Morse-Smale flow, Milnor metric, and dynamical zeta function

With the help of interactions between the fixed points and the closed orbits of a Morse-Smale flow, we introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem.


Introduction
The study of the relation between the combinatorial/analytic torsion of a flat vector bundle and the Morse-Smale flow was initiated by Fried [Fri87] and Sánchez-Morgado [SM96].In this paper, we give a formula relating -a spectral invariant: the Ray-Singer metric associated with a flat vector bundle with a Hermitian metric on a closed Riemannian manifold; -a dynamical invariant: the Milnor metric which reflects the interactions between the fixed points and the closed orbits of the Morse-Smale flow, and generalizes the absolute value at zero point of the Ruelle dynamical zeta function; -a transgressed Euler class: the Mathai-Quillen current.
0.1.Background.-Let X be a connected closed smooth manifold of dimension m.Let (F, ∇ F ) be a complex flat vector bundle of rank r on X with flat connection ∇ F .Let ρ : π 1 (X) → GL r (C) be the holonomy representation of the fundamental group π 1 (X).Denote by H • (X, F ) the cohomology of the sheaf of locally constant sections of F , and by λ = m i=0 (det H i (X, F )) (−1) i the determinant line of H • (X, F ). Assume that H • (X, F ) = 0 and that F is equipped with a flat metric, which is equivalent to say that its holonomy representation ρ is unitary.The Reidemeister torsion [Rei35,Fra35,dR50] is a positive real number defined by means of of a triangulation on X.However, it does not depend on the triangulation and becomes a topological invariant.It is the first invariant that could distinguish closed manifolds such as lens spaces which are homotopy equivalent but not homeomorphic.
The analytic torsion was introduced by Ray and Singer [RS71] as an analytic counterpart of the Reidemeister torsion.In order to define the analytic torsion one has to choose a Riemannian metric on X.The analytic torsion is a certain weighted alternating product of regularized determinants of the Hodge Laplacians acting on the space of differential forms with values in F .
Let us explain Bismut-Zhang's theorem [BZ92, Th. 0.2] in more detail.Indeed, to formulate their result in the case where the flat vector bundle is not necessarily acyclic or unitarily flat, Bismut and Zhang introduced the so-called Ray-Singer metric, which is a metric on λ defined as the product of the analytic torsion with an L 2 -metric on λ.Also they introduced the Milnor metric on λ which is a combinatorial metric associated with a Morse-Smale gradient flow.It generalizes the Reidemeister torsion J.É.P. -M., 2021, tome 8 to the case where F is neither acyclic nor unitarily flat.In this way, they were able to extend the Cheeger-Müller theorem to a comparison theorem of two metrics on λ, one is analytic and the other one is combinatorial.
The study of the relation between the combinatorial/analytic torsion and the dynamical system can be traced back to Milnor [Mil68].Fried [Fri86] showed that on hyperbolic manifolds the analytic torsion of an acyclic unitarily flat vector bundle is equal to the value at zero point of the Ruelle dynamical zeta function of the geodesic flow.He conjectured [Fri87, p. 66 Conj.] that similar results should hold true for more general flows.In [She18], following previous contributions by , using Bismut's orbital integral formula [Bis11], the author affirmed the Fried conjecture for geodesic flows on closed locally symmetric manifolds.In [SY17], the authors made a further generalization to closed locally symmetric orbifolds.
Besides the gradient flow, Morse-Smale flow is the simplest structurally stable dynamical system which has only two types of recurrent behaviors: closed orbits and fixed points [Pal68,PS70].Fried [Fri87,Th. 3.1] proved his conjecture for the Morse-Smale flows without fixed points.When compared with Bismut-Zhang's theorem [BZ92, Th. 0.2], it seems natural to ask whether there is a relation between the torsion invariant (or more generally the Ray-Singer metric for non acyclic and non unitarily flat vector bundle) and a general Morse-Smale flow which has both fixed points and closed orbits.This is one of the motivations of Sánchez-Morgado's work [SM96].He showed that the heteroclinic orbits have a non trivial contribution in the torsion invariant, and in this way he constructed a counterexample to Fried's conjecture on Seifert manifolds.
In this paper, we introduce a new Milnor metric, which indeed contains the heteroclinic contributions and generalizes the absolute value at zero point of the Ruelle dynamical zeta function, and we give a comparison theorem for the Milnor and Ray-Singer metrics on λ.We believe that in this way we give a complete answer in the affirmative to the above question.
Let us mention that there is another interpretation of the Ruelle dynamical zeta function provided by Dang-Rivière [DR20c].See also [DR19,DR20a,DR20b,DR21] for related works.

0.2.
A new Milnor metric.-A vector field V is called Morse-Smale if V generates a flow whose nonwandering set is the union of a finite set A of hyperbolic fixed points and a finite set B of hyperbolic closed orbits, and if the stable and unstable manifolds of the critical elements in A B intersect transversally.
Let us take a Hermitian metric g F on F .In Section 2.4, we construct on λ a Milnor type metric • M,2 λ,V using long exact sequences associated with a Smale filtration of the Morse-Smale flow.Note that the long exact sequences encode the information about the interactions between the critical elements in A B. If V is a negative gradient of a Morse function, then our Milnor metric is just the classical one as defined in [BZ92, Def.1.9], which generalizes [Mil66].
Our first result says that the Milnor metric • M,2 λ,V is a generalization of the absolute value at zero point of the Ruelle dynamical zeta function.For a closed orbit γ ∈ B, let γ ∈ R * + be its minimal period, and let ind(γ) ∈ N be its index (see (2.3)).Take ∆(γ) to be 1 if γ is untwist and −1 in the contrary case (see (2.4)).The Ruelle dynamical zeta function is defined for s ∈ C by .
Proposition 0.1.-If V does not have any fixed points, and if none of ∆(γ) is an eigenvalue of ρ(γ), then H • (X, F ) = 0, and the norm of the canonical section 1 ∈ C = λ is given by 0.3.The main result of the paper.-Let g T X be a metric on T X.Let ψ(T X, ∇ T X ) be the Mathai-Quillen current associated with the Levi-Civita connection ∇ T X (see Section 3.2).It is a current of degree m − 1 defined on the total space of the tangent bundle T X, which takes values in o(T X), the orientation line bundle of T X.Let • RS,2 λ be the Ray-Singer metric on λ associated with Our main result is the following.Theorem 0.2.-We have If V does not have any closed orbits, Theorem 0.2 reduces to [BZ92, Th. 0.2].Note also that if F is unitarily flat, then the right-hand side of (0.1) varnishes.Therefore, if V does not have any fixed points and if F is unitarily flat, by Proposition 0.1, our theorem corresponds to [Fri87,Th. 3.1].
Our proof of Theorem 0.2 is based on a result of Franks [Fra79, Prop.5.1], who constructed a gradient flow by destroying the closed orbits of the Morse-Smale flow.In Section 2.5, we first establish a comparison formula between our Milnor metric associated with the original Morse-Smale flow and the classical one associated with Franks' gradient flow.In Section 3, to obtain Theorem 0.2, we apply Bismut-Zhang's formula [BZ92, Th. 0.2], which compares the Ray-Singer metric with the Milnor metric for Franks' gradient flow.
Recall that F is said to be unimodular, if its holonomy representation ρ is unimodular, i.e., |det ρ(γ)| = 1 for all γ ∈ π 1 (X).This is equivalent to the fact that there is a Hermitian metric g F such that θ(F, g F ) = 0.By Theorem 0.2, we get λ,V .
J.É.P. -M., 2021, tome 8 0.4.Organization of the paper.-In Section 1, we introduce some conventions on the determinant line, the cohomology of a circle, and also a long exact sequence associated with three manifolds Y 1 ⊂ Y 2 ⊂ Y 3 .In Section 2, we recall some background on Morse-Smale flows.We also introduce the Milnor type metric, and we show Proposition 0.1.In Section 3, we recall the constructions of Mathai-Quillen current and Ray-Singer metric.We show our main result.We use the convention N = {0, 1, 2, . ..} and R * + = (0, ∞).Acknowledgements.-We are indebted to Xiaolong Han and Xiaonan Ma for reading a preliminary version of this paper and for useful suggestions.S.S. would like to thank Nguyen Viet Dang and Gabriel Rivière for fruitful discussions on Morse-Smale flows.

Preliminaries
This section is organized as follows.In Section 1.1, we introduce our convention on the determinant line.In Section 1.2, we give a metric on the determinant line of the cohomology of S 1 .This is our model case near the closed orbits of a flow.In Section 1.3, we explain a long exact sequence associated with a triple of manifolds 1.1.The determinant line.-Let W be a complex finite dimensional vector space.
We denote by W * the dual space.If dim W = 1, we write Λ j (W ) to be the exterior algebra.Set be a complex of finite dimensional vector spaces.By [KM76] or [BGS88, (1.5)], we have the canonical isomorphism of lines 1.2.The cohomology of S 1 .-Let S 1 = R/Z be an oriented circle.Let F be a flat vector bundle of rank r on S 1 .Let ρ : π 1 (S 1 ) → GL r (C) be the holonomy (1) of F .Let a 0 ∈ π 1 (S 1 ) be the generator of π 1 (S 1 ), which is compatible with the orientation on S 1 .Set A = ρ(a 0 ) ∈ GL r (C).
Consider the canonical triangulation on S 1 induced by one 0-simplex σ 0 and one 1-simplex σ 1 as in Figure 1.1.It induces a complex of simplicial cochains with values By (1.1), the canonical element We equip det H ) is just Proposition 0.1 for the rotation flow on S 1 .
Remark 1.2.-Since the flat vector bundle is not necessarily unimodular, i.e., |det (A)| is not necessarily equal to 1, the choice of the orientation on S 1 is very important.
(1) For any flat vector bundle F on a manifold X, the holonomy is a representation ρ : , where X is the universal cover of X and π 1 (X) acts on the left on X by the deck transformation and on C r by ρ.
By (1.1) and (1.7), we get an isomorphism of lines Using the other triples (∅, Y 1 , Y 2 ) and (∅, Y 1 , Y 3 ), we get similar isomorphisms By (1.8) and (1.9), we see that Proof.-As in [BM13, (0.15)], let us take a smooth triangulation of Y 3 such that it induces also smooth triangulations on Y 1 and Y 2 .Denote by the complexes of simplicial cochains with coefficients in F .Then we have an exact sequence of complexes By (1.1) and (1.12), we get an isomorphism of lines We can define f C 2,32 , f C 1,31 and f C 21,32 in a similar way.By an easy calculation, there is ).

Milnor metric
This section is organized as follows.In Sections 2.1 and 2.2, we recall the definitions of Morse-Smale flow and the associated Ruelle dynamical zeta function.In Section 2.3, we recall some results due to Franks [Fra79,Fra82] on the construction of a new gradient flow by destroying the closed orbits of the original Morse-Smale flow.In Section 2.4, using the Smale filtration, we introduce the Milnor metric.In Section 2.5, we establish a comparison formula for the two Milnor metrics, one is associated with the Morse-Smale flow and the other is associated with the gradient flow constructed by Franks.
We refer the reader to the classical textbook of Palis and de Melo [PdM82] for the basic notion on dynamical system.
2.1.Morse-Smale flow.-Let X be a connected closed smooth manifold of dimension m.Let V be a vector field on X.Consider the differential equation Equation (2.1) defines a group of diffeomorphism (φ t ) t∈R of X.If x ∈ X, an orbit of x is defined by the image t ∈ R → φ t (x) ∈ X.We call x ∈ X is a fixed point, if its orbit reduces to a point, i.e, for all t ∈ R, φ t (x) = x.
J.É.P. -M., 2021, tome 8 Clearly, x ∈ X is a fixed point if and only if V (x) = 0. We call an orbit is closed if it is diffeomorphic to S 1 .Denote by A the set of fixed points and by B the set of closed orbits.
x , and there exist C > 0, θ > 0 and a Riemannian metric g T X on X such that for v ∈ E u x , v ∈ E s x , and t > 0, we have The unstable and stable manifolds of the hyperbolic fixed point x are defined by where d X denotes the Riemannian distance on (X, g T X ).
is a negative gradient of a Morse function f with respect to some Riemannian metric, then the index ind(x) of the critical point x is just the Morse index of f at x. Definition 2.2.-A closed orbit γ of the flow φ • is called hyperbolic, if there is a φ t -invariant continuous splitting T X| γ = RV ⊕ E u γ ⊕ E s γ , of C 0 -vector bundles over γ such that (2.1) holds.The associated unstable and stable manifolds are defined by The index ind(γ) ∈ N of γ is defined by Clearly, A ∪ γ∈B γ is contained in the nonwandering set.
Definition 2.4.-A vector field V or a flow φ • is called Morse-Smale if -the sets A and B are finite and contain only hyperbolic elements; -the nonwandering set of φ • is equal to A ∪ γ∈B γ; -the stable and unstable manifold of any critical element in A B intersect transversally.
In the sequel, we assume that V is a Morse-Smale vector field.
γ is orientable along γ, and is called twist otherwise.Put (2.4)For closed orbits we must distinguish the following four cases in establishing the standard forms.Assume γ ∈ B such that ind(γ) = p.
Case 1. -Suppose that T X| γ is orientable and that γ is untwist.In this case, γ is said to be of standard form, if there is a system of coordinates (t, y 1 , . . ., y m−1 ) ∈ S 1 ×D m−1 on a tubular neighborhood U γ of γ such that γ is represented by (t, 0) ∈ S 1 × D m−1 and Case 2. -Suppose that T X| γ is orientable and that γ is twist.In this case, γ is said to be of standard form, if U γ and V can be obtained from Case 1 by the identification (t, x 1 , . . ., x m−1 ) ∼ (t + 1/2, −x 1 , x 2 , . . ., x p , −x p+1 , x p+2 , . . ., x m−1 ).
J.É.P. -M., 2021, tome 8 Case 4. -Suppose that T X| γ is not orientable and that γ is twist.In this case, γ is said to be of standard form, if U γ and V can be obtained from Case 1 by the identification (t, x 1 , . . ., x m−1 ) Note that in [Fra79, §1] the author assumed that X is orientable, so only the first two cases appear.
The following three propositions are [Fra79, Prop.1.6, Th. 2.2, Prop.5.1].Their proofs can be generalized to the non orientable case with some evident modifications.We omit the details.
Proposition 2.6.-For any Morse-Smale vector field V , there is a smooth family of Morse-Smale vector fields (V ) 0 1 such that V 0 = V and that the critical elements of V 1 are all of standard forms and are precisely the same as the critical elements of V .Moreover, V 0 and V 1 are topologically conjugated, i.e., there is a homeomorphism carrying the orbits of V 0 to those of V 1 and preserving their orientations.Remark 2.8.-Following the proof of [Fra79,Prop. 1.6] given by Franks, we can choose the family (V ) 0 1 such that the critical elements are preserved under the deformation.However, in the proof of our main result Theorem 0.2 given in Section 3, we need only choose a family such that all the set of the fixed points of V are in a small neighbourhood of the set of the fixed points of V .Proposition 2.9.-If V is a Morse-Smale vector field whose flow has fixed points in standard form and no closed orbits, then V is a negative gradient of a certain Morse function with respect to some Riemannian metric.
To state the following proposition, let us introduce some notation.For x, y ∈ A such that ind(y x and W s y intersect transversally.Let us fix an orientation on each W u x with x ∈ A. Define n(a) = ±1 as in [BZ92, (1.28)], whose definition does not require the manifold to be orientable.
Proposition 2.10.-For some small neighborhood U = γ∈B U γ of closed orbits γ∈B γ, there is a Morse function f on X whose gradient vector field ∇f with respect to a certain Riemannian metric is Morse-Smale, such that -on X U , we have -on each U γ , the Morse function f has only two critical points x γ , x γ of index ind(γ) + 1 and ind(γ) respectively.Also, Γ(x γ , x γ ) consists of two integral curves a γ , a γ (see Figure 2.1) such that their composition a γ • (a γ ) −1 and the closed orbit γ lie in the same freely homotopy class of loops on X and that (3)  n(a γ )n(a γ ) = −∆(γ).
It is easy to see that the nonwandering set of V 1 in U γ consists of two points (1, 0), (1/2, 0) ∈ S 1 × D m−1 .Then, by a small perturbation on V 1 , we get a Morse-Smale gradient vector field −∇f which has the desired transversality and other properties.We remark that by the above construction, we can find a family of vector fields (V ε ) 0 ε 1 connecting V and −∇f such that near {1/4} × D m−1 , for any ε ∈ [0, 1], V ε does not vanish.Similar remark holds for γ in standard forms of Cases 2-4.In Section 3.5, we will use this fact to simplify the proof of our main theorem.

Smale filtration and Milnor metric.
-Following [Fra82, Def.9.10], let be a Smale filtration on X associated with V .Note that each X p ⊂ X is a submanifold with boundary, and can be constructed by the sublevel set of a smooth Lyapunov function.Also, we have (3) This requires a choice of the orientations on the unstable manifolds of xγ , x γ .Such choice is irrelevant.
J.É.P. -M., 2021, tome 8 -on each ∂X p , V does not vanish and points toward the inside of X p ; -there is only one critical element c ∈ A B contained in X p+1 X p and {c} = t∈R φ t (X p+1 X p ).
Let (F, ∇ F ) be a flat vector bundle on X induced by the representation ρ.Let H • (X, F ) be the cohomology of the sheaf of locally constant sections of F .Put We use the notation in Section 1.3.By (1.10), we get an isomorphism (2.9) σ V : By Proposition 1.3, up to a sign, the morphism σ V does not depend on the way that the cohomologies are fused.By [Fra82, Th. 9.11] (see also [SM96,§2]), if the critical element c ∈ X p+1 X p is a fixed point, then and if the critical element c ∈ X p+1 X p is a closed orbit, then where o(E u c ) is the orientation line bundle of E u c along the closed orbit c.We equip det H ) defined in (1.5).Let g F be a Hermitian metric on F .By (2.9)-(2.11), the restriction g F | A and the metric λ,V on λ.By Proposition 1.3, this metric does not depend on the way that the cohomologies are fused.
Definition 2.12.-The metric • M,2 λ,V on λ is called the Milnor metric associated with V .
coincides with the one constructed by Bismut-Zhang [BZ92, Def.1.9].In fact, there is a small difference with Bismut-Zhang's construction, where they used a filtration [BZ92, (1.37)] induced by sublevel sets of a nice Morse function.Using Proposition 1.3, we can deduce that the two constructions coincide.
Remark 2.14.-For two topologically conjugated Morse-Smale vector fields whose critical elements coincide, we can choose the same Smale filtration.From our construction, the corresponding Milnor metrics coincide.
Remark 2.15.-The Milnor metric for general Morse-Smale flow does not depend on the Smale filtration (2.7).We will not give a direct proof since it is a consequence of our main Theorem 0.2.

2.5.
A comparison formula for Milnor metrics.-In this section, we assume that all the critical elements of V are in standard forms, and that f is chosen as in Proposition 2.10.
Let det τ (a γ ) ∈ det F x γ ⊗ (det F xγ ) −1 be the canonical element induced by the parallel transport with respect to the flat connection along the integral curve a γ (see Figure 2 Proposition 2.17.-The following identity holds, Proof.-We refine the filtration (2.7) by the new critical points of f .By Propositions 1.3 and 2.10, and by (2.9), the following diagram commutes (2.15) where the first vertical arrow is induced by (1.4) and the second vertical arrow is a multiplication by ±1.The Milnor metric • M,2 λ,−∇f is obtained from the metric g F | A∪{xγ ,x γ :γ∈B} via σ −∇f .By (2.15), we get (2.14).

An extension of Bismut-Zhang's theorem to Morse-Smale flow
This section is organized as follows.In Sections 3.1-3.4,following [BZ92], we recall the constructions of the Berezin integral, the Mathai-Quillen current, and the Ray-Singer metric.In Section 3.5, we restate and prove our main theorem.-Let E be a real Euclidean space of dimension n with the scalar product , , and let W be a real vector space of finite dimension.We use the supersymmetric formalism of Quillen [Qui85].Denote by ⊗ the tensor product of super algebras.
Suppose temporarily that E is oriented and that e 1 , . . ., e n is an oriented orthonormal basis of E. Let e 1 , . . ., e n be the corresponding dual basis of E * .We define B to be the linear map from Λ More generally, if o(E) is the orientation line of E, then B defines a linear map from Let A ∈ End anti (E) be an antisymmetric endomorphism of E. We identify A with (3.1) Ȧ = 1 2 1 i,j n e i , Ae j e i ∧ e j ∈ Λ 2 (E * ).

By definition, the Pfaffian
Clearly, Pf[A] vanishes if n is odd.
3.2.Mathai-Quillen formalism.-Recall that X is a connected closed smooth manifold of dimension m.Let E be a Euclidean vector bundle of rank n on X with a Euclidean metric g E and a metric connection ∇ E .Let be its curvature.Denote by o(E) the orientation line bundle of E. The Euler form of (E, ∇ E ) is given by Clearly, e(E, ∇ E ) = 0, if n is odd.Let E be the total space of E, and let π : E → X be the natural projection.We will use the formalism of the Berezin integral developed in Section 3.1 with W Let T V E ⊂ T E be the vertical subbundle of T E , and let T H E ⊂ T E be the horizontal subbundle of T E with respect to ∇ E , so that (3.2) T E = T H E ⊕ T V E .
J.É.P. -M., 2021, tome 8 By identification T V E π * E, the vertical projection with respect to the splitting (3.2) induces a section P E ∈ C ∞ (E , T * E ⊗ π * E).Using the metric g E , we identify Let (α T ) T 0 and (β T ) T >0 be families of forms on E defined by Clearly, Let us recall [BZ92, Th. 3.4, 3.5, & 3.7].
Theorem 3.2.-For T 0, the form α T is closed whose cohomology class does not depend on T .For T > 0, α T represents the Thom class of E so that π * α T = 1, and we have ∂α T ∂T = −dβ T .
We identify X as a submanifold of E by the zero section.The normal bundle to X in E is exactly E and the conormal bundle is E * .Let δ X be the current on E defined by the integration on X.If µ is a smooth compactly supported form on E with values in π * o(T X), then For a current v on E , denote by WF(v) ⊂ T * E its wave front set [Hör90, §8.1].
whose support is contained in K and for any T 1, we have In view of (3.3) and (3.4), is a well-defined current of degree n − 1 on E with values in π * o(E).
J.É.P. -M., 2021, tome 8 3.3.metric.-We use the notation in Section 2. Recall that (F, ∇ F ) is a flat vector bundle on X.Let (Ω • (X, F ), d X ) be the de Rham complex of smooth forms on X with values in F .By de Rham's theory, its cohomology is H • (X, F ).
Take metrics g T X and g F on T X and F .Let , Λ • (T * X)⊗F be the induced metric on Λ • (T * X) ⊗ F .Let dv X ∈ Ω m (X, o(T X)) be the Riemannian volume form on X. For Then (3.9) defines an L 2 -metric on Ω • (X, F ). Let d X * be the formal adjoint of d X with respect to the L 2 -metric •, • Ω • (X,F ) .Put Then X is a formally self-adjoint second order elliptic differential operator acting on Ω • (X, F ).By Hodge theory, we have (3.10)ker X H • (X, F ).
By (2.8) and (3.10), the restriction of the L 2 -metric •, • Ω • (X,F ) to ker X induces a metric | • | RS,2 λ on λ.Let (ker X ) ⊥ be the orthogonal space to ker X in Ω • (X, F ). Then X acts as an invertible operator on (ker X ) ⊥ .Let ( X ) −1 be the inverse of X acting on (ker X ) ⊥ .Let N Λ • (T * X) be the number operator on Λ • (T * X), which is multiplication by p on Λ p (T * X).Let ∇ T X be the Levi-Civita connection on (T X, g T X ).Let ψ(T X, ∇ T X ) be the Mathai-Quillen current.By (2.1) and by Proposition 3.7, for any Morse-Smale vector field V , the pull-back V * ψ(T X, ∇ T X ) is a well-defined current of degree m − 1 on X with values in o(T X).Set θ(F, g F ) = Tr[(g F ) −1 ∇ F g F ] ∈ Ω 1 (X).
Then, θ(F, g F ) is a closed 1-form and its cohomology class θ(F ) = [θ(F, g F )] ∈ H 1 (X) does not depend on the metric g F .Up to a normalization, the class θ(F ) coincides with the first Kamber-Tondeur class [KT74].
J.É.P. -M., 2021, tome 8 By our construction of K, for γ ∈ B, the two points x γ , x γ , and the integral curve a γ are in the same simplex σ m γ ∈ K m , so that (3.18) W 0 (x γ ) − W 0 (x γ ) = log det τ (a γ ) Then W 1 is a closed current of degree 1 on X such that Supp(W 1 ) ⊂ K m−1 .By (3.6) and by A ∩ K m−1 = ∅, (−V ) * ψ(T X, ∇ T X ) is smooth in the neighbourhood of the support of W 1 , so that is a well-defined current on X.By (3.7), (3., ∇f and −V can be connected by a family of vector fields without zero.Using the fact that Supp(W 1 ) ⊂ K m−1 , and by a version of Proposition 3.11 where θ(F, g F ) is replaced by the closed current W 1 , we get (3.22).The proof of our theorem is completed.
J.É.P. -M., 2021, tome 8 2.2.Ruelle dynamical zeta function.-For γ ∈ B, denote by γ be a representation of the fundamental group of X.If γ ∈ B, denote by ρ(γ) the holonomy along γ.Clearly, ρ(γ) is well-defined up to a conjugation.Definition 2.5.-The twist Ruelle dynamical zeta function is a meromorphic function on C defined for s ∈ C by (2.5) R ρ (s) = γ∈B det(1 − ∆(γ)ρ(γ)e −s γ ) (−1) ind(γ) .2.3.Franks' Morse function.-We follow [Fra79, §1].Let D r be the r-dimensional open unit ball of center 0 ∈ R r .A fixed point x ∈ A of index p is said to be of standard form if there is a system of coordinates (y 1 , . . ., y m ) ∈ D m on a neighborhood of x such that x is represented by 0 and Remark 2.7.-The second part of Proposition 2.6 is a consequence of the Structural stability of the Morse-Smale flow [Pal68, PS70].
The relation between the Morse-Smale flow and the Morse function is summarized in the following two propositions.The first one is due to Smale [Sma61, Th.B].