements c, c ∈ det C • , we have

τ (C • (z), c) = [c : c ] • τ (C • (z), c ), (2.5) where [c : c ] ∈ C satisfies c = [c : c ] • c .
For every j = 0, . . . , n, fix a decomposition

C j = A j ⊕ B j ,
where B j = ker ∂ ∩ C j and A j is any complementary of B j in C j . Fix some basis a 1 j , . . . , a j j of A j ; then ∂a 1 j , . . . , ∂a j j is a basis of B j+1 by acyclicity of (C • , ∂). Now let

c j = a 1 j ∧ • • • ∧ a j j ∧ ∂a 1 j-1 ∧ • • • ∧ ∂a j-1 j-1 ∈ det C j , and c = c 0 ⊗ (c 1 ) -1 ⊗ c 2 ⊗ • • • ⊗ (c n ) (-1) n ∈ det C • .
Now by definition of the refined torsion, we have for |z| small enough

τ (C • (z), c) = ± n j=0 det A j (z) (-1) j+1 (2.6)
where the sign ± is independent of z and A j (z) is the matrix sending the basis a 1 j , . . . , a j j , ∂a 1 j-1 , . . . , ∂a j-1 j-1

to the basis a 1 j , . . . , a j j , ∂(z)a 1 j-1 , . . . , ∂(z)a j-1 j-1

(which is indeed a basis of C j for |z| small enough). Let k : C • → C • of degree -1 defined by k∂a m j = a m j , ka m j = 0, for every j and m ∈ {0, . . . , j }. Then k∂ + ∂k = Id C • and detA j (z) = det ∂B j-1 ⊕B j ∂(z)k ⊕ Id . Now (2.2) and (2.6) imply the desired result, because τ (C c) by (2.5).

• (z), Γ) = [c Γ : c] • τ (C • (z),

Geometrical setting and notations

We introduce here our geometrical conventions and notations. In particular, we adopt the formalism of Harvey-Polking [HP + 79] which will be convenient to compute flat traces and relate the variation of the Ruelle zeta function with topological objects.

3.1. Twisted cohomology. We consider M an oriented closed connected manifold of odd dimension n = 2r + 1. Let E → M be a flat vector bundle over M of rank d ≥ 1. For k ∈ {0, . . . , n}, we will denote the bundle Λ k T * M by Λ k for simplicity. We will denote by

Ω k (M, E) = C ∞ (M, Λ k ⊗ E) the space of E valued k-forms. We set Ω • (M, E) = n k=0 Ω k (M, E).
Let ∇ be a flat connection on E. We view the connection as a degree 1 operator (as an operator of the graded vector space Ω • (M, E))

∇ : Ω k (M, E) → Ω k+1 (M, E), k = 0, . . . , n.
The flatness of the connection reads ∇ 2 = 0 and thus we obtain a cochain complex Ω • (M, E), ∇ . We will assume that the connection ∇ is acyclic, that is, the complex Ω • (M, E), ∇ is acyclic, or equivalently, the cohomology groups

H k (M, ∇) = u ∈ Ω k (M, E) : ∇u = 0 ∇v : v ∈ Ω k-1 (M, E)
, k = 0, . . . , n, are trivial.

Currents and Schwartz kernels. Let

D • (M, E) = n k=0 D (M, Λ k ⊗ E)
the space of E-valued currents. Let E ∨ denote the dual bundle of E. We will identify D k (M, E) and the topological dual of Ω n-k (M, E ∨ ) via the non degenerate bilinear pairing

α, β = M α ∧ β, α ∈ Ω k (M, E), β ∈ Ω n-k (M, E ∨ ),
where ∧ is the usual wedge product between E-valued forms and E ∨ -valued forms.

A continuous linear operator

G : Ω • (M, E) → D • (M, E) is called homogeneous if for some p ∈ Z, we have G Ω k (M, E) ⊂ D k+p (M, E)
for every k = 0, . . . , n; the number p is called the degree of G and is denoted by deg G. In that case, the Schwartz kernel theorem gives us a twisted current

G ∈ D n+p (M × M, π * 1 E ∨ ⊗ π * 2 E) satisfying Gu, v M = G, π * 1 u ∧ π * 2 v M ×M , u ∈ Ω k (M, E), v ∈ Ω n-k-p (M, E ∨ ),
where π 1 and π 2 are the projections of M × M onto its first and second factors respectively.

3.3. Integration currents. Let N be an oriented submanifold of M of dimension d, possibly with boundary. The associated integration current

[N ] ∈ D n-d (M ) is given by [N ], ω = N i * N ω, ω ∈ Ω d (M ),
where i N : N → M is the inclusion. We have classically

d[N ] = (-1) d+1 [∂N ]. (3.1) For f ∈ Diff(M ), we will set Gr(f ) = {(f (x), x), x ∈ M } the graph of f . Note that Gr(f ) is a n-dimensional submanifold of M × M which is canonically oriented since M is.
Therefore, we can consider the integration current over Gr(f ). By definition, we have for any

α, β ∈ Ω • (M ) [Gr(f )], π * 1 α ∧ π * 2 β = M f * α ∧ β.
In particular, [Gr(f )] is the Schwartz kernel of f * : Ω

• (M ) → Ω • (M ). 3.4. Flat traces. Let G : Ω • (M, E) → D • (M, E
) be an operator of degree 0. We denote its Schwartz kernel by G and we define

WF (G) = (x, y, ξ, η), (x, y, ξ, -η) ∈ WF(G) ⊂ T * (M × M ),
where WF denotes the classical Hörmander wavefront set, cf [START_REF] Hörmander | The analysis of linear partial differential operators: Distribution theory and Fourier analysis[END_REF]§8]. We will also use the notation WF(G) = WF(G) and WF (G) = WF (G). Assume that where tr denotes the trace on E ∨ ⊗ E. We will also use the notation

WF (G) ∩ ∆(T * M ) = ∅, ∆(T * M ) = {(x, x, ξ, ξ), (x, ξ) ∈ T * M }. (3.2) Let ι : M → M × M, x → (x,
tr gr G = tr s N G,
where

N : Ω • (M, E) → Ω • (M, E) is the number operator, that is, N ω = kω for every ω ∈ Ω k (M, E).
The notation tr s is motivated by the following. Let A : C ∞ (M, F ) → D (M, F ) be an operator acting on sections of a vector bundle F . If A satisfies (3.2), we can also define a flat trace tr

A as in [DZ16, §2.4]. Now if G : Ω • (M, E) → D • (M, E) is an operator of degree 0, it gives rise to an operator G k : C ∞ (M, F k ) → D (M, F k ) for each k = 0, . . . , n, where F k = Λ k ⊗ E.
Then the link between the two notions of flat trace mentioned above is given by

tr s G = n k=0 (-1) k tr G k . If Γ ⊂ T * M is a closed conical subset, we let D • Γ (M, E) = u ∈ D • (M, E), WF(u) ⊂ Γ (3.3)
be the space of E-valued current whose wavefront set is contained in Γ, endowed with its usual topology, cf.

[Hör90, §8]. If Γ is a closed conical subset of T * (M × M ) not intersecting the conormal to the diagonal N * ∆(T * M ) = {(x, x, ξ, -ξ), (x, ξ) ∈ T * M },
then the flat trace is continuous as a map

D • Γ (M × M, π * 1 E ∨ ⊗ π * 2 E) → R. 3.5. Cyclicity of the flat trace. Let G, H : Ω • (M, E) → D • (M, E) be two homogeneous operators. We denote by G, H their respective kernels. If Γ ⊂ T * (M × M ) is a closed conical subset, we define Γ (1) = {(y, η), ∃x ∈ M, (x, y, 0, η) ∈ Γ}, Γ (2) = {(y, η), ∃x ∈ M, (x, y, -η, 0) ∈ Γ}.
Then under the assumption WF(G) (2) ∩ WF(H) (1) = ∅, the operator F = G • H is well defined by [Hör90, Theorem 8.2.14] and its Schwartz kernel F satisfies the wave front set estimate :

WF (F) ⊂ (x, y, ξ, η) | ∃(z, ζ), (x, z, ξ, ζ) ∈ WF (G) and (z, y, ζ, η) ∈ WF (H) .
If both compositions G • H and H • G are defined, we will denote by

[G, H] = G • H -(-1) deg G deg H H • G
the graded commutator of G and H. We have the following The above result follows from the cyclicity of the L 2 -trace, the approximation result [DZ16, Lemma 2.8], the relation

tr s [G, H] = tr (-1) N F, G ,
where N is the number operator and tr is the flat trace with the convention from [START_REF] Dyatlov | Dynamical zeta functions for Anosov flows via microlocal analysis[END_REF], see §3.4, and the fact that the map (G,

H) → G • H is continuous D • Γ (M × M, π * 1 E ∨ ⊗ π * 2 E) × D • Γ (M × M, π * 1 E ∨ ⊗ π * 2 E) → D • Υ (M × M, π * 1 E ∨ ⊗ π * 2 E)
for any closed conical subsets Γ, Γ ⊂ T * (M × M ) such that Γ (2) ∩ Γ (1) = ∅, and where Υ is a closed conical subset given in [Hör90, 8.2.14].

3.6. Perturbation of holonomy. Let γ : [0, 1] → M be a smooth curve and α ∈ Ω 1 (M, End(E)). Let P t (resp. Pt ) be the parallel transport

E γ(0) → E γ(t) of ∇ (resp. ∇ = ∇ + α) along γ| [0,t] . Then Pt = P t exp - t 0 P -τ α( γ(τ ))P τ dτ . (3.4)
The above formula will be useful in some occasion. For simplicity, we will denote for any

A ∈ C ∞ (M, End(E)) γ A = t 0 P -τ A(γ(τ ))P τ dτ ∈ End E γ(0)
so that P1 = P 1 expγ α(X) .

Pollicott-Ruelle resonances

4.1. Anosov dynamics. Let X be a smooth vector field on M and denote by ϕ t its flow. We will assume that X generates an Anosov flow, that is, there exists a splitting of the tangent space

T x M at every x ∈ M T x M = RX(x) ⊕ E s (x) ⊕ E u (x),
where E u (x), E s (x) are subspaces of T x M depending continuously on x and invariant by the flot ϕ t , such that for some constants C, ν > 0 and some smooth metric | • | on T M one has

|(dϕ t ) x v s | ≤ Ce -νt |v s |, t ≥ 0, v s ∈ E s (x), |(dϕ t ) x v u | ≤ Ce -ν|t| |v u |, t ≤ 0, v u ∈ E u (x).
We will use the dual decomposition

T * M = E * 0 ⊕ E * u ⊕ E * s where E * 0 = (E u ⊕ E s ) * , E * s = (E 0 ⊕ E s ) * , E * u = (E 0 ⊕ E u ) * . (4.1) 4.2.
Pollicott-Ruelle resonances. Let ι X denote the interior product with X and

L ∇ X = ∇ι X + ι X ∇ : Ω • (M, E) → Ω • (M, E)
be the Lie derivative along X acting on E-valued forms. Locally, the action of L ∇ X is given by the following. Take U a domain of a chart and write ∇ = d + A where A ∈ Ω 1 (M, End(E)). Take w 1 , . . . , w (resp. e 1 , . . . , e d ) some local basis of Λ k (resp. E) on U . Then for any

1 ≤ i ≤ and 1 ≤ j ≤ d, L ∇ X (f w i ⊗ e j ) = (Xf )w i ⊗ e j + f (L X w i ) ⊗ e j + f w i ⊗ A(X)e j , f ∈ C ∞ (U ),
where L X is the standard Lie derivative acting on forms. In particular, L ∇ X is a differential operator of order 1 acting on sections of the bundle Λ • T * M ⊗ E, whose principal part is diagonal and given by X.

Denote by Φ t

k the induced flow on the vector bundle

Λ k T * M ⊗ E → M , that is, Φ t k (β ⊗ v) = T (dϕ t ) -1 x β ⊗ P ∇ t (x)v, x ∈ M, (β, v) ∈ Λ k (T * x M ) × E x , t ∈ R,
where P ∇ t (x) is the parallel transport induced by ∇ along the curve {ϕ s (x), s ∈ [0, t]}. This induces a map

e tL ∇ X : Ω • (M, E) → Ω • (M, E).
For Re(s) big enough, the operator

L ∇ X + s acting on Ω • (M, E) is invertible with inverse (L ∇ X + s) -1 = ∞ 0 e -tL ∇ X e -st dt. (4.2)
The results of [START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov flows[END_REF] generalize to the flat bundle case as in [DR17c, §3] and the resolvent

L ∇ X + s -1 , viewed as a family of operators Ω • (M, E) → D • (M, E),
L ∇ X + s -1 = Y s 0 (s) + J(s 0 ) j=1 (-1) j-1 L ∇ X + s 0 j-1 Π s 0 (s -s 0 ) j (4.3)
where

Y s 0 (s) is holomorphic near s = s 0 , Π s 0 = 1 2πi Cε(s 0 ) L ∇ X + s -1 ds : Ω • (M, E) → D • (M, E) (4.4)
is an operator of finite rank. Here C ε (s 0 ) = {|z -s 0 | = ε} with ε > 0 small enough is a small circle around s 0 such that Res(L ∇ X ) ∩ {|z -s 0 | ≤ ε} = {s 0 }. Moreover the operators Y s 0 (s) and Π s 0 extend to continuous operators

Y s 0 (s), Π s 0 : D • E * u (M, E) → D • E * u (M, E).
(4.5)

The space

C • (s 0 ) = ran(Π s 0 ) ⊂ D • E * u (M, E) is called the space of generalized resonant states of L ∇
X associated to the resonance s 0 .

4.4. The twisted Ruelle zeta function. Fix a base point x ∈ M and identify π 1 (M ) with π 1 (M, x ). Let Per(X) be the set of periodic orbits of X. For every γ ∈ Per(X) we fix some base point x γ ∈ Im(γ) and an arbitrary path c γ joining x γ to x . This path defines an isomorphism ψ γ : π 1 (M, x γ ) ∼ = π 1 (M ) and we can thus define every γ ∈ Per(X)

ρ ∇ ([γ]) = ρ ∇ (ψ γ [γ]).
The twisted Ruelle zeta function associated to the pair (X, ∇) is defined by

ζ X,∇ (s) = γ∈G X det Id -ρ ∇ ([γ])e -s (γ) , Re(s) > C, (4.6)
where G X is the set of all primitive closed orbits of X (that is, the closed orbits that generate their class in π 1 (M )), (γ) is the length of the orbit γ and C > 0 is some big constant depending on ρ and X satisfying

ρ ∇ ([γ]) ≤ exp(C (γ)), γ ∈ G # X , (4.7)
for some norm • on End(E x ).

For every closed orbit γ, we have

| det(I -P γ )| = (-1) q det(I -P γ ), (4.8)
for some q ∈ Z not depending on γ, where P γ is the linearized Poincaré return map of γ, that is

P γ = d x ϕ -(γ) | Es(x)⊕Eu(x) for x ∈ Im(γ) (if we choose another point in Im(γ)
, the map will be conjugated to the first one. This condition is always true when ϕ t is contact, in which case we have q = dim E s .

Giuletti-Pollicott-Liverani and Dyatlov-Zworski [START_REF] Giulietti | Anosov flows and dynamical zeta functions[END_REF][START_REF] Dyatlov | Dynamical zeta functions for Anosov flows via microlocal analysis[END_REF] showed that ζ X,∇ has a meromorphic continuation to C whose poles and zeros are contained in Res(L ∇ X ); moreover, the order of ζ X,∇ near a resonance

s 0 ∈ Res(L ∇ X ) is given by 1 1 Actually, it follows from [DZ16] that m(s0) = (-1) q n-1 k=0 (-1) k m 0 k (s0), where m 0 k (s0) is the di- mension of Πs 0 Ω k (M, E) ∩ ker ιX .
We can however repeat the arguments using the identity

det(Id -Pγ) = - n k=0 (-1) k k tr Λ k dxϕ -(γ) instead of the identity det(Id -Pγ) = n-1 k=0 (-1) k tr Λ k Pγ (see [DZ16, m(s 0 ) = (-1) q+1 n k=0 (-1) k km k (s 0 ), (4.9)
where m k (s 0 ) is the rank of the spectral projector Π s 0 | Ω k (M,E) .

4.5. Topology of resonant states. Since ∇ commutes with L ∇ X , it induces a differential on the complexes C • (s 0 ) for any s 0 ∈ Res(L ∇ X ). It is shown in [START_REF] Viet | Topology of Pollicott-Ruelle resonant states[END_REF] that the complexes C • (s 0 ), ∇ are acyclic whenever s 0 = 0. Moreover, for s 0 = 0, the map

Π s 0 =0 : Ω • (M, ∇) -→ C • (s 0 = 0)
is a quasi-isomorphism, that is, it induces isomorphisms at the level of cohomology groups. Since we assumed ∇ to be acyclic, the complex C • (s 0 = 0), ∇ is also acyclic.

The dynamical torsion of a contact Anosov flow

From now on, we will assume that the flow ϕ t is contact, that is, there exists a smooth one form ϑ ∈ Ω 1 (M ) such that ϑ ∧ (dϑ) r is a volume form on M , ι X ϑ = 1 and ι X dϑ = 0. The purpose of this section is to define the dynamical torsion of the pair (ϑ, ∇). We first introduce a chirality operator Γ ϑ acting on Ω • (M, E) which is defined thanks to the contact structure. Then the dynamical torsion is a renormalized version of the twisted Ruelle zeta function corrected by the torsion of the finite dimensional space of the generalized resonant states for resonance s 0 = 0 computed with respect to Γ ϑ . This construction was inspired by the work of Braverman-Kappeler on the refined analytic torsion [START_REF] Braverman | Refined analytic torsion as an element of the determinant line[END_REF].

5.1. The chirality operator associated to a contact structure. Let V X → M denote the bundle T * M ∩ ker ι X . Note that for k ∈ {0, . . . , n}, we have the decomposition

Λ k T * M = Λ k-1 V X ∧ ϑ ⊕ Λ k V X .
(5.1)

Indeed, if α ∈ Λ k T * M we may write α = (-1) k+1 ι X α ∧ ϑ ∈Λ k-1 V X ∧ϑ + α -(-1) k+1 ι X α ∧ ϑ. ∈Λ k V X Let us introduce the Lefschetz map L : Λ • V X → Λ •+2 V X u → u ∧ dϑ.
Since dϑ is a symplectic form on V X , the maps L r-k induce bundle isomorphisms ), and study the action of L ∇ X on the bundles Λ k T * M ⊗ E rather than its action on the bundles Λ k T * M ∩ ker ιX ⊗ E, to obtain (4.9).

L r-k : Λ k V X ∼ -→ Λ 2r-k V X , k = 0, . . . ,
Definition 5.1. The chirality operator associated to the contact form ϑ is the operator

Γ ϑ : Λ • T * M → Λ n-• T * M defined by Γ 2 ϑ = 1 and Γ ϑ (f ∧ ϑ + g) = L r-k g ∧ ϑ + L r-k+1 f, f ∈ Λ k-1 V X , g ∈ Λ k V X , k ∈ {0, . . . , r}, (5.3)
where we used the decomposition (5.1).

Note that in particular one has for k ∈ {r + 1, . . . , n},

Γ ϑ (f ∧ ϑ + g) = L k-r -1 g ∧ ϑ + L k-1-r -1 f.
5.2. The refined torsion of a space of generalized eigenvectors. The operator Γ ϑ acts also on Ω • (M, E) by acting trivially on E-coefficients. Since L X ϑ = 0, Γ ϑ and L ∇ X commute so that Γ ϑ induces a chirality operator

Γ ϑ : C • (s 0 ) → C n-• (s 0 )
for every s 0 ∈ Res(L ∇ X ). Recall from §4.5 that the complexes C • (s 0 ), ∇ are acyclic. The following formula motivates the upcoming definition of the dynamical torsion.

Proposition 5.2. Let s 0 ∈ Res(L ∇ X ) \ {0, 1}. We have τ (C • (s 0 ), Γ ϑ ) -1 = (-1) Qs 0 det gr,C • (s 0 ) L ∇ X where Q s 0 = r k=0 (-1) k (r + 1 -k) dim C k (s 0 )
and τ (C • (s 0 ), Γ ϑ ) ∈ C\0 is the refined torsion of the acyclic complex C • (s 0 ), ∇ with respect to the chirality Γ ϑ , cf Definition 2.1.

Let us first admit the above proposition; the proof will be given in § §5.5,5.6.

Spectral cuts. If

I ⊂ [0, 1) is an interval, we set Π I = s 0 ∈Res(L ∇ X ) |s 0 |∈I Π s 0 and C • I = s 0 ∈Res(L ∇ X ) |s 0 |∈I C • (s 0 ). Note that L ∇ X + s acts on C • (s 0 ) for every s 0 ∈ Res(L ∇ X ) as -s 0 Id +J where J is nilpotent. We thus have for s / ∈ Res(L ∇ X ) det gr,C • I L ∇ X + s (-1) q+1 = s 0 ∈Res(L ∇ X ) |s 0 |∈I (s -s 0 ) m(s 0 ) , (5.4)
where det gr is the graded determinant, cf. §2.5.

Let λ ∈ [0, 1) such that Res(L ∇ X ) ∩ {s ∈ C : |s| = λ} = ∅. Now define the meromorphic function ζ (λ,∞) X,∇ (s) = ζ X,∇ (s)det gr,C • [0,λ] L ∇ X + s (-1) q .
(5.5) Then (4.9) and (5.4) show that ζ (λ,∞)

X,∇ has no pole nor zero in {|s| ≤ λ}, so that the number ζ (λ,∞) X,∇ (0) is well defined. 5.4. Definition of the dynamical torsion. Let 0 < µ < λ < 1 such that for every s 0 ∈ Res(L ∇ X ), one has |s 0 | = λ, µ. Using Proposition 2.2 and Proposition 5.2 we obtain, with notations of §5.3,

τ C • [0,λ] , Γ ϑ = (-1) Q (µ,λ] det gr,C • (µ,λ] L ∇ X -1 τ C • [0,µ] , Γ ϑ ,
where for an interval I we set

Q I = s 0 ∈Res(L ∇ X ) |s 0 |∈I Q s 0 .
This allows us to give the following Definition 5.3 (Dynamical torsion). The number

τ ϑ (∇) = (-1) Q [0,λ] ζ (λ,∞) X,∇ (0) (-1) q • τ C • [0,λ] , Γ ϑ ∈ C \ 0 (5.6)
is independent of the spectral cut λ ∈ (0, 1). We will call this number the dynamical torsion of the pair (ϑ, ∇).

Remark 5.4. If c X,∇ s m(0) is the leading term of the Laurent expansion of ζ X,∇ (s) at s = 0, then taking λ small enough actually shows that

τ ϑ (∇) = (-1) Q 0 c (-1) q X,∇ • τ C • (0), Γ ϑ . (5.7) In particular, if 0 / ∈ Res(L ∇ X ), τ ϑ (∇) = ζ X,∇ (0) (-1) q .
(5.8)

Note that we could have taken (5.7) as a definition of the dynamical torsion; however (5.6) is more convenient to study the regularity of the τ ϑ (∇) with respect to ϑ and ∇.

Remark 5.5. This definition actually makes sense even if ∇ is not acyclic; in that case the formula (5.6) defines an element of the determinant line det

H • (M, ∇), under the identifica- tion H • (M, ∇) = H • C • [0,λ] , ∇ given by the quasi-isomorphism Π [0,λ] : Ω • (M, E) → C • [0,λ] , cf §4
.5. However, we will focus here on the acyclic case.

The rest of this section is devoted to the proof of Proposition 5.2. 5.5. Invertibility of the contact signature operator. To prove Proposition 5.2 we shall use §2.4 and introduce the contact signature operator

B ϑ = Γ ϑ ∇ + ∇Γ ϑ : D • (M, E) → D • (M, E),
where Γ ϑ acts trivially on E. We fix in what follows some s 0 ∈ Res(L ∇ X ) \ {0, 1} and we denote C • (s 0 ) by C • for simplicity. We also set

C • 0 = C • ∩ ker(ι X ).
The following result will put us in position to apply Proposition 2.3.

Lemma 5.6. The operator B ϑ is invertible C • → C • . Proof. We set C • even = k even C k , C • odd = k odd C k .
Then B ϑ preserves the decomposition

C • = C • even ⊕ C • odd . Note that because Γ 2 ϑ = 1, we have B ϑ | C • even = Γ ϑ B ϑ | C • odd Γ ϑ . It thus suffices to show that B ϑ is injective on C • even . Let β ∈ C • even such that B ϑ β = 0. Write β = r k=0 β 2k ∈ C • even , with β 2k = f 2k-1 ∧ ϑ + g 2k , f 2k-1 ∈ C 2k-1 0 , g 2k ∈ C 2k 0 , k = 0, . . . , r. Then B ϑ β = 0 writes, since Γ ϑ ∇(C k ) ⊂ C n-k-1 and ∇Γ ϑ (C k ) ⊂ C n-k+1 , Γ ϑ ∇β 2k + ∇Γ ϑ β 2(k+1) = 0, k = 0, . . . , r.
(5.9)

Because ∇ does not leave the decomposition (5.1) stable, we need to introduce an operator Ψ :

C • 0 → C •+1 0
which mimics the action of ∇. We define

Ψµ = ∇µ -(-1) k L ∇ X µ ∧ ϑ, µ ∈ C k 0 .
(5.10)

Because L X dϑ = 0, the map Ψ satisfies the simple relation

Ψ µ ∧ dϑ j = (Ψµ) ∧ dϑ j , µ ∈ C • 0 , j ∈ N, (5.11)
that is, Ψ commutes with L . Also, observe that

Ψ 2 µ = -L ∇ X µ ∧ dϑ, µ ∈ C • 0 .
(5.12) Indeed, using the fact that L ∇ X and ∇ commute,

Ψ 2 µ = ∇ ∇µ -(-1) k L ∇ X µ ∧ ϑ -(-1) k+1 L ∇ X ∇µ -(-1) k L ∇ X µ ∧ ϑ ∧ ϑ = ∇ 2 µ + (-1) k+1 ∇ L ∇ X µ ∧ ϑ + (-1) k L ∇ X ∇µ ∧ ϑ -L ∇ X 2 µ ∧ ϑ ∧ ϑ = (-1) k+1 (-1) k L ∇ X µ ∧ dϑ. Assume first that k ≤ r/2 -1. Then 2k + 2 ≤ r;
we can thus write, with (5.10) in mind,

Γ ϑ ∇β 2k = Γ ϑ ∇f 2k-1 ∧ ϑ -f 2k-1 ∧ dϑ + ∇g 2k = Γ ϑ Ψf 2k-1 ∧ ϑ -L ∇ X f 2k-1 ∧ ϑ ∧ ϑ -f 2k-1 ∧ dϑ + Ψg 2k + L ∇ X g 2k ∧ ϑ = Ψf 2k-1 + L ∇ X g 2k ∧ dϑ r-2k + Ψg 2k -f 2k-1 ∧ dϑ ∧ dϑ r-2k-1 ∧ ϑ.
Similarly we find by (5.11)

∇Γ ϑ β 2k+2 = ∇ f 2k+1 ∧ dϑ r-2k-1 + g 2k+2 ∧ dϑ r-2k-2 ∧ ϑ = Ψf 2k+1 -L ∇ X f 2k+1 ∧ ϑ ∧ dϑ r-2k-1 + Ψg 2k+2 ∧ ϑ + g 2k+2 ∧ dϑ ∧ dϑ r-2k-2 .
(5.13) Thus (5.9) writes, with the decompostion (5.1) in mind,

Ψf 2k+1 + g 2k+2 ∧ dϑ r-2k-1 + Ψf 2k-1 + L ∇ X g 2k ∧ dϑ r-2k = 0 (5.14) and -L ∇ X f 2k+1 ∧ dϑ + Ψg 2k+2 ∧ dϑ r-2k-2 + Ψg 2k -f 2k-1 ∧ dϑ ∧ dϑ r-2k-1 = 0. (5.15)
Then applying Ψ to (5.15) gives, with (5.12) and (5.11),

-ΨL ∇ X f 2k+1 -L ∇ X g 2k+2 ∧ dϑ r-2k-1 -L ∇ X g 2k ∧ dϑ r-2k -Ψf 2k-1 ∧ dϑ r-2k = 0.
Note that Ψ commutes with L ∇ X and thus with L ∇ X -1 (which exists since s 0 = 0). Then applying L ∇ X -1 to the above relation we get

-Ψf 2k+1 -g 2k+2 ∧ dϑ r-2k-1 -g 2k ∧ dϑ r-2k -L ∇ X -1 Ψf 2k-1 ∧ dϑ r-2k = 0.
Injecting this in (5.14), we obtain

L ∇ X -Id g 2k + Id -L ∇ X -1 Ψf 2k-1 ∧ dϑ r-2k = 0. Since L r-2k is injective on C 2k 0 and L ∇ X -Id is invertible (since s 0 = 1), this yields L ∇ X g 2k + Ψf 2k-1 = 0. (5.16) Applying L ∇ X -1 Ψ to the above equation we get Ψg 2k -f 2k-1 ∧ dϑ = 0
(5.17) by (5.11); thus (5.15) gives

Ψg 2k+2 -L ∇ X f 2k+1 ∧ dϑ ∧ dϑ r-2k-2 = 0.
Now repeating this process with k replaced by k -1 we obtain Ψg 2k -L ∇ X f 2k-1 ∧ dϑ ∧ dϑ r-2k = 0. This implies with (5.17) that

Id -L ∇ X f 2k-1 ∧ dϑ r-2k+1 = 0, which leads to f 2k-1 = 0 since L r-(2k-1) is injective on C 2k-1 0 and L ∇ X -Id is invertible on C • ; thus g 2k = 0 by (5.16), since L ∇ X is invertible. We therefore obtained β 2k = 0, k ≤ r/2 -1. Next assume k ≥ (r + 1)/2. Set k = r -k and β2 k+1 = Γ ϑ β 2k ∈ C 2 k+1 0 , β2 k-1 = Γ ϑ β 2k+2 ∈ C 2 k-1 0 . Then (5.9) writes Γ ϑ ∇ β2 k-1 + ∇Γ ϑ β2 k+1 = 0.
Since 2 k + 1 ≤ r and we can do exactly as before to get β2 k-1 = 0 which leads to β 2k+2 = 0. Therefore we obtained

β 2k = 0, k ≥ (r + 1)/2 + 1.
Therefore it remains to show that β 2p = 0 and β 2(p+1) = 0, where p = r/2 . We will assume that r = 2p + 1 is odd and put p = p + 1 (the case r even is similar). Then (5.9) implies, since β 2k = 0 for every k = p, p ,

∇Γ ϑ β 2p + Γ ϑ ∇β 2p = 0, Γ ϑ ∇β 2p = 0, ∇Γ ϑ β 2p = 0.
(5.18)

We can compute, keeping (5.10) in mind,

∇Γ ϑ β 2p = ∇ L -1 g 2p ∧ ϑ + f 2p -1 = ΨL -1 g 2p ∧ ϑ + L ∇ X L -1 g 2p ∧ ϑ ∧ ϑ + L -1 g 2p ∧ dϑ + Ψf 2p -1 -L ∇ X f 2p -1 ∧ ϑ, and
Γ ϑ ∇β 2p = Γ ϑ Ψg 2p + L ∇ X g 2p ∧ ϑ + Ψf 2p-1 ∧ ϑ -L ∇ X f 2p-1 ∧ ϑ ∧ ϑ -f 2p-1 ∧ dϑ = Ψg 2p ∧ ϑ -f 2p-1 ∧ dϑ ∧ ϑ + L ∇ X g 2p ∧ dϑ + Ψf 2p-1 ∧ dϑ.
Therefore the first equation of (5.18) implies, since

L -1 g 2p ∧ dϑ = g 2p , ΨL -1 g 2p -L ∇ X f 2p -1 -f 2p-1 ∧ dϑ + Ψg 2p = 0 (5.19)
and

g 2p + Ψf 2p -1 + Ψf 2p-1 ∧ dϑ + L ∇ X g 2p ∧ dϑ = 0. (5.20) Applying L ∇ X -1 Ψ to (5.19) leads to -g 2p -Ψf 2p -1 -ΨL ∇ X -1 f 2p-1 ∧ dϑ + -g 2p ∧ dϑ = 0. Therefore, Id -L ∇ X -1 Ψf 2p-1 + L ∇ X -Id g 2p ∧ dϑ.
(5.21)

As before this gives Ψf 2p-1 + L ∇ X g 2p = 0 and thus with (5.20) one gets

L ∇ X g 2p + Ψf 2p-1 = 0, g 2p + Ψf 2p -1 = 0. (5.22) Next compute ∇Γ ϑ β 2p = g 2p ∧ dϑ 2 + Ψf 2p-1 ∧ dϑ 2 + Ψg 2p ∧ ϑ ∧ dϑ -L ∇ X f 2p+1 ∧ ϑ ∧ dϑ 2
Therefore the third part of (5.18) gives (we take the ∧ϑ component of the above equation)

-L ∇ X f 2p-1 ∧ dϑ 2 + Ψg 2p ∧ dϑ = 0. Applying L ∇ X -1 Ψ to (5.22) we get Ψg 2p = f 2p-1 ∧ dϑ;
we therefore obtain that f 2p-1 = 0 by injectivity of L 2 on C r-2 0 . Thus g 2p = 0 by (5.21). Finally compute

∇β 2p = Ψf 2p -1 ∧ ϑ + Ψg 2p + L ∇ X g 2p ∧ ϑ = 0.
Therefore the second part of (5.18) implies (since Γ ϑ ∇β 2p = 0 is equivalent to

∇β 2p = 0) Ψf 2p -1 + L ∇ X g 2p = 0.
Therefore by (5.22) we get L ∇ X -Id g 2p = 0, and thus g 2p = 0. Using (5.19) we conclude that L ∇ X f 2p -1 = 0 which leads to f 2p -1 = 0.

5.6. Proof of Proposition 5.2. We start from Proposition 2.3 which gives us, in view of Lemma 5.6,

τ (C • , Γ ϑ ) = (-1) r dim C r + det Γ ϑ ∇| C r + (-1) r r-1 j=0 det Γ ϑ ∇| C j + ⊕C n-j-1 + (-1) j .
(5.23)

where we set as in §2.4

C • + = C • ∩ ker(∇Γ ϑ ), C • -= C • ∩ ker(Γ ϑ ∇).
We first note that for k ∈ {0, . . . , r} and β ∈ Ω k (M, E), one has

∇Γ ϑ β = L r-k ∇β -(-1) k ι X ∇β ∧ ϑ + L ι X ∇ι X β -ι X β ∧ ϑ + (-1) k L r-k+1 β -∇ι X β + (-1) k ι X (β -∇ι X β) ∧ ϑ , Γ ϑ ∇β =L r-k-1 ∇β -(-1) k ι X ∇β ∧ ϑ ∧ ϑ + (-1) k L r-k ι X ∇β , (5.24)
where L j-r = (L r-j | Λ j V X ) -1 for 0 ≤ j ≤ r. Indeed, using the decomposition (5.1),

Γ ϑ β = (-1) k+1 ι X β ∧ dϑ r-k+1 + β + (-1) k ι X β ∧ ϑ ∧ dϑ r-k ∧ ϑ = (-1) k+1 ι X β ∧ dϑ r-k+1 + β ∧ dϑ r-k ∧ ϑ,
which leads to

∇Γ ϑ β = (-1) k+1 ∇ι X β ∧ dϑ r-k+1 + ∇β ∧ dϑ r-k ∧ ϑ + (-1) k β ∧ dϑ r-k+1 = (-1) k+1 (-1) k+1 ι X ∇ι X β ∧ ϑ ∧ dϑ r-k+1 + (-1) k+1 ∇ι X β + (-1) k ι X ∇ι X β ∧ ϑ ∧ dϑ r-k+1 + ∇β -(-1) k ι X ∇β ∧ ϑ ∧ dϑ r-k ∧ ϑ + (-1) k β + (-1) k ι X β ∧ ϑ ∧ dϑ r-k+1 -ι X β ∧ dϑ r-k+1 ∧ ϑ,
which is exactly the first part of (5.24). The second part follows directly from the decomposition (5.1).

Let us introduce, for k ∈ {0, . . . , r}, the operator J k : C k → C k defined by

J k β = f ∧ ϑ-(-1) k Ψf (5.25) for any β = f ∧ ϑ + g ∈ C k with f ∈ C k-1 0 and g ∈ C k 0 ,
and where Ψ is defined in (5.10). Then we claim that J k takes it values in C k + . Indeed, we have for any

f ∈ C k-1 0 and g ∈ C k 0 , ∇Γ ϑ (f ∧ ϑ + g) = ∇ g ∧ dϑ r-k ∧ ϑ + f ∧ dϑ r-k+1 = Ψg ∧ dϑ r-k ∧ ϑ + (-1) k g ∧ dϑ r-k+1 + Ψf ∧ dϑ r-k+1 + (-1) k+1 L ∇ X f ∧ dϑ r-k+1 ∧ ϑ, which implies that β = f ∧ ϑ + g lies in C k + if and only if Ψg + (-1) k+1 L ∇ X f ∧ dϑ ∧ dϑ r-k = 0 and Ψf + (-1) k g ∧ dϑ r-k+1 = 0. (5.26) But now note that if β = f ∧ ϑ + g = J k β = f ∧ ϑ -(-1) k Ψf for some β = f ∧ ϑ + g then f = f and g = -(-1) k
Ψf , and thus β satisfies the second part of (5.26). We also obtain Ψg = -(-1) k Ψ 2 f = -(-1) k L ∇ X f ∧ dϑ by (5.12), so the first part of (5.26) is also satisfied. Therefore J k : C k → C k + ; moreover it is obvious that J k is a projector. Therefore we can consider the restricted projection

J k | C k + : C k + → C k + ,
we will still denote by J k . The next lemma will be helpful to compute the determinants lying in the product (5.23).

Lemma 5.7.

Take k ∈ {0, • • • , r -1}. Then for any β = f ∧ ϑ + g ∈ C k + with f ∈ C k-1 0 and g ∈ C k 0 , one has (Γ ϑ ∇) 2 β = L ∇ X L ∇ X -Id β -L ∇ X -Id J k β.
Proof. Since k < r we can write, thanks to (5.24),

Γ ϑ ∇β = ∇β ∧ ϑ ∧ dϑ r-k-1 + (-1) k ι X ∇β ∧ dϑ r-k .
Therefore

∇Γ ϑ ∇β = -(-1) k ∇β ∧ dϑ r-k + (-1) k ∇ι X ∇β ∧ dϑ r-k = (-1) k L ∇ X -Id ∇β ∧ dϑ r-k = ι X ∇ι X ∇β -ι X ∇β ∧ ϑ ∧ dϑ r-k + (-1) k (L ∇ X -Id) ∇β -(-1) k ι X ∇β ∧ ϑ ∧ dϑ r-k ,
where we used ∇ι X ∇β = L ∇ X ∇β and ι X ∇ι X ∇β = L ∇ X ι X ∇β. Since β ∈ C k + one has with (5.24)

∇β -(-1) k ι X ∇β ∧ ϑ ∧ dϑ r-k = ι X β -ι X ∇ι X β ∧ dϑ r-k+1 .

This leads to

∇Γ ϑ ∇β = ι X ∇ι X ∇β -ι X ∇β ∧ ϑ ∧ dϑ r-k + (-1) k L ∇ X -Id ι X β -ι X ∇ι X β ∧ dϑ r-k+1 . Since ι X ∇ι X ∇β -ι X ∇β = L ∇ X -Id ι X ∇β and ι X β -ι X ∇ι X β = Id -L ∇ X ι X β, we obtain ∇Γ ϑ ∇β = L ∇ X -Id ι X ∇β ∧ ϑ ∧ dϑ r-k + (-1) k L ∇ X -Id Id -L ∇ X ι X β ∧ dϑ r-k+1 ,
and thus by definition of Γ ϑ

Γ ϑ ∇Γ ϑ ∇β = -(-1) k Id -L ∇ X 2 ι X β ∧ ϑ + L ∇ X -Id ι X ∇β.
(5.27)

Now, writing β = f ∧ ϑ + g where ι X f = 0 and ι X g = 0, we have

∇β = ∇f ∧ ϑ -(-1) k f ∧ dϑ + ∇g, ι X ∇β = L ∇ X f ∧ ϑ + (-1) k ∇f + L ∇ X g, ι X β ∧ ϑ = -(-1) k f ∧ ϑ.
(5.28) Injecting those relations in (5.27) we get

Γ ϑ ∇Γ ϑ ∇β = L ∇ X L ∇ X -Id (f ∧ ϑ + g) -L ∇ X -Id f ∧ ϑ -(-1) k ∇f + (-1) k L ∇ X f ∧ ϑ ,
which concludes in view of (5.10) and (5.25).

We now deal with the case k = r.

Lemma 5.8. One has, for β ∈ C r + ,

Γ ϑ ∇β = (-1) r L ∇ X -Id β + (Id -J r )β .
Proof. We have

Γ ϑ ∇β = L -1 ∇β -(-1) r ι X ∇β ∧ ϑ + (-1) r ι X ∇β.
Since β ∈ C r + we have with (5.24) that ∇β -

(-1) r ι X ∇β ∧ ϑ = (ι X β -ι X ∇ι X β) ∧ dϑ. Therefore, Γ ϑ ∇β = (ι X β -ι X ∇ι X β) ∧ ϑ + (-1) r ι X ∇β.
We now conclude as in the previous lemma, using (5.28).

We are now in position to finish the proof of Proposition 5.2. We will set, for 0

≤ k ≤ n, m k = dim C k , m 0 k = dim C k 0 , m ± k = dim C k ± . First take k ∈ {0, • • • , r -1}. First take k ∈ {0, • • • , r -1}. Because B ϑ is invertible on C • , Γ ϑ ∇ induces an isomorphism C k + → C n-k-1 + . Take any basis γ of C k + . Then Γ ϑ ∇γ is a basis of C n-k-1 + and the matrix of Γ ϑ ∇| C k + ⊕C n-k+1 + in the basis γ ⊕ Γ ϑ ∇γ is 0 (Γ ϑ ∇) 2 γ Id 0 , (5.29)
where

(Γ ϑ ∇) 2 γ is the matrix of (Γ ϑ ∇) 2 | C k + in the basis γ. Define Jk = Id -J k : C k + → C k + .
Then Jk is a projector (since J k is) and Lemma 5.7 implies that J k (and thus Jk ) commutes with L ∇ X . Moreover one has

(Γ ϑ ∇) 2 | ker Jk = L ∇ X -Id 2 , (Γ ϑ ∇) 2 | ran Jk = L ∇ X L ∇ X -Id . As a consequence, det (Γ ϑ ∇) 2 | C k + = s 0 (1 + s 0 ) m + k -m 0 k-1 (1 + s 0 ) 2m 0 k-1 = s 0 m + k -m 0 k-1 (1 + s 0 ) m + k +m 0 k-1 ,
because on C • (and in particular on C k + ), one has L ∇ X = -s 0 Id +ν where ν is nilpotent, and one has dim ker Jk = dim ranJ k = m 0 k-1 . Indeed, by (5.25) we can view J k as a map C k-1 0 → C k + , which is obviously injective. We finally obtain with (5.29)

det Γ ϑ ∇| C k + ⊕C n-k+1 + = (-1) m + k s 0 m + k -m 0 k-1 (1 + s 0 ) m + k +m 0 k-1 .
(5.30)

We now deal with the case k = r. Lemma 5.8 gives

Γ ϑ ∇| ker Jr = (-1) r L ∇ X -Id , Γ ϑ ∇| ran Jr = (-1) r L ∇ X . As before, we obtain det Γ ϑ ∇| C r + = (-1) rm + r (-1) m + r s 0 m + r -m 0 r-1 (1 + s 0 ) m 0 r-1 .
(5.31)

Combining (5.23) with (5.30) and (5.31) we finally obtain

τ (C • , Γ ϑ ) = (-1) J s 0 K (1 + s 0 ) L (5.32)
where

J = r k=0 (-1) k m + k , K = r k=0 (-1) k (m + k -m 0 k-1 ), L = r-1 k=0 (-1) k (m + k -m 0 k ).
Note that for 0 ≤ k ≤ r -1 one has by acyclicity and because Γ ϑ induces isomorphisms

C k + C n-k - (since B ϑ is invertible), m + k = m - n-k = dim ker (∇| C n-k ) = dim ran (∇| C n-k-1 ) = m n-k-1 -m - n-k-1 = m k+1 -m + k+1 . Therefore m + k + m + k+1 = m k+1 , 0 ≤ k ≤ r -1, (5.33) which leads to m + k + m + k+1 = m 0 k + m 0 k+1 . As a consequence, since m + 0 = m 0 = m 0 0 , we get m + r -m 0 r = -(m + r-1 -m 0 r-1 ) = • • • = (-1) r (m + 0 -m 0 0 ) = 0. This implies m 0 k = m + k , 0 ≤ k ≤ r, (5.34)
which leads to L = 0. Moreover, since m 0 k = m 0 2r-k , we get

K = r k=0 (-1) k (m 0 k -m 0 k-1 ) = 2r k=0 (-1) k m 0 k = - n k=0 (-1) k km k = (-1) q m(s 0 ),
where we used (4.9) in the last equality. Finally, again because

m 0 k = m 0 2r-k , 2J = (-1) r m 0 r + 2r k=0 (-1) k m 0 k = (-1) r m 0 r - n k=0 (-1) k km k .
We have

(-1) r m 0 r = r k=0 (-1) k m k , n k=0 (-1) k km k = r k=0 (-1) k (2k -n)m k ,
where the first equality comes from (5.33) and (5.34) and the second from the fact that m k = m n-k . We thus obtained

J = r k=0 (-1) k (r + 1 -k)m k = Q s 0 ,
and finally by (5.32) τ (C • , Γ ϑ ) = (-1) Qs 0 (-s 0 ) (-1) q m(s 0 )

But now recall from (5.4) that det gr,C • L ∇ X (-1) q+1 = (-s 0 ) m(s 0 ) . This completes the proof.

6. Invariance of the dynamical torsion under small perturbations of the contact form

In this section, we are interested in the behaviour of the dynamical torsion when we deform the contact form. Namely, we prove here the Theorem 7. Assume that (ϑ t ) t∈(-δ,δ) is a smooth family of contact forms such that their Reeb vector fields X t generate a contact Anosov flow for each t. Let (E, ∇) be an acyclic flat vector bundle. Then the map t → τ ϑt (∇) is real differentiable and we have

d dt τ ϑt (∇) = 0.
We will thus consider a family of contact forms and set ϑ = ϑ 0 and X = X 0 . We also fix an acyclic flat vector bundle (E, ∇).

6.1. Anisotropic spaces for a family of vector fields. To study the dynamical torsion when the dynamics is perturbed, we construct with the help of [Bon18] some anisotropic Sobolev spaces on which each X t has nice spectral properties. We refer to Appendix B where we briefly recall the construction of these spaces.

By §B.4, the set (t, s), s / ∈ Res(L ∇ Xt ) is open in (-δ, δ) × C. Fix λ ∈ (0, 1) such that Res(L ∇ X ) ∩ {|s| ≤ λ} ⊂ {0}. (6.1)
Then for t close enough to 0, we have Res(L ∇ Xt )∩{|s| = λ} = ∅ so that the spectral projectors

Π t = 1 2iπ |s|=λ (L ∇ Xt + s) -1 ds : Ω • (M, E) → D • (M, E) (6.2)
are well defined. The next proposition is a brief summary of the results from Appendix B. We will denote for any C, ρ > 0,

Ω(c, ρ) = {Re(s) > c} ∪ {|s| ≤ ρ} ⊂ C. (6.3)
Proposition 6.1. There is c, ε 0 > 0 such that for any ρ > 0 there exists anisotropic Sobolev spaces

Ω • (M, E) ⊂ H • 1 ⊂ H • ⊂ D • (M, E)
, each inclusion being continuous with dense image, such that the following holds.

(1) For each t ∈ [-ε 0 , ε 0 ], the family s → L ∇ Xt + s is a holomorphic family of (unbounded) Fredholm operators

H • 1 → H • 1 and H • → H • of index 0 in the region Ω(c, ρ). Moreover L ∇ Xt ∈ C 1 [-ε 0 , ε 0 ], L(H • 1 , H • ) .
(2) For every relatively compact open region Z ⊂ int Ω(c, ρ) such that Res(L ∇ X ) ∩ Z = ∅, there exists t Z > 0 such that

L ∇ Xt + s -1 ∈ C 0 [-t Z , t Z ] t , Hol Z s , L(H • 1 , H • ) .
(

) Π t ∈ C 1 [-ε 0 , ε 0 ] t , L(H • , H • 1 ) . 3
We will thus fix such Hilbert spaces for some ρ > c + 1. We denote

C • t = ran Π t ⊂ H • , Π = Π t=0 and C • = ran Π. 6.2. Variation of the torsion part. Let Γ t : C • t → C n-• t
be the chirality operator associated to X t , c.f. §5.1. The next lemma allows us to compute the variation of the finite dimensional torsion part of the dynamical torsion. Lemma 6.2. We have that t

→ τ (C • t , Γ t ) is real differentiable and d dt τ (C • t , Γ t ) = -tr s,C • t Π t ϑ t ι Ẋt τ (C • t , Γ t ),
where Ẋt = d dt X t .

Proof. By Proposition 6.1, the operator Π t | C • : C • → C • t is invertible for t close enough to 0 and we will denote by Q t its inverse. Then for t close enough to 0, one has

τ (C • t , Γ t ) = τ (C • , Γt ), where Γt = Q t Γ t Π t | C • : C • → C • because
d dt τ (C • t , Γ t ) = 1 2 tr s,C • Γt Γt τ (C • t , Γ t ),
where Γt = d dt Γt : C • → C • . Since Γ t and Π t commute, and by the two first points of Proposition 6.1, we can apply (A.2) to get

Γt = ΠΓ t Π| C • + tΠ ΓΠ + o C • →C • (t). This leads to ΓΓ = Π ΓΓ| C • ,
where we removed the subscripts t to signify that we take all the t-dependent objects at t = 0. Therefore,

1 2 tr s,C • ΓΓ = 1 2 tr s,C • Π ΓΓ ,
Now notice that Γ 2 t = 1 implies Γ Γ + ΓΓ = 0. Therefore, for every k ∈ {0, . . . , r},

tr C n-k Γ Γ = tr C k ΓΓ ΓΓ = tr C k ΓΓ = -tr C k Γ Γ.
Therefore we only need to compute tr C k Γ Γ for k ∈ {0, . . . , r} to get the full super trace tr s,C • ΓΓ . Since n is odd we have

1 2 tr s,C • ΓΓ = 1 2 tr C • (-1) N +1 ΠΓ Γ = r k=0 (-1) k+1 tr C k ΠΓ Γ .
Let k ∈ {0, . . . , r} and α ∈ Ω k (M ). Using the decomposition

α = (-1) k-1 ι Xt α ∧ ϑ t + α + (-1) k ι Xt α ∧ ϑ t ,
we get by definition of Γ t

Γ t α = (-1) k-1 ι Xt α ∧ (dϑ t ) r-k+1 + α + (-1) k ι Xt α ∧ ϑ t ∧ (dϑ t ) r-k ∧ ϑ t . Therefore, Γt α = (-1) k-1 ι Ẋt α ∧ (dϑ t ) r-k+1 + (r -k + 1)(-1) k-1 ι Xt α ∧ d θt ∧ (dϑ t ) r-k + (-1) k ι Ẋt α ∧ ϑ t + ι Xt α ∧ θt ∧ (dϑ t ) r-k ∧ ϑ t + α + (-1) k ι Xt α ∧ ϑ t ∧ (dϑ t ) r-k ∧ θt + (r -k) α + (-1) k ι Xt α ∧ ϑ t ∧ d θt ∧ (dϑ t ) r-k-1 ∧ ϑ t
Now we use the decompositions

d θt = -ι Xt d θt ∧ ϑ t + d θt + ι Xt d θt ∧ ϑ t , θt = θt (X t )ϑ + θt -θt (X t )ϑ , ι Ẋt α = (-1) k ι Xt ι Ẋt α ∧ ϑ t + ι Ẋt α + (-1) k+1 ι Xt ι Ẋt α ∧ ϑ t
to get, again by definition,

Γ Γα = (-1) k-1 ι Ẋ α + (-1) k+1 ι X ι Ẋ α ∧ ϑ ∧ ϑ + (-1) k-1 L r-k -1 (-1) k ι X ι Ẋ α ∧ (dϑ) r-k+1 + (r -k + 1) L r-k+1 -1 (-1) k-1 ι X α ∧ d θ + ι X d θ ∧ ϑ ∧ (dϑ) r-k ∧ ϑ -(r -k + 1) (-1) k-1 ι X α ∧ ι X d θ + (-1) k ι X α ∧ θ -θ(X)ϑ + L r-k+1 -1 α + (-1) k ι X α ∧ ϑ ∧ (dϑ) r-k ∧ θ -θ(X)ϑ ∧ ϑ + α + (-1) k ι X α ∧ ϑ θ(X) + (r -k) L r-k -1 α + (-1) k ι X α ∧ ϑ ∧ d θ + ι X d θ ∧ ϑ ∧ (dϑ) r-k-1 , (6.4)
where again we removed the subscripts t to signify that we take everything at t = 0. Now let A k : C k 0 → C k 0 (note that here C k 0 is C k ∩ ker ι X , cf §5.1, and not C k t at t = 0) defined by

A k u = (r -k) L r-k -1 u ∧ d θ + ι X d θ ∧ (dϑ) r-k-1 .
Note that the maps defined by the second, the fourth, the fifth and the sixth terms of the right hand side of (6.4) are anti-diagonal, that is they have the form 0 0 in the

decomposition C • = C •-1 0 ∧ ϑ ⊕ C • 0 . Therefore, since A r = 0 (we also set A -1 = 0), r k=0 (-1) k+1 tr C k ΠΓ Γ = r k=0 (-1) k+1 tr C k Πϑι Ẋ + tr C k 0 Π θ(X) + r k=0 (-1) k+1 tr C k-1 0 ΠA k-1 + tr C k 0 ΠA k = r k=0 (-1) k+1 tr C k Πϑι Ẋ + tr C k 0 Π θ(X) . (6.5) But now note that if α = f ∧ ϑ + g ∈ C k-1 0 ∧ ϑ ⊕ C k 0 then ϑ ∧ ι Ẋ α = ϑ( Ẋ)(f ∧ ϑ) + ϑ ∧ ι Ẋ g.
This shows that for every k ∈ {0, . . . , n} one has

tr C k Πϑι Ẋ = tr C k-1 0 Πϑ( Ẋ).
(6.6) Injecting this relation in (6.5) we obtain, with ϑ( Ẋ) = -θ(X) and the formula θ(X)

| C 2r-k 0 L r-k = L r-k θ(X)| C k 0 , r k=0 (-1) k+1 tr C k ΠΓ Γ = r k=0 (-1) k+1 tr C k-1 0 Πϑ( Ẋ) -tr C k 0 Πϑ( Ẋ) = 2r k=0 (-1) k tr C k 0 Πϑ( Ẋ).
But this concludes since by (6.6) we have Xt,∇ (0), cf. §5.3. For t close enough to 0, let P t : T M → T M be defined by

P t : ker ϑ ⊕ RX → ker ϑ ⊕ RX t , v + µX → v + µX t .
For simplicity, we will still denote Λ k ( T P t ) : Λ k T * M → Λ k T * M by P t . Then formula (5.4) of [START_REF] Nguyen | The Fried conjecture in small dimensions[END_REF] gives that for Re(s) big enough, t → ζ Xt,∇ (s) is differentiable and we have for every ε > 0 small enough

d dt t=0 log ζ X,∇ (s) = (-1) q s tr s Ṗ (L ∇ X + s) -1 e -ε(L ∇ X +s) ,
where Ṗ = d dt t=0 P t . One can show that for every k ∈ {0, . . . , n} and β ∈ Λ k T * M one has Ṗ β = ϑ ∧ ι Ẋ β. (6.7) Therefore (we differentiated at t = 0 but we can do the same for small t)

d dt log ζ Xt,∇ (s) = (-1) q s tr s ϑ t ι Ẋt (L ∇ Xt + s) -1 e -ε(L ∇ X t +s) . (6.8)
Now let us compute the variation of the [0, λ] part of ζ (λ,∞) (s).

Lemma 6.3. We have

d dt log det gr,C • t L ∇ Xt + s (-1) q+1 = (-1) q+1 tr s,C • t ϑ t ι Ẋt L ∇ Xt (L ∇ Xt + s) -1 .
Proof. We are in a position to apply Lemma A.2 which gives

d dt log det gr,C • t L ∇ Xt + s (-1) q+1 = (-1) q+1 tr gr,C • t Π t L ∇ Ẋt (L ∇ Xt + s) -1 . Denote A t = P -1 t Ṗt .
Then one can verify that

ι Xt = P -1 t ι X P t , which leads to L ∇ Ẋt = -∇A t ι Xt + ∇ι Xt A t -A t ι Xt ∇ + ι Xt A t ∇.
Using (-1) N N ∇ = ∇(-1) N +1 (N + 1), (-1) N N ι Xt = ι Xt (-1) N -1 (N -1), and the cyclicity of the trace, we get since (L ∇ Xt + s) -1 commute with ι Xt and ∇,

tr C • t (-1) N +q+1 N Π t L ∇ Ẋt (L ∇ Xt + s) -1 = (-1) q+1 tr C • t Π t A t (-1) N (N + 1)ι Xt ∇ + (-1) N N ∇ι Xt -(-1) N N ι Xt ∇ -(-1) N (N -1)∇ι Xt (L ∇ Xt + s) -1 = (-1) q+1 tr C • t (-1) N Π t A t L ∇ Xt (L ∇ Xt + s) -1
Therefore using (6.7) again this concludes, because P t=0 = Id.

6.4. Proof of Theorem 7. Combining this lemma and (6.8) we obtain that for Re(s) big enough and t small enough

ζ (λ,∞) Xt,∇ (s) ζ (λ,∞) X 0 ,∇ (s) = exp -s t 0 tr s ϑ t ι Ẋτ (L ∇ Xτ + s) -1 e -ε(L Xτ +s) dτ - t 0 tr s,C • τ Π τ ϑ t ι Ẋτ L ∇ Xτ (L ∇ Xτ + s) -1 dτ (-1) q+1
. (6.9)

Note that for every s / ∈ Res(L ∇ Xt ) we have

L ∇ Xt (L ∇ Xt + s) -1 = Id -s(L ∇ Xt + s) -1 , so that tr s,C • t Π t ϑ t ι Ẋt L ∇ Xt (L ∇ Xt + s) -1 = tr s,C • t Π t ϑ t ι Ẋt -str s,C • t Π t ϑ t ι Ẋt (L ∇ Xt + s) -1 . (6.10)
We now fix s 0 ∈ C with Re(s 0 ) big enough so that (6.9) is valid and a smooth path c :

[0, 1] → C with c(0) = 0, c(1) = s 0 and c(u) / ∈ Res(L ∇ X ), u ∈ (0, 1]. Let δ, t 0 > 0 small enough so that dist {|s| = λ} ∪ (V δ ∩ {|s| ≥ λ}), Res(L ∇ Xt ) ≥ 2δ, |t| ≤ t 0 , (6.11)
where V δ is the open δ-neighborhood of Im c. We moreover ask that

(Res(L ∇ Xt ) ∩ {|s| ≤ λ}) ⊂ {|s| ≤ δ} and (V δ ∩ {|s| ≥ λ}) ∩ Res(L ∇ Xt ) = ∅. For t ∈ [-t 0 , t 0 ] and s / ∈ Res(L ∇ Xt ) we define Y t (s) = L ∇ Xt + s -1 (Id -Π t ). ( 6
(s) = L ∇ Xt + s -1 e -ε L ∇ X t +s .
Then [DGRS18, Proposition 6.3] gives that the map

[-t 0 , t 0 ] × {|s| = λ} (t, s) → Q t (s) ∈ D n Γ (M × M, E ∨ E) is bounded for some closed conic subset Γ ⊂ T * (M × M ) not intersecting the conormal of the diagonal. Moreover by §B.7, we have that [-t 0 , t 0 ] t → Π t is bounded in D n Ws×Wu (M × M, E ∨ E), and so is the map [-t 0 , t 0 ] × {|s| = λ} → L ∇ Xt + s -1 Π t .
As a consequence (6.12), (6.13) and (6.14) imply that the map Since the flat trace is the usual trace for operators of finite rank,

[-t 0 , t 0 ] × {|s| ≤ 3δ/2} (t, s) → Y t (s) ∈ D n Γ (M × M, E ∨ E), ( 6
tr s ϑ t ι Ẋt Q t (s) -tr s,C • Π t ϑ t ι Ẋt (L ∇ Xt + s) -1 = tr s ϑ t ι Ẋt L ∇ Xt + s -1 (Id -Π t )e -ε(L ∇ X t +s) + tr s,C • Π t ϑ t ι Ẋt (L ∇ Xt + s) -1 e -ε(L ∇ X t
+s) -Id .

Then (6.16), (6.17) and (6.12) imply that the right hand side of the last equation is continuous with respect to t with values in holomorphic functions on (

V δ ∩ {|s| ≥ 5δ/4}) ∪ {|s| ≤ 3δ/2} (indeed s → (L ∇ Xt + s) -1 e -ε(L ∇ X t +s) -Id is holomorphic of C • t )
, and so is the left hand side. As a consequence, (6.10) shows that both members of (6.9) are holomorphic on this region and

ζ (λ,∞) Xt,∇ (0) = ζ (λ,∞) X 0 ,∇ (0) exp - t 0 tr s,C • τ Π τ ϑι Ẋτ dτ (-1) q+1
.

Comparing this with Lemma 6.2 we obtain Theorem 7 by definition of the dynamical torsion, cf §5.4.

Variation of the connection

In this section we compute the variation of the dynamical torsion when the connection is perturbed. This formula will be crucial to compare the dynamical torsion and Turaev's refined combinatorial torsion.

7.1. Real-differentiable families of flat connections. Let U ⊂ C be some open set and consider ∇(z), z ∈ U , a family of flat connections on E. We will assume that the map z → ∇(z) is C 1 , that is, there exists continuous maps z → µ z , ν z ∈ Ω 1 (M, End(E)) such that for any z 0 ∈ U one has

∇(z) = ∇(z 0 ) + Re(z -z 0 )µ z 0 + Im(z -z 0 )ν z 0 + o(z -z 0 ), (7.1)
where o(z -z 0 ) is understood in the Fréchet topology of Ω 1 (M, End(E)). We will denote for any σ ∈ C

α z 0 (σ) = Re(σ)µ z 0 + Im(σ)ν z 0 ∈ Ω 1 (M, End(E)). (7.2)
Note that since the connections ∇(z) are assumed to be flat, we have

[∇(z), α z (σ)] = ∇(z)α z (σ) + α z (σ)∇(z) = 0. (7.3) 7.2. A cochain contraction induced by the Anosov flow. For z ∈ U let L ∇(z) X + s -1 = J(0) j=1 -L ∇(z) X j-1 Π 0 (z) s j + Y (z) + O(s) (7.4)
be the development (4.3) for the resonance s 0 = 0. Let C • (0; z) = ran Π 0 (z). Recall from §4.5 that since ∇(z) is acyclic, the complex (C • (0; z), ∇(z)) is acyclic. Therefore there exists a cochain contraction k(z) : C • (0; z) → C • (0; z), i.e. a map of degree -1 such that

∇(z)k(z) + k(z)∇(z) = Id C • (0;z) . (7.5)
We now define

K(z) = ι X Y (z)(Id -Π 0 (z)) + k(z)Π(z) : Ω • (M, E) → D • (M, E). (7.6)
A crucial property of the operator K is that it satisfies the chain homotopy equation

∇(z)K(z) + K(z)∇(z) = Id Ω • (M,E) .
(7.7) 7.3. The variation formula. For simplicity, we will set for every z ∈ U τ (z) = τ ϑ (∇(z)).

The operators K(z) defined above are involved in the variation formula of the dynamical torsion, as follows.

Proposition 7.1. The map z → τ (z) is real differentiable; we have for every z ∈ U and ε > 0 small enough

d(log τ ) z σ = -tr s α z (σ)K(z)e -εL ∇(z) X , σ ∈ C. (7.8)
The proof of the previous proposition is similar of that of the last subsection, i.e. we compute the variation of each part of the dynamical torsion. The rest of this section is devoted to the proof of Proposition 7.1. 7.4. Anisotropic Sobolev spaces for a family of connections. Fix some z 0 ∈ U . Recall from §6.1 that we chose some anisotropic Sobolev spaces

H • 1 ⊂ H • . Notice that L ∇(z) X = L ∇(z 0 ) X + β(z)(X), (7.9)
where β(z) ∈ Ω 1 (M, End(E)) is defined by

∇(z) = ∇(z 0 ) + β(z). Therefore (7.1) implies that L ∇(z) X -L ∇(z 0 ) X
is a C 1 family of pseudo-differential operators of order 0, and thus forms a C 1 family of bounded operators

H • → H • and H • 1 → H •
1 by construction of the anisotropic spaces and standard rules of pseudo-differential calculus (see for example [START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov flows[END_REF]). As a consequence and thanks to Proposition 6.1, we are in position to apply [START_REF] Kato | Perturbation theory for linear operators[END_REF]Theorem 3.11]; thus if δ is small enough we have that

R ρ = (z, s) ∈ C 2 , |z -z 0 | < δ, s ∈ Ω(c, ρ), s / ∈ σ H • (L ∇(z) X ) is open, (7.10) where σ H • (L ∇(z) X ) denotes the resolvent set of L ∇(z) X
on H • , and Ω(c, ρ) is defined in (6.3). Moreover (7.1) and (7.9) imply that for any open set Z ⊂ Ω(c, ρ) such that Res L ∇(z 0 ) X ∩Z = ∅, there exists δ Z > 0 such that for any j ∈ {0, 1}, to obtain that for some closed conic set Γ ⊂ T * (M × M ) not intersecting the conormal to the diagonal and any ε > 0 small enough, the map (s,

L ∇(z) X + s -1 ∈ C 1 |z -z 0 | < δ Z , Hol Z s , L H • j , H • j . ( 7
z) → K(s, z) is bounded from Z × {|z -z 0 | < δ Z } with values D Γ (M × M, π * 1 E ∨ ⊗ π * 2 E), where K(s, z) is the Schwartz kernel of the shifted resolvent L ∇(z) X + s -1 e -εL ∇(z) X . 7.5. A family of spectral projectors. Fix λ ∈ (0, 1) such that {s ∈ C, |s| ≤ λ} ∩ Res L ∇(z 0 ) X ⊂ {0}.
(7.12)

Thanks to (7.10), if z is close enough to z 0 ,

{s ∈ C, |s| = λ} ∩ Res L ∇(z) X = ∅, (7.13)
by compacity of the circle. For z ∈ U we will denote by

Π(z) = 1 2iπ |s|=λ L ∇(z) X + s -1 ds (7.14) the spectral projector of L ∇(z) X
on generalized eigenvectors for resonances in {s ∈ C, |s| ≤ λ}, and C • (z) = ran Π(z). It follows from (7.11), (7.13) and (7.14) that the map

z → Π(z) ∈ L(H • j , H • j )
is C 1 for j = 0, 1. We can therefore apply A.3 to get, for δ small enough,

Π(z) ∈ C 1 {|z -z 0 | < δ} z , L(H • , H • 1 ) . (7.15)
7.6. Variation of the finite dimensional part. Because (C • (z 0 ), ∇(z 0 )) is acyclic, there exists a cochain contraction k(z 0 ) : C • (z 0 ) → C •-1 (z 0 ), cf §2.6. The next lemma computes the variation of the finite dimensional part of the dynamical torsion.

Lemma 7.2. The map z → c(z) = τ (C • (z), Γ) is real differentiable at z = z 0 and d(log c) z 0 σ = -tr s,C • Π(z 0 )α z 0 (σ)k(z 0 ), σ ∈ C.
Proof. By continuity of the family z → Π(z), we have that Π(z

)| C • (z 0 ) : C • (z 0 ) → C • (z) is an isomorphism for |z -z 0 | small enough, of inverse denoted by Q(z).
For those z we denote by C • (z) the graded vector space C • (z 0 ) endowed with the differential

∇(z) = Q(z)∇(z)Π(z) : C • (z 0 ) → C • (z 0 ).
Then because Γ commutes with every Π(z) one has

τ ( C • (z), Γ) = τ (C • (z), Γ) (7.16)
By (7.15) we can apply (A.2) in the proof of Lemma A.2 which gives for any h small enough

∇(z 0 + σ)Π(z 0 ) = Π(z 0 )∇(z 0 )Π(z 0 ) + Π(z 0 )α z 0 (σ)Π(z 0 ) + o C • (z 0 )→C • (z 0 ) (σ).
Therefore the real differentiable version of Lemma 2.4 implies the desired result. 7.7. Variation of the zeta part. We give a first Proposition which computes the variation of the Ruelle zeta function in its convergence region.

Proposition 7.3 (Variation of the dynamical zeta function). For Re(s) big enough, the map z → g s (z) = ζ X,∇(z) (s) is C 1 near z = z 0 and we have for every ε > 0 small enough

d(log g s ) z 0 σ = (-1) q+1 e -εs tr s α z 0 (σ)ι X L ∇(z 0 ) X + s -1 e -εL ∇(z 0 ) X .
Proof. Let ϕ t denote the flow of X. For γ ∈ G X , dϕ -(γ) |γ will denote dϕ -(γ) taken at any point of the image of γ; this ambiguity will not stand long since another choice of base point will lead to a conjugated linear map, and we aim to take traces. We have the standard factorization, for Re(s) big enough and any z near z 0 ,

g s (z) = exp n k=0 (-1) k k γ∈G X # (γ) (γ) tr ρ ∇(z) (γ)e -s (γ) tr Λ k (dϕ -(γ) )| γ det(I -P γ ) , (7.17)
where

P γ = dϕ -(γ)
Eu⊕Es is the linearized Poincaré map of γ, and # (γ) is the primitive period of γ. Now (3.4) implies

tr ρ ∇(z 0 +σ) (γ) = tr ρ ∇(z 0 ) (γ) -tr ρ ∇(z 0 ) (γ) γ α z 0 (σ)(X) + o(σ) (γ).
As a consequence, the sum in (7.17) is C 1 near z = z 0 for Re(s) big enough, and

d(log g s ) z 0 σ = - n k=0 (-1) k k γ∈G X # (γ) (γ) tr ρ ∇(z 0 ) (γ) γ α z 0 (σ)(X) e -s (γ) tr Λ k (dϕ -(γ) )| γ det(I -P γ ) .
Now a slight extension of Guillemin trace formula [START_REF] Victor Guillemin | Lectures on spectral theory of elliptic operators[END_REF] gives, in D (R >0 ),

tr α z 0 (σ)(X)e -tL ∇ X Ω k (M,E) = γ # (γ) (γ) tr ρ ∇(z 0 ) (γ) γ α z 0 (σ)(X) tr Λ k dϕ -(γ) |det(I -P γ )| δ(t-(γ)),
where δ is the Dirac distribution. But now recall from §4.4 that | det(I -P γ )| = (-1) q det(I -P γ ). Therefore, if ε > 0 satisfies ε < (γ) for all γ, arguing exactly as in [DZ16, §4], with (4.2) in mind,

d(log g s ) z 0 σ = e -εs (-1) q+1 tr gr α z 0 (σ)(X) L ∇(z 0 ) X + s -1 e -εL ∇(z 0 ) X .
Now it remains to turn the graded trace tr gr into a super trace tr s keeping in mind the relation tr gr = tr s (N •) where N is the number operator, cf. §3.4. Note that

α z 0 (σ)(X) = [α z 0 (σ), ι X ] = α z 0 (σ) • ι X + ι X • α z 0 (σ). We therefore have N α z 0 (σ)(X) = N [α z 0 (σ), ι X ] = N α z 0 (σ)ι X + ι X (N -1)α z 0 (σ) = N α z 0 (σ)ι X -(N -1)α z 0 ι X + [(N -1)α, ι X ]. Since ι X commutes with L ∇(z 0 ) X + s -1 e -εL ∇(z 0 ) X one finally obtains N α z 0 (σ)(X) = α z 0 (σ)ι X L ∇(z 0 ) X + s -1 e -εL ∇(z 0 ) X + (N -1)α z 0 (σ) L ∇(z 0 ) X + s -1 e -εL ∇(z 0 ) X , ι X .
This concludes by cyclicity of the flat trace.

The following lemma is a direct consequence of Lemma A.2 and the fact that Π 0 (z 0 ) = Π(z 0 ) by (7.12).

Lemma 7.4. For Re(s) big enough, the map

z → h s (z) = det gr,C • (z) L ∇(z) X + s (-1) q+1 is C 1 near z = z 0 and d(log h s ) z 0 σ = (-1) q+1 tr s,C • (z 0 ) Π 0 (z 0 )α z 0 (σ)ι X L ∇(z 0 ) X + s -1 .
7.8. Proof of Proposition 7.1. Combining the two lemmas of the preceding subsection we obtain for Re(s) big enough, the map z → ζ (λ,∞) X,∇(z) (s) = g s (z)/h s (z) is real differentiable at z = z 0 (and therefore on U since we may vary z 0 ). Moreover for every ε > 0 small enough d log

g s h s z σ = (-1) q+1 e -εs tr s α z (σ)ι X L ∇(z) X + s -1 e -εL ∇(z) X -tr s,C • (z) Π 0 (z)α z (σ)ι X L ∇(z) X + s -1 . ( 7
(-1) q+1 d (log b) z σ = tr s α z (σ)ι X Y (z)(Id -Π 0 (z))e -εL ∇(z) X +tr s,C • (z) Π 0 (z)α z (σ)ι X Q z (ε) , where Q z (ε) = n≥1 (-ε) n n! L ∇(z) X n-1 : C • (z) → C • (z). Recall that if c(z) = τ (C • (z), Γ) one has τ (z) = c(z)b(z) (-1) q . Therefore Lemma 7.2 gives, with what precedes, d(log τ ) z σ = -tr s α z (σ)K(z)e -εL ∇(z) X -tr s,C • (z) Π 0 (z)α z (σ) k(z) Id -e -εL ∇(z) X + ι X Q z (ε) . (7.19)
We have Id -e -εL ∇(z)

X = -L ∇(z) X Q z (ε), which leads to ι X Q z (ε) + k(z) Id -e -εL ∇(z) X = ι X -k(z)L ∇(z) X Q z (ε).
But now since k(z) is a cochain contraction, we get

ι X -k(z)L ∇(z) X = [∇(z), k(z)ι X ].
Because ∇(z) commutes with Π 0 (z) and L ∇(z)

X , we obtain with (7.3)

∇(z), Π 0 (z)α z (σ)k(z)ι X Q z (ε) = Π 0 α ι X Q z (ε) + k(z) Id -e -εL ∇(z) X .
This concludes by (7.19) and the cyclicity of the trace.

Euler structures, Chern-Simons classes

The Turaev torsion is defined using Euler structures, introduced by Turaev [Tur90], whose purpose is to fix sign ambiguities of combinatorial torsions. We shall use however the representation in terms of vector fields used by Burghelea-Haller [START_REF] Burghelea | Euler structures, the variety of representations and the Milnor-Turaev torsion[END_REF]. The goal of the present section is to introduce these Euler structures, in view of the definition of the Turaev torsion.

8.1. The Chern-Simons class of a pair of vector fields. If X ∈ C ∞ (M, T M ) is a vector field with isolated non degenerate zeros, we define the singular 0-chain

div(X) = - x∈Crit(X) ind X (x)[x] ∈ C 0 (M, Z),
where Crit(X) is the set of critical points of X and ind X (x) denotes the Poincaré-Hopf index of x as a critical point of X2 . Note also that div (-X) = -div(X) since M is odd dimensional.

Let X 0 , X 1 be two vector fields with isolated non degenerate zeros. Let p : M × [0, 1] → M be the projection over the first factor and choose a smooth section H of the bundle p * T M → M × [0, 1], transversal to the zero section, such that H restricts to X i on {i} × M for i = 0, 1. Then the set H -1 (0) ⊂ M × [0, 1] is an oriented smooth submanifold of dimension 1 with boundary (it is oriented because M and [0, 1] are), and we denote by [H -1 (0)] its fundamental class.

Definition 8.1. The class

p * [H -1 (0)] ∈ C 1 (M, Z)/∂C 2 (M, Z),
where p * is the pushforward by p, does not depend on the choice of the homotopy H relating X 0 and X 1 , cf. [BH06, §2.2]. This is the Chern-Simons class of the pair (X 0 , X 1 ), denoted by cs(X 0 , X 1 ).

We have the fundamental formulae ∂cs(X 0 , X 1 ) = div(X 1 ) -div(X 0 ), cs(X 0 , X 1 ) + cs(X 1 , X 2 ) = cs(X 0 , X 2 ), (8.1)

for any other vector field with non degenerate zeros X 2 . Notice also that if X 0 and X 1 are nonsingular vector fields, then cs(X 0 , X 1 ) defines a homology class in H 1 (M, Z).

8.2. Euler structures. Let X be a smooth vector field on M with non degenerate zeros. An Euler chain for X is a singular one-chain e ∈ C 1 (M, Z) such that ∂e = div(X). Euler chains for X always exist because M is odd-dimensional hence χ(M ) = 0.

Two pairs (X 0 , e 0 ) and (X 1 , e 1 ), with X i a vector field with non degenerate zeros and e i an Euler chain for X i , i = 0, 1, will be said to be equivalent if

e 1 = e 0 + cs(X 0 , X 1 ) ∈ C 1 (M, C)/∂C 2 (M, Z).
(8.2) Definition 8.2. An Euler structure is an equivalence class [X, e] for the relation (8.2). We will denote by Eul(M ) the set of Euler structures.

There is a free and transitive action of H 1 (M, Z) on Eul(M ) given by

[X, e] + h = [X, e + h], h ∈ H 1 (M, Z).
8.3. Homotopy formula relating flows. Let X 0 , X 1 be two vector fields with non degenerate zeros. Let H be a smooth homotopy between X 0 and X 1 as in §8.1 and set

X t = H(t, •) ∈ C ∞ (M, T M ). For ε > 0 we define Φ ε : M × [0, 1] → M × M × [0, 1] via Φ ε (x, t) = e -εXt (x), x, t , x ∈ M, t ∈ [0, 1].
Set also, with notations of §3.3,

H ε = Gr(Φ ε ) ⊂ M × M × R. Then H ε is a submanifold with boundary of M × M × R which is oriented (since M and R are). Define [H ε ] = (Φ ε ) * ([M ] × [[0, 1]]) ∈ D n (M × M × R)
to be the associated integration current, cf. §3.3. Let g be any metric on M and let ρ > 0 be smaller than its injectivity radius. Then for any x, y ∈ M with dist(x, y) ≤ ρ, we denote by P (x, y) ∈ Hom(E x , E y ) the parallel transport by ∇ along the minimizing geodesic joining x to y.

Then P ∈ C ∞ (M × M, π * 1 E ∨ ⊗ π * 2 E) and we can define R ε = -π * [H ε ] ⊗ P ∈ D n-1 (M × M, π * 1 E ∨ ⊗ π * 2 E), where π : M × M × R → M × M is the projection over the two first factors. Note that R ε is well defined if ε is small enough so that dist x, e -sXt (x) ≤ ρ, s ∈ [0, ε], t ∈ [0, 1], x ∈ M, (8.3) which implies supp π * [H ε ] ⊂ {(x, y), dist(x, y) ≤ ρ}. Now, let R ε : Ω • (M, E) → D •-1 (M, E)
be the operator of degree -1 whose Schwartz kernel is R ε .

Lemma 8.3. We have the following homotopy formula

[∇, R ε ] = ∇R ε + R ε ∇ = e -εL ∇ X 1 -e -εL ∇ X 0 . (8.4)
Proof. First note that because M is odd dimensional, the boundary (computed with orientations) of the manifold H ε is

∂H ε = Gr(e -εX 0 ) × {0} -Gr(e -εX 1 ) × {1}, Therefore we have, cf. (3.1), (-1) n d M ×M π * [H ε ] = π * [∂H ε ] = Gr(e -εX 0 ) -Gr(e -εX 1 )
where Gr(e -εX i ) denotes the integration current on the manifold Gr(e -εX i ) for i = 0, 1. Now note that we have by construction ∇ E ∨ E P = 0. Therefore

∇ E ∨ E R ε = (-1) n
Gr(e -εX 1 ) -Gr(e -εX 0 ) ⊗ P.

Note that by definition of e

-L ∇ X i (cf §4.2), the formula (8.3) and the flatness of ∇ imply that the Schwartz kernel of e

-εL ∇ X i is Gr(e -εX i ) ⊗ P . This concludes because the Schwartz kernel of [∇, R ε ] is (-1) n ∇ E ∨ E R ε , cf. [HLJ01, Lemma 2.2].
The next formula follows from the definition of the flat trace and the Chern-Simons classes. It will be crucial for the topological interpretation of the variation formula obtained in §7.

Lemma 8.4. We have for any α ∈ Ω • (M, End(E)) such that tr α is closed and ε > 0 small enough tr s αR ε = tr α, cs(X 0 , X 1 ) .

(

Note that because H is transverse to the zero section, we have

WF(R ε ) ∩ N * ∆ = ∅, (8.6)
where N * ∆ denotes the conormal to the diagonal ∆ in M × M , so that the above flat trace is well defined.

Proof. We denote by i :

M → M × M the diagonal inclusion. Note that the Schwartz kernel of αR ε is (-1) n π * 2 α ∧ R ε = -π * 2 α ∧ R ε since n is odd.
From the definition of the super flat trace tr s , we find that

tr s αR ε = tr i * (π * 2 α ∧ π * [H ε ] ⊗ P ) , 1 , (8.7)
where π 2 : M × M → M is the projection over the second factor. Of course we have

i * P = Id E ∈ C ∞ (M, End(E)). We therefore have tr i * (π * 2 α ∧ π * [H ε ] ⊗ P ) = tr α ∧ i * π * [H ε ] = tr α ∧ p * j * [H ε ]
where

j : M × [0, 1] → M × M × [0, 1], (x, t) → (x, x, t). This leads to tr s αR ε = tr α ∧ p * j * [H ε ], 1 = p * tr α, j * [H ε ] . Now if ε is small enough, we can see that j * [H ε ] = [H -1 (0)].
Therefore tr s αR ε = tr α, p * [H -1 (0)] = tr α, cs(X 0 , X 1 ) .

Morse theory and variation of Turaev torsion.

We introduce here the Turaev torsion which is defined in terms of CW decompositions. In the spirit of the seminal work of Bismut-Zhang [START_REF] Bismut | An extension of a theorem by Cheeger and Müller[END_REF] based on geometric constructions of Laudenbach [START_REF] Laudenbach | On the Thom-Smale complex[END_REF], we use a CW decomposition which comes from the unstable cells of a Morse-Smale gradient flow induced by a Morse function. This allows us to interpret the variation of the Turaev torsion as a supertrace on the space of generalized resonant states for the Morse-Smale flow. This interpretation will be convenient for the comparison of the Turaev torsion with the dynamical torsion. 9.1. Morse theory and CW-decompositions. Let f be a Morse function on M and X = -grad g f be its associated gradient vector field with respect to some Riemannian metric g (the tilde notation is used to make the difference with the Anosov flows we studied until now). For any a ∈ Crit(f ), we denote by

W s (a) = y ∈ M, lim t→∞ e t X y = a , W u (a) = y ∈ M, lim t→∞ e -t X y = a ,
the stable and unstable manifolds of a. Then it is well known that W s (a) (resp.

W u (x)) is a smooth embedded open disk of dimension n -ind f (a) (resp. ind f (a)), where ind f (a) is the index of a as a critical point of f , that is, in a Morse chart (z 1 , . . . , z n ) near a, f (z 1 , . . . , z n ) = f (a) -z 2 1 -• • • -z 2 ind f (a) + z 2 ind f (a)+1 + • • • + z 2 n .
For simplicity, we will denote

|a| = ind f (a) = dim W u (a),
and we fix an orientation of every W u (a).

We assume that X satisfies the Morse-Smale condition, that is, for any a, b ∈ Crit(f ), the manifolds W s (a) and W u (b) are transverse. Also, we assume that for every a ∈ Crit(f ), the metric g is flat near a. Let us summarize some results from [Qin10, Theorems 3.2,3.8,3.9] ensured by the unstable manifolds of f . We would like to mention that such results can be found in slightly different form in the work of Laudenbach [START_REF] Laudenbach | On the Thom-Smale complex[END_REF] and are used in [START_REF] Bismut | An extension of a theorem by Cheeger and Müller[END_REF] 3 . First, W u (a) admits a compactification to a smooth |a|-dimensional manifold with corner W u (a), endowed with a smooth map e a : W u (a) → M that extends the inclusion

W u (a) ⊂ M . Then the collection W = W u (a) a∈Crit(f )
and the applications e a induce a CW-decomposition on M . Moreover, the boundary operator of the cellular chain complex is given by

∂W u (a) = |b|=|a|-1 #L(a, b)W u (b),
where L(a, b) is the moduli space of gradient lines joining a to b and #L(a, b) is the sum of the orientations induced by the orientations of the unstable manifolds of (a, b), see [Qin10, Thm 3.9]. 9.2. The Thom-Smale complex. For k = 0, . . . , n, we set

C • (W, E ∨ ) = n k=0 C k (W, E ∨ ), C k (W, E ∨ ) = a∈Crit(f ) |a|=k E ∨ a , k = 0, . . . , n.
We endow the complex C • (W, E ∨ ) with the boundary operator ∂ ∇ ∨ defined by

∂ ∇ ∨ u = |b|=|a|-1 γ∈L(a,b) ε γ P γ (u), a ∈ Crit(f ), u ∈ E ∨ a ,
where for γ ∈ L(a, b),

P γ ∈ End(E ∨ a , E ∨ b )
is the parallel transport of ∇ ∨ along the curve γ and ε γ = ±1 is the orientation number of γ ∈ L(a, b).

Then by [START_REF] Laudenbach | On the Thom-Smale complex[END_REF] (see also [START_REF] Viet | Topology of Pollicott-Ruelle resonant states[END_REF] for a different approach), there is a canonical isomorphism

H • (M, ∇ ∨ ) H • (W, ∇ ∨ ),
where H • (M, ∇ ∨ ) is the singular homology of flat sections of (E ∨ , ∇ ∨ ) and H • (W, ∇ ∨ ) denotes the homology of the complex C • (W, E ∨ ) endowed with the boundary map ∂ ∇ ∨ . Therefore this complex is acyclic since ∇ (and thus ∇ ∨ ) is. 9.3. The Turaev torsion. Fix some base point x ∈ M and for every a ∈ Crit(f ), let γ a be some path in M joining x to a. Define

τ e,o (∇) -1 = ϕ C•(W,∇ ∨ ) (c) ∈ C \ 0, where ϕ C•(W,∇ ∨ )
is the homology version of the isomorphism (2.1). Note that ∇ ∨ (and not ∇) is involved in the definition of τ e,o (∇). Indeed, we use here the cohomological version of Turaev's torsion, which is more convenient for our purposes, and which is consistent with [START_REF] Braverman | Ray-Singer type theorem for the refined analytic torsion[END_REF], [BK + 08, p. 252]. 9.4. Resonant states of the Morse-Smale flow. In [START_REF] Viet | Topology of Pollicott-Ruelle resonant states[END_REF], it has been shown that we can define Ruelle resonances for the Morse-Smale gradient flow L ∇ X as described in §4 in the context of Anosov flows. More precisely, we have that the resolvent

L ∇ X + s -1 : Ω • (M, E) → D • (M, E),
is well defined for Re(s) 0, has a meromorphic continuation to all s ∈ C. The poles of this continuation are the Ruelle resonances of L ∇ X and the set of those will be denoted by Res(L ∇ X ). In fact, the set Res(L ∇ X ) does not depend on the flat vector bundle (E, ∇). Let λ > 0 be such that Res

(L ∇ X ) ∩ {|s| ≤ λ} ⊂ {0}; set Π = 1 2πi |s|=λ L ∇ X + s -1 ds (9.3)
the spectral projector associated to the resonance 0, and denote by

C • = ran Π ⊂ D • (M, E)
the associated space of generalized eigenvectors for L ∇ where α z (σ) is given by (7.2) and e is given by (9.1).

The rest of this section is devoted to the proof of Proposition 9.1. For convenience, we will first study the variation of z → τ e,o (∇(z) ∨ ). 9.6. A preferred basis. Let a ∈ Crit(f ) and k = |a|. We denote by [W u (a)] ∈ D n-k Γ (M ) the integration current over the unstable manifold W u (a) of X, it is a well defined current far from ∂W u (a). We also pick a cut-off function χ a ∈ C ∞ (M ) valued in [0, 1] with χ a ≡ 1 near a and χ a is supported in a small neighborhood Ω a of a, with Ω a ∩ ∂W u (a) = ∅. Recall from 9.3 that we have a basis u 1,a , . . . , u d,a of E a . Using the parallel transport of ∇, we obtain flat sections of E over W u (a) that we will still denote by u 1,a , . . . , u d,a . Define ũj,a = Π χ a [W u (a)] ⊗ u j,a ∈ C n-k , j = 1, . . . , d.

(9.4)

By [START_REF] Viet | Spectral analysis of Morse-Smale flows II: resonances and resonant states[END_REF] we have that ũj,a , a ∈ Crit(f ), 1 ≤ j ≤ d is a basis of C • . Adapting the proof of [DR17a, Theorem 2.6] to the bundle case, we obtain the following proposition which will allow us to compute the Turaev torsion with the help of the complex C • .

Proposition 9.2. The map Φ : C • (W, ∇) → C n-• defined by Φ u j,a = ũj,a , a ∈ Crit(f ), j = 1, . . . , d, is an isomorphism and satisfies4

Φ • ∂ ∇ = (-1) • ∇ • Φ.
An immediate corollary is that (using the notation of §2.2)

τ e,o (∇ ∨ ) = ϕ C•(W,∇) (u) -1 = τ ( C • , ũ), (9.5)
where u ∈ det C • (W, ∇) (resp. ũ ∈ det C • ) is the element given by the basis {u j,a } (resp. {ũ j,a }) and the ordering of the cells W u (a). 9.7. Proof of Proposition 9.1. For any a ∈ Crit(f ) we denote by P γa (z) ∈ Hom(E x , E a ) the parallel transport of ∇(z) along γ a . We set u j,a (z) = P γa (z)P -1 γa (z 0 )u j,a and ũj,a (z

) = Π(z) χ a [W u (a)] ⊗ u j,a (z)
,

where again we consider u j,a (z) as a ∇(z)-flat section of E over W u (a) using the parallel transport of ∇(z). The construction of Ruelle resonances for Morse-Smale gradient flow follows from the construction of anisotropic Sobolev spaces [START_REF] Viet | Spectral analysis of Morse-Smale gradient flows[END_REF], on which L ∇ X + s is a holomorphic family of Fredholm operators of index 0 in the region {Re(s) > -2}, and such that ∇(z) is bounded H • 1 → H • . Every argument made in §7.4 also stand here and z → Π(z) is a C 1 family of bounded operators H • → H • 1 . Note that by continuity, Π(z) induces an isomorphism C • (z 0 ) → C • (z) for z close enough to zero. Let ũ(z) ∈ det C • (z) be the element given by the basis {ũ j,a (z)} and the ordering of the cells W u (a). Then by (9.5) and (2.5) we have

Ω • (M, E) ⊂ H • 1 ⊂ H • ⊂ D • (M, E), see
τ e,o (∇(z) ∨ ) = τ C • (z), ũ(z) = ũ(z) : Π(z)ũ(z 0 ) τ C • (z), Π(z)ũ(z 0 ) , (9.6) where Π(z)ũ(z 0 ) ∈ det C • (z) is the image of ũ by the isomorphism det C • (z 0 ) → det C • (z)
induced by Π(z), and ũ(z) = ũ(z) : Π(z)ũ(z 0 ) Π(z)ũ(z 0 ). Doing exactly as in §7.6, we obtain

that z → τ (z) = τ C • (z), Π(z)ũ is C 1 and d(log τ ) z 0 σ = -tr s, C • Π(z 0 )α z 0 (σ) k(z 0 ). (9.7)
Therefore it remains to compute the variation of ũ(z) : Π(z)ũ(z 0 ) . This is the purpose of the next formula.

Lemma 9.3. We have

ũ(z) : Π(z)ũ(z 0 ) = a∈Crit(f ) det P γa (z)P -1 γa (z 0 ) (-1) n-|a| .
Proof. By definition of the basis {u a,j } in §9.3 it suffices to show that for z small enough

Π(z)ũ a,i = d j=1 A j a,i (z)ũ a,j (z), a ∈ Crit(f ), 1 ≤ i, j ≤ d, (9.8)
where the coefficients A j a,i (z) are defined by u a,i (z 0 )(a) = d j=1

A j a,i (z)u a,j (z)(a).

Consider the dual operator L

∇(z) ∨ -X : Ω • (M, E ∨ ) → Ω • (M, E ∨ ).
The above constructions, starting from a dual basis s 1 , . . . , s d ∈ E ∨

x of u 1 , . . . , u d , give a basis {s a,i (z)} of each Γ(W s (a), ∇(z) ∨ ) (the space of flat section of ∇(z) ∨ over W s (a)), since the unstable manifolds of -X are the stable ones of X. Let C • ∨ (z) be the range of the spectral projector Π ∨ (z) from (9.3) associated to the vector field -X and the connection ∇(z) ∨ . We have a basis {s a,i (z)} of C • ∨ (z) given by sa,i (z

) = Π ∨ (z) χ a [W s (a)] ⊗ s a,i (z)
.

We will prove that for any a, b ∈ Crit(f ) with same Morse index we have for any 1 ≤ i, j ≤ d, sa,j (z), ũa,i (z 0 ) = s a,j (z)(a), u a,i (z 0 )(a

) E ∨ a ,Ea if a = b, 0 if a = b . (9.9)
First assume that a = b. Then W u (a) ∩ W s (b) = ∅ by the transversality condition, since a and b have same Morse index. Therefore for any t 1 , t 2 ≥ 0, we have e

-t 1 L ∇(z) ∨ -X χ b [W s (b)] ⊗ s b,j (z) , e -t 2 L ∇(z 0 ) X χ a [W u (a)] ⊗ u a,i (z)
= 0, (9.10) since the currents in the pairing have disjoint support because they are respectively contained in W s (b) and W u (a). Now notice that for Re(s) big enough, one has

L ∇(z) ∨ -X + s -1 = ∞ 0 e -tL ∇(z) ∨ -X e -ts dt and L ∇(z 0 ) X + s -1 = ∞ 0 e -tL ∇(z 0 ) X e -ts dt.
Therefore the representation (9.3) of the spectral projectors and the analytic continuation of the above resolvents imply with (9.10) that sb,j (z), ũa,i = 0.

Next assume that a = b. Then W u (a) ∩ W s (a) = {a}. Since the support of sa,i (z) (resp. ũa,i (z 0 )) is contained in the closure of W s (a) (resp. W u (a)), we can compute

Π ∨ (z) χ a [W s (a)] ⊗ s a,j (z) , Π χ a [W u (a)] ⊗ u a,i (z 0 ) = χ a [W s (a)] ⊗ s a,j (z), χ a [W u (a)] ⊗ u a,i (z 0 ) = [a], s a,j (z), u a,i (z 0 ) E ∨ ,E ,
where the first equality stands because sa (z) = [W s (a)] ⊗ s a,j (z) near a by [DR17b, Proposition 7.1]. This gives (9.9). This identity immediately yields (9.8) with A j a,i (z) = s a,j (z)(a), u a,i (z 0 )(a) E ∨ a ,Ea since we have Π(z) = This equation combined with (9.6) and (9.7) yields, if τ

∨ (z) = τ e,o (∇(z) ∨ ) d(log τ ∨ ) z 0 σ = -tr s, C • Π(z 0 )α z 0 (σ) k(z 0 ) + e tr α z 0 (σ).
The proof is almost finished. But since we need to formulate our results in terms of the cohomological torsion, we still have to make some tedious formal manipulations to pass to the cohomological formalism. The first step is to replace ∇ by the dual connection ∇ ∨ in the above formula. We also introduce some notation. The operator Π was the spectral projector on the kernel of L ∇

X

. Now we need to work with the spectral projector on ker(L

∇(z 0 ) ∨ X ) (resp ker(L ∇(z 0 ) -X
)) that we denote by Π ∨ + (z 0 ) (resp Π -(z 0 )) where the + (resp -) sign emphasizes the fact that we deal with + X (resp -X). Now note that

∇(z) ∨ = ∇(z 0 ) ∨ -T α z 0 (z -z 0 ) + o(z -z 0 ).
Therefore, applying what precedes to τ (z) we get

d(log τ ) z 0 σ = -tr s, C • ∨,+ Π ∨ + (z 0 ) -T α z 0 (σ) k∨ + (z 0 ) + e tr -T α z 0 (σ) , (9.12)
where Π ∨ + (z 0 ) is the spectral projector (9.3) associated to ∇(z 0 ) ∨ and + X, C • ∨,+ = ran Π ∨ + (z 0 ), and k ∨ + (z 0 ) is any cochain contraction on the complex ( C • ∨,+ , ∇(z 0 ) ∨ ). Now, we have the identification

C k ∨,+ ∨ C n-k -,
where C • -is the range of Π -(z 0 ), the spectral projector (9.3) associated to ∇(z 0 ) and -X. It is easy to show that under this identification, one has

Π ∨ + T α z 0 (σ) k(z 0 ) ∨ = Π -(z 0 )α z 0 (σ)k -(z 0 ) + Π -(z 0 )α z 0 (σ), k -(z 0 ) ,
where for any j ∈ {0, . . . , n}, we set

k -(z 0 )| C n-j - = (-1) j+1 k∨ + (z 0 )| Cj+1 ∨ : C n-j - → C n-j-1 - .
Then k -(z 0 ) is a cochain contraction on the complex ( C • -, ∇(z 0 )). As a consequence, since n is odd,

tr s, C • ∨,+ Π ∨ + (z 0 ) -T α z 0 (σ) k∨ + (z 0 ) = tr s, C • - Π -(z 0 )α z 0 (σ)k -(z 0 ).
This concludes by (9.12) since tr(-T β) = -tr β for any β ∈ Ω 1 (M, End(E)).

Comparison of the dynamical torsion with the Turaev torsion

In this section we see the dynamical torsion and the Turaev torsion as functions on the space of acyclic representations. This is an open subset of a complex affine algebraic variety. Therefore we can compute the derivative of τ ϑ /τ e,o along holomorphic curves, using the variation formulae obtained in § §7,9. From this computation we will deduce Theorem 4. Proposition 10.1 relies on the variation formulae given by Propositions 7.1 and 9.1, and Lemma 8.4 which gives a topological interpretation of those. 10.3. An adapted family of connections. Following [BV17, §4.1], there exists a flat vector bundle E over M and a C 1 family of connections ∇(z), |z| < δ, in the sense of §7.1, such that 5 ρ ∇(z) = ρ(z) (10.1)

for every z; we can moreover ask the family ∇(z) to be complex differentiable at z = 0, that is,

∇(z) = ∇ + zα + o(z), (10.2)
where ∇ = ∇(0) and α ∈ Ω 1 (M, End(E)). Note that flatness of ∇(z) implies

L ∇ -X + s -1 = Π - s + Y + O(s) be the Laurent expansion of L ∇ -X + s -1
near s = 0. The fact that s = 0 is a simple pole comes from [START_REF] Viet | Spectral analysis of Morse-Smale gradient flows[END_REF]. As in 7.2, we consider the operator

K = ι -X Y (Id -Π -) + k-Π -: Ω • (M, E) → D • (M, E),
where kis any cochain contraction on C • -= ran Π -. Note that we have the identity [∇, K] = ∇ K + K∇ = Id .

(10.3)

The next proposition will allow us to interpret the term tr s, C • Π -(z)α z (σ) k-(z) appearing in Proposition 9.1 as a flat trace similar to the one appearing in Proposition 7.1. This will be crucial for the comparison between τ ϑ and τ e,o .

Proposition 10.2. For ε > 0 small enough, the wavefront set of the Schwartz kernel of the operator ι -X Y (Id -Π -)e -εL ∇ -X does not meet the conormal to the diagonal in M × M and we have for any α ∈ Ω 1 (M, End(E))

tr s αι -X Y (Id -Π -)e -εL ∇ -X = 0.
We refer to appendix C for the proof. An immediate corollary is the formula

tr s, C • - Π -α k-= tr s α Ke -εL ∇ -X . (10.4) Indeed, since L ∇ -X Π -= 0, we have Π -e -εL ∇ -X = Π -.
Moreover, since the trace of finite rank operators coincides with the flat trace, we have tr s, C

• - Π -α k-= tr s, C • - Π -α k-e -εL ∇ -X = tr s α k-Π -e -εL ∇ -X .
Therefore we obtain with Proposition 10.2

tr s, C • Π -α k-= tr s αι -X Y (Id -Π -)e -εL ∇ -X + tr s α k-Π -e -εL ∇ -X ,
which gives (10.4).

5 It is actually stated in [BV17, §4.1] that one can find a C 1 family of connections satisfying (10.1); however looking carefully at the proofs one can choose the family ∇(z) to be C 1 in z.

10.5. Proof of Proposition 10.1. Note that we have by (10.1)

τ ϑ (ρ(z)) = τ ϑ (∇(z)), τ e,o (ρ(z)) = τ e,o (∇(z)).
We will set f (z) = τ ϑ (∇(z))/τ e,o (∇(z)) for simplicity. Now we apply Proposition 7.1, Proposition 9.1 to obtain that z → f (z) is real differentiable (since z → ∇(z) is); moreover it is complex differentiable at z = 0 by (10.2) and for ε > 0 small enough we have

d dz z=0 log f (z) = -tr s αKe -εL ∇ X +tr s α Ke -εL ∇ -X + tr α, e , (10.5)
where we used (10.4). Let

∆ = ∇∇ + ∇ ∇ : Ω • (M, E) → Ω • (M, E)
be the Hodge-Laplace operator induced by any metric on M and any Hermitian product on E. Because ∇ is acyclic, ∆ is invertible and Hodge theory gives that its inverse ∆ -1 is a pseudo-differential operator of order -2. Define

J = ∇ ∆ -1 : D • (M, E) → D •-1 (M, E).
We have of course

[∇, J] = ∇J + J∇ = Id D • (M,E) . (10.6)
Let R ε be the interpolator at time ε defined in §8.3 for the pair of vector fields (-X, X). This implies with (8.4)

[∇, R ε ] = e -εL ∇ X -e -εL ∇ -X . (10.7) Now define G ε = J Ke -εL ∇ X -Ke -εL ∇ -X -R ε : Ω • (M, E) → D •-2 (M, E).
Let us compute, having (10.6) in mind,

[∇, G ε ] = ∇J Ke -εL ∇ X -Ke -εL ∇ -X -R ε -J Ke -εL ∇ X -Ke -εL ∇ -X -R ε ∇ = (Id -J∇) Ke -εL ∇ X -Ke -εL ∇ -X -R ε -J K∇e -εL ∇ X -K∇e -εL ∇ -X -R ε ∇ = Ke -εL ∇ X -Ke -εL ∇ -X -R ε -J [∇, K]e -εL ∇ X -[∇, K]e -εL ∇ -X -[∇, R ε ] ,
where we used that e -εL ∇ X and e -εL ∇ -X commute with ∇. Now note that (7.7), (8.4) and (10.3) imply

[∇, K]e -εL ∇ X -[∇, K]e -εL ∇ -X -[∇, R ε ] = e -εL ∇ X -e -εL ∇ -X -e -εL ∇ X -e -εL ∇ -X = 0. Therefore we obtained [∇, G ε ] = Ke -εL ∇ X -Ke -εL ∇ -X -R ε . Because [∇, α] = 0 we have [∇, αG ε ] = -α Ke -εL ∇ X -Ke -εL ∇ -X -R ε .
Using the notations of §3.4, WF(J), WF(α), WF(∇) are contained in the conormal bundle of the diagonal N * ∆ since J, α, ∇ are pseudodifferential operators; moreover, equation (8.6) shows that WF Ke -εL ∇ X -Ke The identity [∇, α] = 0 also implies that d tr α = tr ∇ E⊗E ∨ α = tr[∇, α] = 0. As a consequence we can apply (8.5) to obtain tr s αR ε = tr α, cs(-X, X) .

-εL ∇ -X -R ε ∩ N * ∆ = ∅.
Now note that ∂ -cs(-X, X)+e = -div(X) -div(-X) + div( X) = 0 by (8.1) and (9.2) since X is non singular. Therefore we obtain Proposition B.2. [Bon18, Lemma 9] Let Q be a pseudo-differential operator micro-locally supported near the zero section in T * M and elliptic there. There exists c, ε 0 > 0 such that for any ρ > 0 and J ∈ N, there is µ 0 , h 0 > 0 such that the following holds. For each µ ≥ µ 0 , 0 < h < h 0 , j ∈ Z such that |j| ≤ J and s ∈ Ω(c, ρ) the operator

d dz z=0 log f (z) = tr α, h where h = [-cs(-X, X) + e] ∈ H 1 (M, Z).
L ∇ Xt -h -1 Q + s : H • µG,j+1 → H • µG,j
is invertible for |t| ≤ ε 0 and the inverse is bounded H • µG,j → H • µG,j independently of t.

B.4. Continuity of the Pollicott-Ruelle spectrum. We fix ρ, J ≥ 4 and µ 0 , µ, h 0 , h, j as in Proposition B.2. We first observe that

L ∇ Xt + s L ∇ Xt -h -1 Q + s -1 = Id +h -1 Q L ∇ Xt -h -1 Q + s -1 . (B.1)
Since Q is supported near 0 in T * M , it is smoothing and thus trace class on any H • µG,j . By analytic Fredholm theory, the family s

→ K(t, s) = h -1 Q L ∇ Xt -h -1 Q + s -1 is a
holomorphic family of trace class operators on H • µG,j in the region Ω(c, ρ). We can therefore consider the Fredholm determinant D(t, s) = det H • µG,j Id +K(t, s) . It follows from [Sim05, Corollary 2.5] that for each t, s → D(t, s) is holomorphic on Ω(c, ρ). Moreover (B.1) shoes that its zeros coincides, on Ω(c, ρ), with the Pollicott-Ruelle resonances of L ∇ Xt . Moreover we have for any s ∈ Ω(c, ρ),

L ∇ Xt -h -1 Q + s -1 -L ∇ X t -h -1 Q + s -1 = -L ∇ Xt -h -1 Q + s -1 L ∇ Xt -L ∇ X t L ∇ X t -h -1 Q + s -1 . (B.2) We have L ∇ Xt -L ∇ X t t -t -→ t→t L ∇ Ẋt in L(H • µG,j+1 , H • µG,j
D(t, s) ∈ C 0 [-ε 0 , ε 0 ] t , Hol Ω(c, ρ) s .
(B.4) B.5. Regularity of the resolvent. Let Z be an open set of C whose closure is contained in the interior of Ω(c, ρ). We assume that Z ∩ Res(L ∇ X 0 ) = ∅. Up to taking ε 0 smaller, Rouché's theorem and (B.4) imply that there exists δ > 0 such that dist Z, Res(L ∇ Xt ) > δ for any |t| ≤ ε 0 . As a consequence we obtain that for every |j| ≤ J the map L ∇ Xt + s -1 :

H • µG,j → H • µG,j is bounded independently of (t, s) ∈ [-ε 0 , ε 0 ] × Z. Noting that L ∇ Xt + s -1 -L ∇ X t + s -1 t -t = -L ∇ Xt + s -1 L ∇ Xt -L ∇ X t t -t L ∇ X t + s -1 , (B.5) we obtain by (B.3) that t → L ∇ X t + s -1 is continuous in L(H • µG,j+1 , H • µG,j
). Therefore applying again (B.5) again, we get that

L ∇ Xt + s -1 ∈ C 1 [-ε 0 , ε 0 ] t , Hol(Z s , L(H • µG,j+1 , H • µG,j-2 ) . (B.6)
Note that here we need |j -2|, |j + 1| ≤ J.

B.6. Regularity of the spectral projectors. Let 0 < λ < 1 such that {|s| = λ} ∩ Res(L ∇ X 0 ) = ∅. Applying the last subsection with Z = {|s| = λ}, we get {|s| = λ} ∩ Res(L ∇ Xt ) = ∅ for any |t| ≤ ε 0 . We can therefore define for those t

Π t = 1 2πi |s|=λ L ∇ Xt + s -1 ds : H • µG,j → H • µG,j .
Then (B.6) gives that Π t ∈ C 1 [-ε 0 , ε 0 ] t , Z s , L(H • µG,j+1 , H • µG,j-2 . This is true for j = 3 and j = -1 because J ≥ 4. Moreover by Rouché's theorem, the number m of zeros of s → D(t, s) does not depend on t. Noting that ∂ s K(t, s)(1 + K(t, s)) -1 = -K(t, s) L ∇ Xt -h -1 Q + s) -1 (1 + K(t, s) -1 , we obtain by [START_REF] Dyatlov | Mathematical Theory of Scattering Resonances[END_REF]Theorem C.11] and the cyclicity of the trace that m is equal to

1 2πi tr |s|=λ ∂ s K(t, s)(1 + K(t, s)) -1 ds = - 1 2πi tr |s|=λ L ∇ Xt -h -1 Q + s -1 (1 + K(t, s)) -1 K(t, s)ds = 1 2πi tr |s|=λ L ∇ Xt -h -1 Q + s -1 (1 + K(t, s)) -1 ,
where we used that s → L ∇ Xt -h -1 Q + s -1 is holomorphic on {|s| ≤ λ}.

L ∇ Xt + s -1 u, v = u, L ∇ ∨ -Xt + s -1 v , u ∈ Ω k (M, E), v ∈ Ω n-k (M, E ∨ ), (B.8)
where •, • is the pairing from §3.2. This shows that Res(L ∇ ∨ -Xt ) = Res(L ∇ Xt ). Therefore we can apply the preceding construction with the escape function g replaced by -g (the unstable bundle of -X t is the stable one of X t and reciprocally) and we obtain that

Π ∨ t = 1 2πi |s|=λ L ∇ ∨ -Xt + s -1 ds ∈ C 1 [-ε 0 , ε 0 ] t , L(H • -µG,0 , H • -µG,1
) .

Note that (B.8) implies

Π t u, v = u, Π ∨ t v , u ∈ Ω k (M, E), v ∈ Ω n-k (M, E ∨ ). (B.9)
We denote C • t = ran Π t , C ∨• t = ran Π ∨ t and m = rank Π t = rank Π ∨ t . Take ϕ 1 , . . . , ϕ m , ψ 1 , . . . , ψ m ∈ Ω • (M, E) such that Π 0 (ϕ 1 ), . . . Π 0 (ϕ m ) is a basis of C • 0 and Π 0 ϕ i , ψ j = 0 if i = j and Π 0 ϕ i , ψ j = 1 otherwise. For t small enough we set

ϕ i t = Π t ϕ i , ψ t j = Π ∨ t ψ j .
Like in the proof of Lemma A. Ws (M, E ∨ )). To proceed, we note that we can construct two weight functions m u , m s satisfying the properties of Lemma B.1 such that {m u ≤ 0} ∩ {m s ≥ 0} = ∅ (for example by choosing well the χ from [Bon18, p. 6]). Let G u , G s ∈ Ψ 0+ (M ) be the associated operators from §B.2. Up to take ε 0 smaller, we obtain with (B.7) that the map t → ϕ i t is bounded in H • µGu,0 for µ > 0 big enough. For any χ ∈ C ∞ (T * M, [0, 1]) such that supp χ ⊂ {m u ≥ δ} for some δ > 0 we have by classical rules of pseudo-differential operators Op(χ)ϕ i t H δµ (M,Λ • ⊗E) ≤ C µ ϕ i t H • µGu,0 ≤ C µ , t ∈ [-ε 0 , ε 0 ], for some constants C µ , C µ independent of t. As a consequence we obtain (for example using [DR17a, Lemma 7.4]) that [-ε 0 , ε 0 ] t → ϕ i t is bounded in D • Wu (M, E) where W u = {m u ≤ 0}. Doing exactly the same with -m s and -X t we obtain that [-ε 0 , ε 0 ] t → ψ i t is bounded in D • Ws (M, E ∨ ) with W s = {-m s ≥ 0}. This shows (B.11).

Proof. Because X is C ∞ -linearizable, we can take U ⊂ R n to be a coordinate patch centered in a so that, in those coordinates, e -t X (x) = e -tA (x) where A is a matrix whose eigenvalues have nonvanishing real parts. Denoting (x 1 , . . . , x n ) the coordinates of the patch, X reads X = 1≤i,j≤n

A j i x i ∂ j .
We have a decomposition R n = W u ⊕ W s stable by A such that A| W u (resp. A| W s ) have eigenvalues with positive (resp. negative) real parts. We will denote A u = A| W u ⊕ Id W s , A s = Id W u ⊕A| W s and δ > 0 such that δ < inf Let χ 1 , χ 2 ∈ Ω • (M ) such that supp χ i ⊂ supp χ for i = 1, 2. For simplicity, we identify e -tA and its action on forms given by the pull-back. Lemma C.2. For any µ > 0, there is ν > 0 with the following property. For every x ∈ M such that dist(x, Crit(f )) ≥ µ, it holds dist x, e -(t+ε) X (x) ≥ ν, t ≥ 0.

Proof. We proceed by contradiction. Suppose that there is µ > 0 and sequences x m ∈ M and t m ≥ ε such that dist x m , e -tm X (x m ) → 0 as m → ∞ and dist(x m , Crit(f )) ≥ µ.

Extracting a subsequence we may assume that x m → x, t m → ∞ (indeed if t m → t ∞ < ∞ then x is a periodic point for X, which does not exist) and for any m, e -t X (x m ) → a and e t X (x m ) → b as t → ∞, for some a, b ∈ Crit(f ). Since the space of broken curves L(a, b) is compact (see [AD]), we may assume that the sequence of curves γ m = e t X (x m ), t ∈ R converges to a broken curve = ( 1 , . . . , q ) ∈ L(a, b) with j ∈ L(c j-1 , c j ) for some c 0 , . . . , c q ∈ Crit(f ) with c 0 = a and c q = b. Because x m → x, the proof of [AD, Theorem 3.2.2] implies x ∈ j for some j so that e -t X x → c j-1 as t → ∞. Therefore replacing x by e -t X for t big enough we may assume that x is contained in a Morse chart Ω(c j-1 ) near c j-1 . Then c j-1 = a. Indeed if it was not the case then we would have e -tm X x m → c as m → ∞ (since x m would be contained in Ω(a) ∩ W s (a) for big enough m and t m → ∞), which is not the case since x = c and dist x m , e -tm X (x m ) → 0 as m → ∞. Therefore the flow line of x m exits Ω(c j-1 ) in the past. We therefore obtain, since e -tm X x m → x, that there is i < j -1 so that c i = c j-1 . This is absurd since the sequence ind f (c i ) i=0,...,q is strictly decreasing. By (9.11) we have supp K Π ∩ ∆ = Crit(f ), where K Π is the Schwartz kernel of Π and ∆ is the diagonal in M × M ; the same holds for e -(t+ε) X Π = Π (see [START_REF] Viet | Pollicott-Ruelle spectrum and Witten laplacians. To appear[END_REF]). Moreover Lemma C.2 implies that if χ ∈ C ∞ (M, [0, 1]) satisfies χ ≡ 1 near ∆ and has support close enough to ∆ we have Therefore for Re(s) 0 it holds G χ,ε,s = χ L X + s -1 (Id -Π)e -ε X χ.

Since both members are holomorphic in the region {Re(s) > -δ} and coincide for Re(s) 0, they coincide in the region Re(s) > -δ. Let β ∈ Ω 1 (M ). We can compute for Re(s) 0, since ι X Π = 0 by [START_REF] Viet | Pollicott-Ruelle spectrum and Witten laplacians. To appear[END_REF], tr s βι X L X + s -1 (Id -Π)e -εL X = tr s βι X G χ,ε,s

  ρ) is locally constant on Rep ac (M, d). Moreover, if dim M = 3 and b 1 (M ) = 0, this map is of modulus one on the connected components of Rep ac (M, d) containing an acyclic and unitary representation. (2) The map

  x) be the diagonal inclusion. Then by [Hör90, Theorem 8.2.4] the pull back ι * G ∈ D n (M, E ∨ ⊗ E) is well defined and we define the super flat trace of G by tr s G = tr ι * G, 1 ,

Proposition 3. 1 .

 1 Let G, H be two homogeneous operators with deg G + deg H = 0 and such that both compositions G • H and H • G are defined and satisfy the bound (3.2). Then we have tr s [G, H] = 0.

  admits a meromorphic continuation to s ∈ C with poles of finite multiplicites; we will still denote by L ∇ X + s -1 this extension. Those poles are the Pollicott-Ruelle resonances of L ∇ X , and we will denote this set by Res(L ∇ X ). 4.3. Generalized resonant states. Let s 0 ∈ Res(L ∇ X ). By [DZ16, Proposition 3.3] we have a Laurent expansion

  k tr C k 0 Πϑ( Ẋ) = tr C • (-1) N +1 Πϑι Ẋ 6.3.Variation of the rest. Let us now interest ourselves in the variation of t → ζ (λ,∞)

  .11) For all z, the map s → L the region Ω(c, ρ) with poles (of finite multiplicity) which coincide with the resonances of L ∇(z) X in this region. Moreover, the arguments from the proof of [DZ16, Proposition 3.4] can be made uniformly for the family z → L

  .18) This gives the variation of ζ (λ,∞) X,∇(z) (s) for Re(s) big enough. To obtain the variation of b(z) = ζ (λ,∞) X,∇(z) (0), we can reproduce the arguments made in §6.4 to obtain

  |a| γ a ∈ C 1 (M, Z). (9.1) Note that the Poincaré-Hopf index of X near a ∈ Crit(f ) is -(-1) |a| so that ∂e = div( X) (9.2) because a∈Crit(f ) (-1) |a| = χ(M ) = 0 by the Poincaré-Hopf index theorem. Therefore e is an Euler chain for X and e = [ X, e] defines an Euler structure. Choose some basis u 1 , . . . , u d of E ∨ x . For each a ∈ Crit(f ), we propagate this basis via the parallel transport of ∇ along γ a to obtain a basis u 1,a , . . . , u d,a of E a . We choose an ordering of the cells W u (a) ; this gives us a cohomology orientation o (see [Tur90, §6.3]). Moreover this ordering and the chosen basis of E ∨ a give us (using the wedge product) an element c k ∈ det C k (W, E ∨ ) for each k, and thus an element c ∈ det C • (W, E ∨ ). The Turaev torsion of ∇ with respect to the choices e, o is then defined by [FT00, §9.2 p. 218]

X.

  Since ∇ and L ∇ X commute, ∇ induces a differential on the complex C • . Moreover, Π maps D • Γ (M, E) to itself continuously where Γ = a∈Crit(f ) N * W u (a) ⊂ T * M. 9.5. A variation formula for the Turaev torsion. Assume that we are given a C 1 family of acyclic connections ∇(z) on E as in §7. We denote by Π -(z) the spectral projector (9.3) associated to ∇(z) and -X, and set C • -(z) = ran Π -(z). By [DR17c] we have that all the complexes ( C • (z), ∇(z)) are acyclic and there exists cochain contractions k-(z) : C • -(z) → C •-1 -(z). As in §7.3 we have a variation formula for the Turaev torsion. Proposition 9.1. The map z → τ (z) = τ e,o (∇(z)) is real differentiable on U and for any z ∈ U d(log τ ) z σ = -tr s, C • (z) Π -(z)α z (σ) k-(z)e tr α z (σ), σ ∈ C

  (z), • ũa,j (z) (9.11) Using the lemma, we obtain, ifµ(z) = ũ(z) : Π(z)ũ(z 0 ) , d(log µ) z 0 σ = a∈Crit(f ) (-1) n-|a| tr A γa (z 0 , σ)P γa (z 0 ) -1where A γa (z 0 , σ) = d (P γa ) z 0 σ. Since n is odd, we obtain by definition of e and (3.4) d(log µ) z 0 σ = a∈Crit(f ) (-1) |a| γa tr α z 0 (σ) = e tr α z 0 (σ).

10. 1 .

 1 The algebraic structure of the representation variety. We describe here the analytic structure of the space Rep(M, d) = Hom(π 1 (M ), GL(C d ))of complex representations of degree d of the fundamental group. Since M is compact, π 1 (M ) is generated by a finite number of elements c 1 , . . . , c L ∈ π 1 (M ) which satisfy finitely many relations. A representation ρ ∈ Rep(M, d) is thus given by 2L invertible d × d matrices ρ(c 1 ), . . . , ρ(c L ), ρ(c -1 1 ), . . . ρ(c -1 L ) with complex coefficients satisfying finitely many polynomial equations. Therefore the set Rep(M, d) has a natural structure of a complex affine algebraic set. We will denote the set of its singular points by Σ(M, d). In what follows, we will only consider the classical topology of Rep(M, d). We will say that a representation ρ ∈ Rep(M, d) is acyclic if ∇ ρ is acyclic. We denote by Rep ac (M, d) ⊂ Rep(M, d) the space of acyclic representations. This is an open set (in the Zariski topology, thus in the classical one) in Rep(M, d), see [BH06, §4.1]. For any ρ ∈ Rep ac (M, d) we set τ ϑ (ρ) = τ ϑ (∇ ρ ), τ e,o (ρ) = τ e,o (∇ ρ ), for any Euler structure e and any cohomological orientation o. 10.2. Holomorphic families of acyclic representations. Let ρ 0 ∈ Rep ac (M, d)\Σ(M, d) be a regular point. Take δ > 0 and ρ(z), |z| < δ, a holomorphic curve in Rep ac (M, d)\Σ(M, d) such that ρ(0) = ρ 0 . Theorem 4 will be a consequence of the following Proposition 10.1. Let X be a contact Anosov vector field on M . Let e = [ X, e] be the Euler structure defined in §9.3. Note that -cs(-X, X) + e is a closed cycle and defines a homology class h ∈ H 1 (M, Z). Then z → τ ϑ (ρ(z))/τ e,o (ρ(z)) is complex differentiable and d dz τ ϑ (ρ(z)) τ e,o (ρ(z)) det ρ(z), h = 0 for any cohomological orientation o.

[

  ∇, α] = ∇α + α∇ = 0. 10.4. A cochain contraction induced by the Morse-Smale gradient flow. Let

  It follows from wave front composition [Hör90, Theorem 8.2.14] that WF(αG ε ) ∩ N * ∆ = ∅. The operators ∇, αG ε satisfy the assumptions of Proposition 3.1 which gives tr s [∇, αG ε ] = 0 and therefore (10.5) reads d dz z=0 log f (z) = -tr s αR ε + tr α, e . (10.8)

  Finally, let us note that by (3.4), d dz z=0 log det ρ(z), h = -tr α, h , since ρ(z) = ρ ∇(z) . Therefore the proposition is proved for z = 0. However the same argument holds for every z close enough to 0, which concludes. 10.6. Proof of Theorem 4. By Hartog's theorem and Proposition 10.1, we have that the map ρ → τ ϑ (ρ) τ e,o (ρ) det ρ, h (10.9) is locally constant on Rep ac (M, d) \ Σ(M, d). Moreover, we can reproduce all the arguments we made in the continuous category to obtain that ρ → τ ϑ (ρ)/τ e,o (ρ) is actually continuous on Rep ac (M, d). Because Rep ac (M, d) \ Σ(M, d) is open and dense in Rep ac (M, d), we get that the map 10.9 is locally constant on Rep ac (M, d). By [FT00, p. 211] we have, if e is another Euler structure, τ e ,o (ρ) = det ρ, e -e τ e,o (ρ). As a consequence, if we set e ϑ = [-X, 0] which defines an Euler structure since X is nonsingular (see §8.2), we have ee ϑ = h and we obtain that ρ → τ ϑ (ρ)/τ e ϑ ,o (ρ) is locally constant on Rep ac (M, d). Now let η be another contact form inducing an Anosov Reeb flow and denote by X η its Reeb flow. Then if e η = [-X η , 0], we have e η -e ϑ = cs(X, X η ) B.3. Uniform parametrices. Let us consider a smooth family of vector fields X t , |t| < ε, perturbing X 0 . For any c, ρ > 0 we will denote Ω(c, ρ) = {Re(s) > c} ∪ {|s| ≤ ρ} ⊂ C. The spaces defined in the last subsection yields an uniform version of [DZ16, Proposition 3.4], as follows.

|

  Re(λ)|where sp(A) is the spectrum of A.

K∂∂ 0 1+

 0 χ,t , π * 1 χ 1 ∧ π * 2 χ 2 = χ 2 , e -tA (Id -Π)χ 1 = χ 2 , e -tA χ 1 -δ(x u )dx u W sπ * s,0 χ 1 = e tAs χ 2 , e -tAu χ 1τ e tAs π * u,τ χ 2 ∧ e -tAu π * s,τ χ 1 dτ, where π u,τ , π s,τ : U → U are defined by π u,τ (x u , x s ) = (x u , τ x s ) and π s,τ (x u , x s ) = (τ x u , x s ). Now write χ 2 = |I|=k β I dx Is s ∧ dx Iu u . We have∂ τ π * u,τ χ 2 (x u , x s ) = ∂ τ I τ |Is| β I (x u , τ x s )dxIu u ∧ dx Is s = I |I s |τ |Is|-1 β I (x u , τ x s )dx Iu u ∧ dx Is s -I τ |Is| (∂ xs β I ) (xu,τ xs) (x s )dx Iu u ∧ dx Is s . Therefore ∂ τ e tAs π * u,τ χ 2 = I |I s |τ |Is|-1 β I (x u , τ e tAs x s ) -τ |Is| (∂ xs β I ) (xu,τ xs) (e tAs x s ) e tAs dx I . Because |e tAs x s | = O(e -tδ ) and |I s |e tAs dx I = O(|I s |e -δt|Is| ), and repeating the argument for ∂ τ e -tAu π * s,τ χ 1 we obtain∂ τ e tAs π * u,τ χ 2 ∧ e -tAu π * s,τ χ 1 = O χ 1 ,χ 2 (e -tδ ). (C.1)Replacing χ 1 and χ 2 by χ 1 e i ξ,• and χ 2 e i η,• with ξ, η ∈ R n one getsK χ,t , π * 1 χ 1 e i ξ,• ∧ π * 2 χ 2 e i η,• τ e tAs π * u,τ χ 2 ∧ e -tAu π * s,τ χ 1 e i e tAs (xu,τ xs),η e i e -tAu (τ xu,xs),ξ dτ+ 1 0 Ue tAs π * u,τ χ 2 ∧ e -tAu π * s,τ χ 1 ∂ τ e i e tAs (xu,τ xs),η e i e -tAu (τ xu,xs),ξ dτ.Denoting g(τ, x u , x s ) = e i e tAs (xu,τ xs),η e i e -tAu (τ xu,xs),ξ we have∂ τ g(τ, x u , x s ) = i e tAs x s , η s + e -tAu x u , ξ u g(τ, x u , x s ) = O C ∞ (M ) (e -tδ ), because |e tAs x s |, |e -tAu x u | = O(e -tδ ). By classical arguments we therefore obtain for any N ∈ N (repeating the process that led to (C.1))K χ,t , π * 1 χ 1 e i ξ 1 ,• ∧ π * 2 χ 2 e i ξ 2 ,• ≤ C N,χ 1 ,χ 2 e -tδ 1 + |e tAs η s | + |e -tAu ξ u | 1 |τ e tAs η s + ξ s | + |τ e -tAu ξ u + η u | -N dτ,where ξ = (ξ u , ξ s ) and η = (η u , η s ). Now assume (ξ, η) is close to N * ∆(T * M ), sayξ |ξ| + η |η| < ν and 1 -ν < |ξ| |η| < 1 + νfor some ν > 0. Then we have for any τ ∈ [0, 1] we get|τ e tAs η s + ξ s | + |τ e -tAu ξ u + η u | ≥ 1 -e -tδ (1 + ν) (|ξ s | + |η u |)As a consequence if ν > 0 is small enough so that (1 + ν)e -(t+ε)δ < 1 for every t ≥ 0 we obtainK χ,t+ε , π * 1 χ 1 e i ξ,• ∧ π * 2 χ 2 e i η,• ≤ C N,χ 1 ,χ 2 (1 + |ξ| + |η|) -N ,whichconcludes. Proof of Proposition 10.2. Fix ε > 0. For a ∈ Crit(f ), take δ a , Γ a , χ a as in Lemma C.1. The proof of Lemma C.1 actually shows that for Re(s) > -δ the integral G χa,ε,s = ∞ 0 e -ts χ a e -(t+ε) X (Id -Π)χ a dt converges as an operator Ω • (M ) → D • (M ). Moreover its Schwartz kernel G χa,ε,s is locally bounded in D n Γa (M × M ) in the region {Re(s) > -δ}. We will need the following lemma.

0 e

 0 χe -(t+ε) X χ = a χ a e -(t+ε) X χ a . Let δ = min a∈Crit(f ) δ a . For Re(s) > -δ, G χ,ε,s = ∞ 0 e -ts χe -(t+ε) X (Id -Π)χdt defines an operator Ω • (M ) → D • (M ), whose Schwartz kernel G χ,ε,s is locally bounded in D n Γ (M × M ) in the region {Re(s) > -δ}, where Γ = a∈Crit(f ) Γ a . Now for Re(s) 0 we haveL X + s -1 =∞ -ts e -t X dt.

= ∞ 0 e 0 e

 00 -ts tr s βι X e -(t+ε) X (Id -Π) = ∞ -ts tr s βι X e -(t+ε) X , where we could interchange the integral and the flat trace thanks to the bound obtained in Lemma C.1. Now the Atiyah-Bott trace formula[START_REF] Francis | A Lefschetz Fixed Point Formula for Elliptic Complexes: I[END_REF] gives tr s βι X e -(t+ε) X = 0 since X vanishes at its critical points. By holomorphy this holds true for any s such that Re(s) > -δ. In particular if λ > 0 is small enoughtr s βι X Y (Id -Π)e -ε X = 1 2iπ |s|=λ tr s βι X L X + s -1 s (Id -Π)e -εL X ds = 0,whereL X + s -1 = Y + Π s + O(s). Therefore Proposition 10.2 is proved in the case where (E, ∇) is the trivial bundle. The general case is handled similarly.

  But now apply [DGRS18, Theorem 4] to obtain that trs ϑι Ẋt Q t (s) ∈ C 0 [-t 0 , t 0 ], Hol V δ ∩ {|s| ≥ 5δ/4} .

		(6.17)
		.15)
	is bounded, where Y t (s) is the Schwartz kernel of the operator Y t (s)e	-ε L ∇ X t

+s . We also know that this map is actually continuous, because it is when valued in D n thanks to the last point of Proposition 6.1. Therefore we obtained that tr s ϑι Ẋt Y t (s) ∈ C 0 [-t 0 , t 0 ], Hol {|s| ≤ 3δ/2} . (6.16)

  The last integral is equal to tr Π t = rank Π t by (B.1). As a consequence we can apply Lemma A.3 to obtain thatΠ t ∈ C 1 [-ε 0 , ε 0 ] t , L(H • µG,0 , H • µG,1 . (B.7) B.7. Wavefront set of the spectral projectors. Let (E, ∇ ∨ ) be the dual bundle of (E, ∇). Then (4.2) implies, for any Re(s) 0,

  → m ij (t) is continuous near t = 0 and m ij (0) = δ ij .Next we show that there exists open conic neighborhoods of N u and N s such that, uni-formly in t ∈ [-ε 0 , ε 0 ], WF(ϕ i t ) ⊂ W u , WF(ψ i t ) ⊂ W s , W u ∩ W s = ∅, i = 1, . . . , m. (B.11) This means that the map [-ε 0 , ε 0 ] t → ϕ i t (resp. ψ i t ) is bounded in D • Wu (M, E) (resp. D •

	3, (B.9) implies that
	m
	Π t =	m ij (t)ϕ i t ψ j t , • ,	(B.10)
	i=1
	where t

indX (x) = (-1) dim(Es)(x) if x is hyperbolic and Es(x) is the stable subspace of x

A difference is that Laudenbach only needs to compactify the unstable cells as C 1 -manifolds with conical singularities whereas Qin proves smooth compactification as manifolds with corners

(-1) • comes from ∂ = (-1) deg +1 d comparing the boundary ∂ and De Rham differential d

Acknowledgements. We warmly thank Nalini Anantharaman, Yannick Bonthonneau, Mihajlo Cekić, Alexis Drouot, Semyon Dyatlov, Malo Jézéquel, Thibault Lefeuvre, Julien Marché, Marco Mazzucchelli, Claude Roger, Nicolas Vichery, Jean Yves Welschinger, Steve Zelditch, for asking questions about this work or for interesting discussions related to the paper. Particular thanks are due to Colin Guillarmou who went through the whole paper, helped us correct many errors and is always a source of inspiration. We thank the organizers of the microlocal analysis program in MSRI for the invitation to speak about our result. N.V.D is very grateful to Gabriel Rivière for his friendship, many inspiring maths discussions, his many advices and for the series of works which made the present paper possible.

by definition. Therefore τ e ϑ ,o (ρ) = τ eη,o (ρ) det ρ, e ϑ -e η = τ eη,o (ρ) det ρ, cs(X η , X) and we obtain that

is locally constant on Rep ac (M, d). By Theorem 7 we thus obtain the second point of Theorem 4.

Finally assume that dim M = 3 and b 1 (M ) = 0. Take R a connected component of Rep ac (M, d) and assume that it contains an acyclic and unitary representation ρ 0 . We invoke [DGRS18, Theorem 1] and the Cheeger-Müller theorem [START_REF] Cheeger | Analytic torsion and the heat equation[END_REF][START_REF] Müller | Analytic torsion and R-torsion of riemannian manifolds[END_REF] to obtain that 0 / ∈ Res(L ∇ρ 0 X ) and

where the first equality comes from (5.8) (we have q = 1 since dim M = 3) and τ RS (ρ 0 ) is the Ray-Singer torsion of (M, ρ 0 ), cf. Lemma A.1. We have

Proof. Using (A.3) and Π 2 t = Π t we obtain (i). This implies

where all the o(t) are taken in L(H, G).

which proves the first part of the Lemma. The second part is very similar.

where

Proof. We start from tr

Now since A t commutes with Π t we have by the second part Lemma A.1

But now the first part of Lemma A.1 gives ΠP Π = 0. We therefore obtain, because A and Π commute,

which concludes.

A.2. Gain of regularity. Assume that we are given four Hilbert spaces E ⊂ F ⊂ G ⊂ H with continuous and denses inclusions. Take also Π t , |t| < δ a family of finite rank projectors on H which is differentiable at t = 0 as family of bounded operators G → H (note that this differs from the last subsection where we had H → G instead), that is

for some P ∈ L(G, H). We will denote C t = ran(Π t ) ⊂ H and C = ran(Π).

Lemma A.3. Assume that Π t is bounded E → F and that Π t is differentiable at t = 0 as a family of L(E, F). Assume also that rank Π t does not depend on t. Then P is actually bounded G → F and

Proof. Because E is dense in H we know that C ⊂ F. There exists ϕ 1 , . . . , ϕ m ∈ E such that ϕ 1 t , . . . , ϕ m t is a basis of C t for t small enough where we set

we finally obtain that t → Π t ∈ L(G, F) is differentiable at t = 0.

Appendix B. Continuity of the Pollicott-Ruelle spectrum

We describe here the spaces used in §6,7. In what follows, M is a compact manifold, (E, ∇) a flat vector bundle on M and X 0 is a vector field on M generating an Anosov flow, cf. §4.1. We denote by T * M = E * u,0 ⊕ E * s,0 ⊕ E * 0,0 its (dual) splitting.

B.1. Bonthonneau's uniform weight function. We state here a lemma from Bonthonneau which is [Bon18, Lemma 3]. This gives us an escape function having uniform good properties for a family of vector fields. A consequence is that one can define some uniform anisotropic Sobolev spaces on which each vector field of the family has good spectral properties. In what follows, | • | is a smooth norm on T * M .

Lemma B.1. There exists conical neighborhoods N u and N s of E * u,0 and E * s,0 , some constants C, β, T, η > 0, and a weight function m ∈ C ∞ (T * M, [0, 1]) such that the following holds. Let X be any vector field satisfying X -X 0 C 1 < η, and denote by Φ t its induced flow on T * M and by E * u and E * s its (dual) unstable and stable bundles. Then

(1) E * • ⊂ N • , for • = s, u and for any t > 0, ξ u ∈ E * u and ξ s ∈ E * s one has

(2) For every t ≥ T it holds

where We set G = Op(g) ∈ Ψ 0+ (M ) for any quantization procedure Op. Then by [Zwo12, § §8.3,9.3,14.2] we have exp(±µG) ∈ Ψ µ+ (M ) for any µ > 0. For any µ > 0 and j ∈ Z we define the spaces

where H j (M, Λ • ⊗ E) is the usual Sobolev space of order j on M with values in the bundle Λ • ⊗ E. Note that any pseudo-differential operator of order m is bounded H

for any µ, m, j.

Appendix C. The wave front set of the Morse-Smale resolvent

The purpose of this section is to prove Proposition 10.2. For simplicity we prove it with for X instead of -X. We will denote Π the spectral projector (9.3) for the trivial bundle (C, d). We will need the following Lemma C.1. Let ε > 0 and a ∈ Crit(f ). There exists δ > 0, a closed conic set Γ ⊂ T * (M × M ) with Γ ∩ N * ∆(T * M ) = ∅ and χ ∈ C ∞ (M, [0, 1]) such that χ ≡ 1 near a such that K χ,t+ε = O D n Γ (M ×M ) (e -tδ ), where for t ≥ 0, K χ,t is the Schwartz kernel of the operator χe -tL X Id -Π χ.