Colin Guillarmou
email: colin.guillarmou@universite-paris-saclay.fr

AND Thibault Lefeuvre
email: tlefeuvre@imj-prg.fr

Gabriel P Paternain
email: g.p.paternain@dpmms.cam.ac.uk

MARKED LENGTH SPECTRUM RIGIDITY FOR ANOSOV SURFACES

Let Σ be a smooth closed oriented surface of genus ě 2. We prove that two metrics on Σ with same marked length spectrum and Anosov geodesic flow are isometric via an isometry isotopic to the identity. The proof combines microlocal tools with the geometry of complex curves.

1. Introduction 1.1. Main results. On a smooth closed connected oriented manifold Σ, a metric g is Anosov if its geodesic flow on the unit tangent bundle SΣ satisfies the Anosov property defined in (2.9). While negatively curved metrics are typical examples of Anosov metrics, the set of Anosov metrics is a considerably larger open set compared to that of negatively curved metrics (cf. [START_REF] Eberlein | When is a geodesic flow of Anosov type? I[END_REF]). However, this set is less well understood: for instance it is still unknown whether the set of Anosov surfaces is path-connected or whether any manifold (of dimension ě 4) that admits an Anosov metric also admits a negatively curved metric. Anosov metrics coincide with the C 2 -interior of metrics without conjugate points [START_REF] Oswaldo | On the creation of conjugate points[END_REF], and they can also exist isometrically embedded in R 3 [START_REF] Victor | Anosov geodesic flows for embedded surfaces[END_REF], unlike negatively curved metrics.

On Σ, we shall denote by M Anosov pΣq the set of all Anosov metrics (assuming it is non-empty) and by M Anosov pΣq :" M Anosov pΣq{Diffeo 0 pΣq the moduli space of Anosov metrics, where we denote by Diffeo 0 pΣq the group of diffeomorphisms isotopic to the identity. Let C be the set of free homotopy classes on Σ. This set is countable and in natural correspondence with conjugacy classes of π 1 pΣ, ‹q. It is well-known that for an Anosov metric g there exists a unique closed geodesic γ g pcq in each free homotopy classes c P C. The marked length spectrum is then defined as the map (1.1) L : M Anosov pΣq Ñ p0, 8q C , L g pcq :" g pγ g pcqq, where g pγq denotes the length of the curve γ computed with respect to g. It is conjectured that the marked length spectrum map (1.1) is injective. For negatively curved metrics, this is known as the Burns-Katok conjecture [START_REF] Burns | Manifolds with nonpositive curvature. Ergodic Theory Dynam[END_REF]. The purpose of the present paper is to establish this conjecture when dim Σ " 2, that is, when Σ is a surface.

Theorem 1.1. Let Σ be a smooth closed connected oriented surface. Then the marked length spectrum L : M Anosov pΣq Ñ p0, 8q C is injective. In other words, if g 1 and g 2 are two Anosov metrics on Σ with same marked length spectrum (that is, L g 1 " L g 2 ), then there exists a smooth diffeomorphism φ : Σ Ñ Σ, isotopic to the identity, such that φ ˚g1 " g 2 .

Remark 3.12 below shows that the main theorem also applies to metrics with C 4 -regularity. The theorem can be seen as the closed surface analogue of Pestov and Uhlmann's celebrated boundary rigidity result for simple surfaces [START_REF] Pestov | Two dimensional compact simple Riemannian manifolds are boundary distance rigid[END_REF]. A surface is simple if it has strictly convex boundary, is nontrapping, and has no conjugate points (like the Anosov property, this is a C 2 -open condition in the metric). In this case, there is a unique geodesic joining any two boundary points, similar to the boundary at infinity of the universal cover of an Anosov surface (which is a disk). The boundary rigidity problem seeks to determine the metric (up to an isometry that is identity on the boundary) from the boundary distance function.

Equality of boundary distance functions for two simple metrics induces a conjugacy between their geodesic flows fixing the boundary of the unit tangent bundle. Similarly, equality of marked length spectra for Anosov surfaces induces a conjugacy between their geodesic flows homotopic to the identity (i.e. fixing the boundary at infinity of the universal cover). A potential approach to both problems is to show that the conjugacy is the lift of an isometry (up to an innocuous action of one of the two flows). Croke [START_REF] Christopher | Rigidity for surfaces of nonpositive curvature[END_REF] and Otal [START_REF] Otal | Sur les longueurs des géodésiques d'une métrique à courbure négative dans le disque[END_REF][START_REF] Otal | Le spectre marqué des longueurs des surfaces à courbure négative[END_REF] used this approach to prove both boundary rigidity and marked length spectrum rigidity for non-positively curved surfaces. In contrast, Pestov and Uhlmann [START_REF] Pestov | Two dimensional compact simple Riemannian manifolds are boundary distance rigid[END_REF] showed that the scattering relation determines the boundary values of holomorphic functions, and relied on the solution of the Calderón problem to prove boundary rigidity for all simple metrics. The problem of marked length spectrum rigidity for Anosov surfaces in the closed setting has remained open until the present work. A posteriori, we will also prove that conjugacies must have the expected form, see Corollary 1.3 below.

Our proof draws inspiration from Pestov-Uhlmann's approach that depends on utilizing "fiberwise holomorphic" smooth invariant functions for the geodesic flow to recover the conformal structure of the surface. However, in the case of closed surfaces, there is no natural Calderón problem connected to the marked length spectrum as in [START_REF] Pestov | Two dimensional compact simple Riemannian manifolds are boundary distance rigid[END_REF], and the smooth invariant functions need to be replaced by suitable singular invariant distributions, which has made it difficult to apply this strategy in our context until now. We take a different route and use the period matrix to recover the conformal structure via the Torelli theorem, see the strategy outline below. We believe that boundary rigidity for simple surfaces in fact follows from Theorem 1.1 in conjunction with the embedding theorem [START_REF] Chen | Riemannian Anosov extension and applications[END_REF]; this will be discussed elsewhere.

As mentioned above, prior to Theorem 1.1, the only known cases of injectivity of the marked length spectrum for closed surfaces were essentially limited to metrics with non-positive curvature ([Cro90, Ota90a] and the subsequent generalization by Croke, Fathi, and Feldman [START_REF] Croke | The marked length-spectrum of a surface of nonpositive curvature[END_REF]). These proofs do not extend to the Anosov setting as they rely crucially on the assumption that the Gauss curvature is non-positive (or that the Morse correspondence preserves angles); thus a new approach is needed to establish Theorem 1.1. For dimensions greater than or equal to three, fewer results exist. Given two Anosov metrics in the same conformal class, Katok's argument in [START_REF] Katok | Four applications of conformal equivalence to geometry and dynamics[END_REF] provides injectivity of the marked length spectrum. Almost all other results require non-positive sectional curvature: Hamenstädt [START_REF] Hamenstädt | Cocycles, symplectic structures and intersection[END_REF] proved the conjecture when one of the two metrics is locally symmetric, relying on the entropy paper of Besson-Courtois-Gallot [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]. Recently, the first two authors of this paper proved the conjecture in any dimension when the curvature is non-positive [START_REF] Guillarmou | The marked length spectrum of Anosov manifolds[END_REF] and the metrics are assumed to be close. An alternative proof based on the concept of geodesic stretch can be found in [START_REF] Guillarmou | Geodesic stretch, pressure metric and marked length spectrum rigidity[END_REF]. Finally, in the Anosov setting and without requiring curvature assumption, the second author and Cekić proved in [CL21, Corollary 1.4] that the marked length spectrum is locally injective near a generic Anosov metric in any dimension.

As a direct consequence of Theorem 1.1, the proof of [CD04, Theorem 1.1] provides a more general version of Theorem 1.1 related to the minimal filling problem of Gromov, see [START_REF] Christopher | Rigidity theorems in Riemannian geometry[END_REF]Question 6.8] where this is discussed.

Corollary 1.2. Let Σ be a smooth closed connected oriented surface and g 1 , g 2 P M Anosov pΣq. If L g 1 ě L g 2 , then vol g 1 pΣq ě vol g 2 pΣq with equality of volumes if and only if there exists a smooth diffeomorphism φ : Σ Ñ Σ, isotopic to the identity, such that φ ˚g1 " g 2 .

In Corollary 1.2, the volumes of the metrics are computed with respect to the Riemannian measure. Corollary 1.2 follows from the positive Livšic Theorem of Lopes and Thieullen [LT05] and Theorem 1.1, see [START_REF] Croke | Lengths and volumes in Riemannian manifolds[END_REF] for a proof. Finally, Theorem 1.1 also implies a classification result for smooth conjugacies of Anosov geodesic flows over surfaces: Corollary 1.3. Let Σ be a smooth closed connected oriented surface, let g 1 , g 2 P M Anosov pΣq and denote by pϕ g i t q tPR the geodesic flow over the unit tangent bundle SΣ i of pΣ, g i q for i " 1, 2. If φ : SΣ 1 Ñ SΣ 2 is a diffeomorphism such that φ ˝ϕg 1 t " ϕ g 2 t ˝φ, @t P R, then there exists a time t 0 P R and an isometry F : pΣ, g 1 q Ñ pΣ, g 2 q such that φ " ϕ g 2 t 0 ˝F 1 , where F 1 : SΣ 1 Ñ SΣ 2 is defined as F 1 px, vq :" pF pxq, dF x pvqq.

Taking g 1 " g 2 in Corollary 1.3, we thus find that the set of self-conjugacies of an Anosov geodesic flow over a surface is given (after taking the quotient by the flow) by the finite group of isometries of the metric. In the case of non-positively curved surfaces, Corollary 1.3 was first obtained in [START_REF] Christopher | Rigidity for surfaces of nonpositive curvature[END_REF]Theorem B]. In [CFF92, Theorem C] an argument is given that shows that in fact the claim in Corollary 1.3 follows from the marked length spectrum rigidity. The only point to note is that [CFF92, Lemma 4.3] holds for an Anosov metric: indeed, a self-conjugacy isotopic to the identity must map a closed geodesic to itself (since there is a unique closed geodesic in each free homotopy class) and by density of closed orbits and transitivity it must be of the form ϕ t 0 . In dimensions three and higher, finiteness of the group of self-conjugacies (after modding out by the flow itself) is known in some cases such as 1{4-pinched negatively curved manifolds, we refer to [START_REF] Damjanović | Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms[END_REF] for further details.

1.2. Strategy. The proof strategy involves demonstrating that the marked length spectrum L g captures the complex structure of the metric rgs up to biholomorphisms isotopic to the identity, and hence determines the class of the underlying complex structure in Teichmüller space (Proposition 3.10). Once this is established, the injectivity of the marked length spectrum in the same conformal class, as shown by Katok [Kat88], leads to a straightforward proof of Theorem 1.1.

To establish that L g determines the complex structure, we show that it encodes the period matrix of the underlying Riemann surface (Proposition 3.1) and then observe that the argument may be repeated on any finite cover to recover the structure in Teichmüller space. This relies on the fact that given two different points in Teichmüller space, there is a finite cover of the surface where the lifted complex structures are in different orbits of the mapping class group (Lemma 3.11).

To recover the period matrix, we introduce an algebra A `pSΣq of fiberwise holomorphic flow-invariant distributions on SΣ that are distributions invariant under the geodesic flow and have a specific Fourier decomposition in the circle fibers of SΣ (see §2.2.3). Their first Fourier mode corresponds to genuine holomorphic differentials on the Riemann surface. It is worth noting that the existence of such a well-behaved algebra of distributions is a non-trivial fact and relies on advances in tensor tomography on surfaces [START_REF] Gabriel | Spectral rigidity and invariant distributions on Anosov surfaces[END_REF][START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF].

Finally, we introduce a generalized intersection number of elements in A `pSΣq with closed geodesics, extending the usual definition of the intersection number for currents à la Bonahon [START_REF] Bonahon | The geometry of Teichmüller space via geodesic currents[END_REF], Otal [START_REF] Otal | Le spectre marqué des longueurs des surfaces à courbure négative[END_REF], and others. This allows for the recovery of integrals of holomorphic 1-forms on closed geodesics, and thus the period matrix, from the conjugacy class of the flow (Lemma 3.8 and Remark 3.9). 1.3. Organization of the paper. The proof of Theorem 1.1 relies on several tools, which we introduce in Section 2. Specifically, §2.1 provides a brief review of the geometry of complex curves, while §2.2 is devoted to the geometry and harmonic analysis of the unit tangent bundle of a surface. In §2.3, we delve into hyperbolic dynamics and tensor tomography. Finally, we present the proof of Theorem 1.1 in Section 3.

Acknowledgements. We warmly thank J. Marché, B. Petri, I. Smith and M. Wolff for very helpful discussions related to the proof of Lemma 3.11. We also thank C. Matheus for pointing out an error in an earlier draft.

Tools

Complex geometry.

In what follows, Σ is a smooth closed oriented surface of genus ě 2.

2.1.1. General facts. Let g be a smooth Riemannian metric on Σ. The conformal class rgs of g (and the orientation of Σ) induces a complex structure J P C 8 pΣ, EndpT Σqq on Σ, turning it into a Riemann surface which we shall denote by pΣ, Jq.

We denote by T pΣq the Teichmüller space of Σ, that is, the space of complex structures J on Σ modulo the equivalence relation that J " J 1 iff there exists a diffeomorphism ψ : Σ Ñ Σ, isotopic to the identity, such that ψ ˚J " J 1 . Such an equivalence class of complex structures will be denoted by rJs. The mapping class group MCGpΣq is defined as the quotient of orientation preserving diffeomorphisms Diff `pΣq modulo isotopy.

There is a well-defined action of MCGpΣq on T pΣq (by pullback). The quotient space MpΣq :" T pΣq{MCGpΣq is called the moduli space of (complex structures on) Σ. We refer to [START_REF] Farb | A primer on mapping class groups[END_REF] for a general introduction to the mapping class group of closed surfaces.

2.1.2. Period matrix. Jacobian. Let ta i , b j u be a canonical basis of the homology H 1 pΣ, Zq on the surface Σ. Let pΣ, Jq be a Riemann surface structure on Σ and denote by H 0 J pΣ, K m q the space of holomorphic sections of the mth power of the canonical bundle K :" T C Σ 1,0 of pΣ, Jq, for m ě 1. It is well-known that there exists a unique basis tζ i u of holomorphic Abelian differentials in H 0 J pΣ, Kq such that ż

a j ζ k " δ jk ,
see [FK92, Proposition, page 63] for instance. The period matrix of pΣ, Jq is then defined as the matrix ΠpJq whose jk-entry is

ΠpJq jk " ż b j ζ k .
It is a symmetric matrix with positive definite imaginary part. The space of symmetric matrices with positive definite imaginary part and size given by the genus of Σ is called the Siegel upper half-space HpΣq. Hence, we get a well-defined period matrix map Π : T pΣq ÝÑ HpΣq.

We will need the Torelli theorem in the following form:

Theorem 2.1. Assume that Σ has genus ě 2. If ΠpJ 1 q " ΠpJ 2 q, then there exists an orientation-preserving diffeomorphism ψ : Σ Ñ Σ such that ψ ˚J2 " J 1 .

We refer to [FK92, Theorem III.12.3] for a proof. Actually, it can be proved that rψs P MCGpΣq lives in a degree 2 extension of the Torelli group that is, it acts as ˘1 on homology H 1 pΣ, Zq but this will not be needed in what follows.

2.2. Unit tangent bundle of the surface. Let SΣ :" tpx, vq P T Σ | |v| g " 1u be the unit tangent bundle of pΣ, gq and π : SΣ Ñ Σ the projection.

2.2.1. Geometry of SΣ. Let pϕ t q tPR be the geodesic flow on SΣ and X its infinitesimal generator. Let V be the vertical vector field generating the SOp2q-rotation group pR θ q θPr0,2πs in the fibers and let V :" RV . Define H :" ´rX, V s and H :" RH. The vector fields tX, H, V u form an orthonormal basis on SM for the Sasaki metric (the natural lift of g to SΣ). We also define (2.1) η ˘:" 1 2 pX ¯iHq.

These operators are called the raising (`) and lowering (´) operators.

The Liouville 1-form λ P C 8 pSΣ, T ˚pSΣqq is defined by λpXq " 1 and λpHq " λpV q " 0. It is invariant by the geodesic flow, that is, L X λ " 0. Moreover, dλ is a 2-form that is non-degenerate on the contact plane RH ' RV and such that ι X dλ " 0. Hence µ :" ´λ ^dλ is a volume-form, invariant by the geodesic flow, called the Liouville volume form. Equivalently, µ is the Riemannian volume form induced by the Sasaki metric on SΣ. From now on, the L 2 space on SΣ is defined as L 2 pSΣq :" L 2 pSΣ, µq.

We define the 1-forms β, ψ on SΣ by βpHq " 1 " ψpV q and βpXq " βpV q " 0 " ψpXq " ψpHq. It can then be checked that (2.2) dλ " ψ ^β, µ " λ ^β ^ψ.

We set pE 0 q ˚:" Rλ, H ˚:" Rβ and V ˚:" Rψ. We refer to [START_REF] Gabriel | Geodesic flows[END_REF] and [PSU23, Chapter 3] for further details on the geometric structure on SΣ, see also Figure 1 below for a representation of the bundles introduced above.

2.2.2. Powers of the canonical line bundle. We introduce the complex line bundle Ω 1 Ñ Σ whose fiber over x P Σ is given by:

pΩ 1 q x :" tupx, ¨q | u P C 8 pSΣq, V u " iuu .
The line bundle Ω 1 is isomorphic to the canonical line bundle K of the underlying Riemann surface pΣ, Jq, that is, there exists a fiberwise linear map π 1 : K Ñ Ω 1 given for all x P Σ by

π 1 : K x Q f Þ Ñ pS x Σ Q v Þ Ñ f pvqq P pΩ 1 q x .
The powers Ω n :" Ω bn 1 for n P Z, correspond to pΩ n q x " tupx, ¨q | u P C 8 pSΣq, V u " inuu and π n :

K bn x Q f Þ Ñ pS x Σ Q v Þ Ñ f pv, .
.., vqq P pΩ n q x is an isomorphism. For n ě 0 (resp. n ď 0), it can be checked that π n : K bn Ñ Ω n intertwines the operators B and η ´(resp. B and η `), see [PSU14, Lemma 2.1] and the discussion below. Hence, from now on, we will freely identify K bn with Ω n via π n. We denote its adjoint by π n˚( with respect to the natural inner products on K bn and Ω n ). Note that p2πq ´1π n˚π n is the identity on C 8 pSΣ, K bn q and (2.3)

1 2π π n ˚πn˚: L 2 pSΣq Ñ L 2 pSΣq
is the L 2 -orthogonal projection onto the n-th Fourier mode. Any function f P L 2 pSΣq can thus be decomposed as f " ř nPZ f n , where f n :" p2πq ´1π n ˚πn˚f . Define

(2.4) H 0 pΣ, Ω n q :" tu P C 8 pSΣq, V u " inu, η ´u " 0u .

We shall denote by H 0 J pΣ, K bn q the complex vector space of holomorphic differentials of degree n. The subscript J indicates that this is computed with respect to the complex structure J. Observe that by the previous discussion, π n identifies H 0 J pΣ, K bn q with H 0 pΣ, Ω n q for n ě 0. By the Riemann-Roch theorem, the following holds true: Lemma 2.2. As complex vector spaces, the dimensions of ker η ˘|C 8 pΣ,Ωnq are given by the following: ' Kernels of η `: For n ď ´2, dim ker η `" ´p2n `1qpg ´1q; for n " ´1, dim ker η `" g; for n " 0, dim ker η `" 1; for n ě 1, η `is injective. ' Kernels of η ´: For n ě 2, dim ker η ´" p2n ´1qpg ´1q; for n " 1, dim ker η ´" g; for n " 0, dim ker η ´" 1; for n ď ´1, η ´is injective.

We refer to [PSU14, Lemma 2.1] for a proof. We let K :" ' ně0 K bn . Observe that H 0 J pΣ, Kq :" ' ně0 H 0 J pΣ, K bn q is naturally a graded Abelian unital algebra (the constant function 1 Σ is the unit) and that there is a natural algebra homomorphism

H 0 J pΣ, Kq Ñ C 8 pSΣq.
Here, sections of H 0 J pΣ, Kq are understood in the algebraic sense, that is, they correspond to finite sums of holomorphic sections. 2.2.3. Fiberwise holomorphic distributions. Notion of degree. Denote by D 1 pSΣq the space of distributions on SΣ seen as the topological dual of volume forms, that is:

D 1 pSΣq :" pC 8 pSΣ, Λ 3 T ˚SΣqq 1 .
Elements in D 1 pSΣq are "generalized" functions which can be paired intrinsically against smooth volume forms. The wavefront set WFpf q of a distribution f P D 1 pSΣq describes the (co)directions in T ˚pSΣq in which the distribution is irregular, see [H 03, Chapter 8] for a detailed account.

The vector field X acts on D 1 pSΣq by duality, namely, given u P D 1 pSΣq and ω P C 8 pSΣ, Λ 3 T ˚SΣq, pXu, ωq :" ´pu, L X ωq. As for L 2 -functions, any f P D 1 pSΣq can be decomposed as a sum of Fourier modes

(2.5) f " ÿ nPZ f n ,
where f n P D 1 pSΣq satisfies V f n " inf n and f n :" 1 2π π n ˚πn˚f . A distribution (or a function) is said to have finite Fourier degree if the sum (2.5) only contains a finite number of terms. It is said to be fiberwise holomorphic (resp. antiholomorphic) if (2.5) only contains f n terms for n ě 0 (resp. non-positive). Equivalently, defining the Szegö projections S ˘: D 1 pSΣq Ñ D 1 pSΣq by (2.6) S

`˜ÿ nPZ

f n ¸:" ÿ ně0 f n , S ´˜ÿ nPZ f n ¸:" ÿ nď0 f n ,
a distribution f is fiberwise holomorphic iff S `f " f and antiholomorphic iff S ´f " f . A distribution is even (resp. odd ) if its decomposition (2.5) only contains even (resp. odd) terms. The operators η ˘act as raising/lowering operators on the Fourier decomposition (2.5), that is:

η ˘: C 8 pΣ, Ω n q Ñ C 8 pΣ, Ω n˘1 q
is continuous. Hence, the operator X " η ´`η `acts on the decomposition (2.5) as

X ˜ÿ nPZ f n ¸" ÿ nPZ η `fn´1 `η´fn`1 . Let (2.7) A ˘pSΣq :" f P D 1 pSΣq | Xf " 0, S ˘f " f ( ,
be the algebra of flow-invariant fiberwise holomorphic (resp. antiholomorphic) distributions. It can be checked that A ˘pSΣq is indeed an algebra, regardless of the underlying Riemannian geometry of the surface, that is, multiplication is well-defined and continuous with respect to the D 1 pSΣq topology, see [BLP23, Theorem 1.1]. Nevertheless, the algebra property will not be needed in the proof of Theorem 1.1.

Definition 2.3. We define deg : A `pSΣq Ñ Z ě0 and σ : A `pSΣq Ñ H 0 J pΣ, Kq as follows: given f " ř ně0 f n P A `pSΣq, σpf q P H 0 J pΣ, Kq is the first non-zero Fourier mode f n 0 and degpf q :" n 0 .

Similarly, a nonpositive degree map deg : A ´pSΣq Ñ Z ď0 can be defined for fiberwise antiholomorphic invariant distributions. The function deg on A `pSΣq is a valuation map (although, we do not have a field but rather a ring, which is why the valuation takes values in Z ě0 , not Z). We have the obvious properties:

(2.8) degpf `gq ě minpdegpf q, degpgqq, degpf gq " degpf q `degpgq.

In the following, we will denote by A ěn pSΣq the fiberwise holomorphic invariant distributions of degree ě n.

2.3. Hyperbolic dynamics. We now further assume that pΣ, gq is an Anosov metric, that is, the geodesic flow on SΣ is Anosov (or uniformly hyperbolic).

2.3.1. Definition. First properties. Recall that the geodesic flow pϕ t q tPR is Anosov if there exists a flow-invariant continuous splitting

T pSΣq " RX ' E s ' E u
and uniform constants C, λ ą 0 such that (2.9) |dϕ t pwq| ď Ce ´λt |w|, @t ě 0, @w P E s , |dϕ ´tpwq| ď Ce ´λt |w|, @t ě 0, @w P E u .

By [START_REF] Klingenberg | Riemannian manifolds with geodesic flow of Anosov type[END_REF] (see also [START_REF] Mañé | On a theorem of Klingenberg[END_REF] for an alternative proof), the bundles E s and E u are known to intersect the vertical bundle V trivially, that is

(2.10) E s X V " E u X V " t0u.
Moreover, there exists C 2´p SΣq functions r ˘solutions to the Ricatti equations Xr ˘`r 2 ˘`π ˚κ " 0, where κ denotes the Gauss curvature on pΣ, gq, such that

E s " RpH `r´V q, E u " RpH `r`V q.
If κ ă 0, then r ´ă 0 while r `ą 0. We set Y s :" H `r´V , Y u :" H `r`V . Note that the basis tX, Y s , Y u u is positively-oriented.

The dual bundles are defined by pE 0 q ˚pE s ' E u q " 0 " pE s q ˚pE 0 ' E s q " pE u q ˚pE 0 ' E u q, where E 0 :" RX. It can be checked that pE s q ˚(resp. pE u q ˚) satisfies similar estimates (2.9) to E s (resp. E u ) with dϕ t being replaced by dϕ ´J t (inverse transpose). Equation (2.10) becomes in the cotangent bundle (2.11)

pE s q ˚X H ˚" pE u q ˚X H ˚" t0u. H V Y u Y s X C β λ E s E u (E u ) * (E s ) * H H * V V * B Figure 1.
The various subbundles in tangent and cotangent space. Theorem 2.4 (Tensor tomography). Assume that pΣ, gq is Anosov. If Xu " f with f, u P C 8 pSΣq and f has degree n, then u has degree maxpn ´1, 0q, and u " 0 if f has degree 0.

Theorem 2.4 was first proved in negative curvature in [START_REF] Guillemin | Some inverse spectral results for negatively curved 2-manifolds[END_REF] and generalized to the Anosov setting in [START_REF] Dairbekov | Some problems of integral geometry on Anosov manifolds[END_REF] for n " 0, 1, [START_REF] Gabriel | Spectral rigidity and invariant distributions on Anosov surfaces[END_REF] for n " 2, and [START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF] for n ě 3. Note that Theorem 2.4 is equivalent to the fact that the X-ray transform operator I g n : C 8 pΣ, S n T ˚Σq Ñ 8 pCq defined by

I g n hpcq :" 1 L g pcq ż Lgpcq 0 h γptq p 9
γptq, ..., 9 γptqq dt is injective when restricted to divergence-free symmetric n-tensors, see [Lef, Chapter 2] for instance. For n " 2, this operator is the linearization (at g) of the marked length spectrum operator g Þ Ñ L g .

Proofs

In this section, we first prove that L g determines the class rJ g s P MpΣq in the moduli space. Note that all the objects defined above (complex structures, stable and unstable bundles, etc.) now depend on the metric g 1 or g 2 and we shall add a subscript to distinguish them. The following is key to Theorem 1.1: Proposition 3.1. Let g 1 , g 2 be two smooth Anosov metrics on Σ. If L g 1 " L g 2 then rJ 1 s " rJ 2 s in MpΣq. Equivalently, there exists an orientationpreserving diffeomorphism ψ : Σ Ñ Σ such that ψ ˚J2 " J 1 and ψ ˚g2 " e 2f g 1 for some f P C 8 pΣq.

This should be compared with the case of a surface with boundary, see [START_REF] Pestov | Two dimensional compact simple Riemannian manifolds are boundary distance rigid[END_REF][START_REF] Guillarmou | Lens rigidity for manifolds with hyperbolic trapped sets[END_REF]. The proof of Proposition 3.1 is the content of §3.1 and §3.2. We then prove in Proposition 3.10 that we can recover from L g the class rJ g s P T pΣq in Teichmüller space, and then the conformal factor of the metric (the latter is a standard argument due to Katok [START_REF] Katok | Four applications of conformal equivalence to geometry and dynamics[END_REF]).

3.1. The algebra isomorphism. Consider an algebra homomorphism Φ : A `pSΣ 1 q ÝÑ A `pSΣ 2 q with A `pSΣ i q defined by (2.7). We say that it is proper if Φp1 SΣ 1 q " 1 SΣ 2 (the constants are mapped to the constants) and if, when f P A `pSΣ 1 q and degpf q ě 1, then degpΦpf qq ě 1. Equivalently, Φ : A ě1 pSΣ 1 q ÝÑ A ě1 pSΣ 2 q is a homomorphism. This definition also naturally extends for algebra homomorphisms Φ : A `pSΣ 1 q ÝÑ A ´pSΣ 2 q (in which case deg ě 1 is mapped isomorphically to deg ď ´1).

There is a natural identification of SΣ 1 with SΣ 2 by simply scaling the fibers via the map

s : SΣ 1 ÝÑ SΣ 2 , v Þ Ñ v{|v| g 2 .
The following holds:

Proposition 3.2. Assume that L g 1 " L g 2 . Then there exists a smooth diffeomorphism φ : SΣ 2 Ñ SΣ 1 such that: (i) φ ˝ϕg 2 t " ϕ g 1 t ˝φ, for all t P R, (ii) s ˝φ : SΣ 2 Ñ SΣ 2 is isotopic to the identity, (iii) φ induces a proper algebra isomorphism: φ ˚: A `pSΣ 1 q ÝÑ A ˘pSΣ 2 q, by pullback of distributions.

At this stage, it is not yet clear that φ ˚maps fiberwise holomorphic invariant distributions of the first metric to those of the second, and it could happen that φ ˚interchanges fibrewise holomorphic with antiholomorphic. We will refer to these cases as the p˘q cases.

Remark 3.3. Actually, one can show that if g 1 and g 2 are connected by a path of metrics that are all Anosov, then φ ˚: A `pSΣ 1 q ÝÑ A `pSΣ 2 q preserves fiberwise holomorphic invariant distributions. This is based on the fact that weak unstable leaves admit a natural orientation and that such a φ would have to preserve this orientation. However, path-connectedness on the space of Anosov metrics on surfaces is an open question. Note that for negatively curved metrics, path-connectedness follows from the uniformization theorem and a straightforward computation (together with the fact that Teichmüller space is path-connected).

We introduce C Ă tpv, ξq P T ˚pSΣq | ξpXpvqq " 0u, the closed cone enclosed by pE s q ˚and pE u q ˚in the half-space tξpV pvqq ě 0u, see Figure 1 for a description. Lemma 3.4. Let f P A ˘pSΣq. Then WFpf q Ă ˘C and π n˚f P C 8 pΣ, K bn q for all n P Z.

Proof. We only treat the p`q case as the p´q one is similar. Observe that f P A `pSΣq is fiberwise holomorphic, i.e. S `f " f so (3.1)

WFpf q " WFpS `f q Ă tpv, ξq P T ˚pSΣq | ξpV pvqq ě 0u , by using the wavefront set description of the Schwartz kernel of S `, see [Gui17a, Lemma 3.10]. Moreover, Xf " 0 so by elliptic regularity one has WFpf q Ă tpv, ξq P T ˚pSΣq | ξpXpvqq " 0u , while by standard propagation of singularities (see [Hör09, Theorem 26.1.1]) WFpf q is invariant by the symplectic lift of pϕ t q tPR . But the maximal flowinvariant subset of T ˚pSΣq contained in tξpV pvqq ě 0, ξpXpvqq " 0u is C, so this proves the claim. Finally, π n˚f P C 8 pΣ, K bn q as the pushforward operator π n˚o nly selects the wavefront set of f in pE 0 q ˚' H ˚(see [START_REF] Friedlander | Introduction to the theory of distributions[END_REF]Proposition 11.3.3] for instance) which is empty as C XpE 0 q ˚'H ˚" t0u.

We now prove Proposition 3.2.

Proof of Proposition 3.2. Two Anosov geodesic flows are orbit equivalent via a Hölder homeomorphism isotopic to the identity (see [START_REF] Ghys | Flots d'Anosov sur les 3-variétés fibrées en cercle[END_REF][START_REF] Gromov | Three remarks on geodesic dynamics and fundamental group[END_REF] and [GKL22, Lemma B.1]). The equality of the marked length spectra L g 1 " L g 2 and the Livšic theorem imply that such orbit equivalence can be upgraded to a conjugacy φ. Moreover, such a conjugacy is necessarily smooth for threedimensional contact Anosov flows: the C 1 -regularity follows from [START_REF] Feldman | Semirigidity of horocycle flows over compact surfaces of variable negative curvature[END_REF] and the C 8 regularity from [START_REF] De | Invariants for smooth conjugacy of hyperbolic dynamical systems[END_REF] (see also [START_REF] Gogolev | Smooth rigidity for 3dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity[END_REF] for a more general smoothness result on conjugacies of three-dimensional volume preserving Anosov flows).

As φ maps E s 2 (resp. E u 2 ) to E s 1 (resp. E u 1 ), we get that ker dλ 2 " E s 2 ' E u 2 is mapped by φ to ker dλ 1 . Hence, φ ˚λ1 " λ 2 and φ ˚p´λ 1 ^dλ 1 q " ´λ2 ^dλ 2 , that is, φ preserves the Liouville volume form hence the orientation. As φ preserves the orientation, Σ is connected, and dφ J pλ 1 q " λ 2 , we deduce that C 1 :" dφ J pC 1 q " ˘C2 . Indeed, since dφ J maps connected sets to connected sets, pE u 1 q ˚to pE u 2 q ˚and pE s 1 q ˚to pE s 2 q ˚, dφ J pC 1 q must be one of the 4 cones in the right Figure 1 (inside the subspace tξpX 2 pvqq " 0u). Now, if C 1 were mapped by φ to one of the other two cones on Figure 1, φ would reverse the orientation, which is absurd.

We now claim that if C 1 " `C2 , then φ ˚: A `pSΣ 1 q Ñ A `pSΣ 2 q is a proper algebra homomorphism, while if C 1 " ´C2 , then φ ˚: A `pSΣ 1 q Ñ A ´pSΣ 2 q is a proper algebra homomorphism. Let f P A `pSΣ 1 q and define f 1 :" φ ˚f P D 1 pSΣ 2 q. By Lemma 3.4, the wavefront set of f is contained in C 1 so the wavefront set of f 1 is contained in the cone C 1 " dφ J pC 1 q " ˘C2 .

The proof being similar in the p´q case, we will now assume that C 1 " `C2 . Then S ´f 1 P C 8 pSΣ 2 q and, since X 2 f 1 " φ ˚pX 1 f q " 0, we also have that X 2 pS ´f 1 q " η `f 1 0 `η`f 1 ´1. By the tensor tomography Theorem 2.4, we deduce that S ´f 1 is of degree 0. Hence f 1 is fiberwise holomorphic. It then remains to show that φ ˚: A `pSΣ 1 q ÝÑ A `pSΣ 2 q is an algebra isomorphism but this is straightforward as it admits an inverse pφ ´1q ˚(and the product property is obviously satisfied). We now show that φ ˚is proper. Obviously, φ ˚p1 SΣ 1 q " 1 SΣ 2 . Now, assume that f P A `pSΣ 1 q has degpf q ě 1 and consider f 1 :" φ ˚f . The term of degree 0 of f 1 " ř kě0 f 1 k is a constant which can be computed as follows:

f 1 0 ˆvolpSΣ 2 q " ż SΣ 2 f 1 µ 2 " ż SΣ 2 φ ˚pf µ 1 q " ż SΣ 1 f µ 1 " f 0 ˆvolpSΣ 1 q " 0.
This proves the claim. The p´q case is treated similarly.

3.2. Extension operator. Period preservation. The aim of this paragraph is to show that, in some sense, the algebra isomorphism φ ˚maps holomorphic differentials of the first surface to the second, and that it preserves periods. We first claim that the map

(3.2) π 1˚: A ě1 pSΣq ÝÑ H J pΣ, Kq
is well-defined. Indeed, a fiberwise holomorphic invariant distribution f P A ě1 pSΣq satisfies Xf " 0 so its first Fourier mode f 1 satisfies η ´f1 " 0 which, in turn, is equivalent to Bπ 1˚f " 0 by §2.2.2.

A key input that we now use is that the map (3.2) is known to be surjective by [START_REF] Gabriel | Spectral rigidity and invariant distributions on Anosov surfaces[END_REF]Theorem 1.5]. This allows to define a right-inverse e 1 such that e 1 : H 0 J pΣ, Kq ÝÑ A ě1 pSΣq, π 1˚˝e1 " 1 H 0 J pΣ,Kq , which we call the extension operator. Remark 3.5. A similar right-inverse e ´1 can also be defined for π ´1˚: A ď´1 pSΣq ÝÑ H J pΣ, K ´1q. More generally, using the spectral theory of Anosov flows, it was proved in [Gui17a, Corollary 3.8] that the map π n˚: A ěn pSΣq ÝÑ H J pΣ, K n q is surjective for all n ě 0 (along with other more general results on the existence of flow-invariant distributions with prescribed n-th Fourier mode). However, this will not be needed in the rest of the paper.

Let

F : H 0 J 1 pΣ, Kq ÝÑ H 0 ˘J2 pΣ, Kq be the map defined by one of the two following commutative diagrams: (i) In the p`q case,

A ě1 pSΣ 1 q φ ˚/ / A ě1 pSΣ 2 q π 1˚ H 0 J 1 pΣ, Kq e 1 O O F / / H 0 J 2 pΣ, Kq (ii
) In the p´q case,

A ě1 pSΣ 1 q φ ˚/ / A ď´1 pSΣ 2 q π ´1˚ H 0 J 1 pΣ, Kq e 1 O O F / / H 0 ´J2 pΣ, Kq
Note that, in the p´q case, if f P A ´pSΣ 2 q, then π ´1˚f P C 8 pΣ, K ´1 J 2 q is a p0, 1q-form for the complex structure J 2 satisfying B J 2 f " 0, and it can be naturally identified with a p1, 0q-form f P C 8 pΣ, K ´J2 q for the complex structure ´J2 satisfying B´J 2 f " 0.

The following result unlocks Theorem 1.1:

Proposition 3.6. The map

F : H 0 J 1 pΣ, Kq ÝÑ H 0 ˘J2 pΣ, Kq is a period-preserving C-linear isomorphism in the following sense: for all rγs P H 1 pΣ, Zq, ω P H 0 J 1 pΣ, Kq, (3.3) ż rγs ω " ż rγs F ω.
That F is an isomorphism follows immediately from the period preservation (3.3). Hence, the only non-trivial part of Proposition 3.6 is (3.3). Here and below, given a homology class rγs P H 1 pΣ, Zq, we shall denote by γ g i a closed oriented geodesic (for the metric g i , i " 1, 2) representing rγs, and γ, when no preferred metric is chosen. We then set S 1 γ " π ´1pγq Ă SΣ to be the circle bundle over γ. If r0, g pγqs Q τ Þ Ñ γpτ q is an arc-length parametrization of γ, we can parametrize S 1 γ by coordinates pτ, θq P R{p g pγqZq ˆR{p2πZq such that: S 1 γ " tpγpτ q, R θ 9 γpτ qq | τ P r0, g pγqs, θ P r0, 2πsu

where R θ denotes rotation by angle θ in SΣ. Note that the tangent space of S 1 γ at the point pτ, θq is spanned by ppcos θqX `psin θqH, V q. We then endow S 1 γ with the orientation given by pcos θ λ `sin θ βq ^ψ. From now on, S 1 γ will always denote the oriented submanifold of SΣ with orientation given by the 2-form above. The following holds:

Lemma 3.7. In the coordinate system pτ, θq, dλ " sin θ dθ ^dτ .

Note that for θ P p0, πq, this is a negative multiple of the 2-form defining the orientation on S 1 γ.

Proof. This follows from (2.2), see also [START_REF] Otal | Le spectre marqué des longueurs des surfaces à courbure négative[END_REF]p. 156].

We next derive the following remarkable formula: Lemma 3.8. Let f P A ˘pSΣq and γ be a closed oriented geodesic. Then the pairing pS 1 γ, f dλq is well-defined and

(3.4) ˘2i ż S 1 γ f dλ " ż rγs π ˘1˚f ,
where rγs P H 1 pΣ, Zq denotes the homology class of the geodesic γ.

Actually, if f P C 8 pSΣq (or if f is merely a distribution but the pairing with S 1 γ makes sense), we will prove the formula

(3.5) i π ż S 1 γ f dλ " ż g pγq 0 f 1 pγpτ q, 9 γpτ qqdτ ´ż g pγq 0 f ´1pγpτ q, 9 γpτ qqdτ,
where τ Þ Ñ γpτ q is an arc-length parametrization of the geodesic. It then suffices to observe that (3.4) is a mere rewriting of (3.5) since for f P A ˘pSΣq, π ˘1˚f is an (anti-)holomorphic differential (and the factor 2π comes from (2.3)).

Remark 3.9 (Generalized intersection number). The pairing pS 1 γ, f dλq should be thought of as a generalized intersection number, following the terminology used by Bonahon [START_REF] Bonahon | The geometry of Teichmüller space via geodesic currents[END_REF], Otal [START_REF] Otal | Le spectre marqué des longueurs des surfaces à courbure négative[END_REF] and others. The difference is that we allow f to be any fiberwise holomorphic invariant distribution in A `pSΣq whereas, in some sense, the aforementioned authors only allowed f to be a constant, which is a (very) particular case of a fiberwise holomorphic distribution. Moreover, we integrate over the full circle bundle S 1 γ whereas these authors integrate over S 1 γ `, that is, for angles θ P p0, πq. The usual intersection number of the Liouville current η (that is, the Liouville form modded out by the flow direction in the universal cover) with the geodesic γ is commonly denoted by ιpη, γq and is equal to 2 g pγq (or g pγq, depending on the convention). It simply corresponds for us to pS 1 γ `, dλq.

Proof of Lemma 3.8. By the wavefront set calculus (see [H 03, Corollary 8.2.7] for instance), the pairing pS 1 γ, f dλq is well-defined as long as

(3.6) N ˚pS 1 γq X WFpf dλq " H.
Now, the conormal N ˚pS 1 γq to S 1 γ is a line contained in pE 0 q ˚' H ˚and by Lemma 3.4 and (2.11), the intersection (3.6) is indeed empty. Hence, the pairing pS 1 γ, f dλq is well-defined and extends the pairing computed for f P C 8 pSΣq. As a consequence, it suffices to establish (3.5) for smooth functions and (3.4) then follows immediately. Now, if f P C 8 pSΣq, we can decompose f in Fourier modes along S 1 γ and write (3.7) f pτ, θq " ÿ kPZ a k pτ qe ikθ .

Then, using (3.7) and sin θ " p2iq ´1pe iθ ´e´iθ q, we get: Note that the minus on the first line comes from the fact that dλ is a negative multiple of the 2-form defining the orientation on S 1 γ. The previous equality corresponds exactly to (3.5).

If ω P H 0 J pΣ, Kq, rγs P H 1 pΣ, Zq and γ is any closed geodesic representative for the metric g whose homology class is rγs, Lemma 3.8 yields

(3.8) 2i ż S 1 γ e 1 ω dλ " ż rγs ω.
We can now complete the proof of Proposition 3.6.

Proof of Proposition 3.6. We first treat the p`q case. Let rγs P H 1 pΣ, Zq and let γ g 1 pcq, γ g 2 pcq be two geodesic representatives for rγs (with respect to g 1 and g 2 ) in the same free homotopy class c P C. Since φ is isotopic to the identity by Proposition 3.2, we get that rφpS 1 γ g 2 pcqqs " rS 1 γ g 1 pcqs in H 2 pSΣ 1 , Zq.

We also claim that the pairing pφpS 1 γ g 2 pcqq, e 1 ω dλ 1 q is well-defined, similarly to Lemma 3.8. Indeed, the tangent space to S 1 γ g 2 pcq does not intersect the closed cone B 2 Ă tpv, wq P T pSΣ 2 q | λ 2 pwq " 0u enclosed by E s 2 and E u 2 , and containing H 2 , see Figure 1. Since φ preserves the orientation, following the same arguments as in Proposition 3.2, the tangent space to φpS 1 γ g 2 pcqq is also a 2-plane avoiding the closed cone B 1 and thus its conormal avoids the closed cone C 1 which contains WFpe 1 ω dλ 1 q by Lemma 3.4, so the pairing is well-defined.

Moreover, given a fiberwise holomorphic invariant distribution f P A `pSΣ 1 q, observe that f dλ is a closed 2-form. We claim that ż

S 1 γg 1 pcq e 1 ω dλ 1 " ż φpS 1 γg 2 pcqq e 1 ω dλ 1 .
Indeed, choose the harmonic representative α P H 2 pSΣ 1 q of f dλ that is, such that rαs " rf dλs in H 2 pSΣ 1 , Cq. By the Hodge decomposition Theorem, we have α " f dλ `du, where u P D 1 pSΣ 1 q and WFpuq " WFpf q. Hence, both pairings pS 1 γ g 1 pcq, duq and pφpS 1 γ g 2 pcqq, duq are well-defined by the wavefront set calculus (same arguments as for f dλ) and equal to 0 since du is exact.

Then, we get:

ż S 1 γg 1 pcq e 1 ω dλ 1 " ż S 1 γg 1 pcq α " ż φpS 1 γg 2 pcqq α " ż φpS 1 γg 2 pcqq e 1 ω dλ 1 ,
where the second equality simply follows from the fact that α is harmonic and rS 1 γ g 1 pcqs " rφpS 1 γ g 2 pcqqs in H 2 pSΣ 1 , Zq.

As a consequence, given ω P H 0 J 1 pΣ 1 , Kq we get by (3.8) and the discussion above that ż

rγs ω " 2i ż S 1 γg 1 pcq e 1 ω dλ 1 " 2i ż φpS 1 γg 2 pcqq e 1 ω dλ 1 " 2i ż S 1 γg 2 pcq φ ˚pe 1 ω dλ 1 q " 2i ż S 1 γg 2 pcq φ ˚pe 1 ωq dλ 2 " ż rγs π 1˚φ ˚pe 1 ωq " ż rγs F ω.
This concludes the proof in the p`q case. In the p´q case, it should be observed that φ actually flips the orientation of rS 1 γg 2 pcqs, that is rφpS 1 γg 2 pcqqs " ´rS 1 γ g 1 pcqs in H 2 pSΣ 1 , Zq, which gives: We can now conclude the proof of Proposition 3.1.

ż rγs ω " 2i ż S 1 γg 1 pcq e 1 ω dλ 1 " ´2i ż φpS 1 γg 2 pcqq e 1 ω dλ 1 " ´2i ż S 1 γg 2 pcq φ ˚pe 1 ω dλ 1 q " ´2i
Proof of Proposition 3.1. If L g 1 " L g 2 , we get by the above results that F : H 0 J 1 pΣ, Kq ÝÑ H 0 ˘J2 pΣ, Kq is a period-preserving C-linear complex isomorphism. Hence, by the Torelli Theorem 2.1, we get that pΣ, J 1 q and pΣ, ˘J2 q have same period matrix, so they are biholomorphic and differ in the Teichmüller space by an element rψs P MCGpΣq. Note that in the p´q case, rψs would reverse the orientation which is absurd as rψs P MCGpΣq, hence only the p`q case survives which is the content of Proposition 3.1.

3.3.

End of the proof. We now prove that L g determines the class of rJ g s P T pΣq in the Teichmüller space of Σ. Proposition 3.10. Let g 1 , g 2 be two smooth Anosov metrics. If L g 1 " L g 2 then rJ g 1 s " rJ g 2 s in T pΣq. Equivalently, there exists a diffeomorphism ψ : Σ Ñ Σ isotopic to the identity such that ψ ˚g2 " e 2f g 1 for some f P C 8 pΣq. Proposition 3.10 will follow from Proposition 3.1 and the following: Lemma 3.11. Let J 1 and J 2 be two complex structures on Σ compatible with orientation such that rJ 1 s ‰ rJ 2 s in T pΣq. Then, there exists a finite cover Σ 1 Ñ Σ such that the lifts rJ 1 1 s, rJ 1 2 s P T pΣ 1 q are not in the same MCGpΣ 1 qorbit.

Let us first derive Proposition 3.10 from that.

Proof of Proposition 3.10. If L g 1 " L g 2 on Σ, then the same equality L g 1 1 " L g 1 2 holds on any finite cover Σ 1 Ñ Σ. Thus, the desired conclusion follows by combining directly Proposition 3.1 with Lemma 3.11.

The proof of Lemma 3.11 was kindly explained to us by M. Wolff.

Proof of Lemma 3.11. In order to simplify notation, given a free homotopy class c P C we write g pcq for the length of the unique g-geodesic in the free homotopy class c. We use the identification of Teichmüller space T pΣq with marked hyperbolic structures on Σ. Let rh 1 s, rh 2 s P T pΣq be the (distinct) hyperbolic structures given by J 1 and J 2 . Let α P CpΣq be a free homotopy class of simple closed curves on Σ such that h 1 pαq ‰ h 2 pαq. Note that such a class always exists otherwise we would have rh 1 s " rh 2 s by the 9g ´9 Theorem (see [START_REF] Farb | A primer on mapping class groups[END_REF]Theorem 10.7]).

We then consider a finite cover Σ 1 Ñ Σ which unfolds every simple closed curve, except α. More precisely, for K ą 0 fixed, we consider a finite cover π : Σ 1 Ñ Σ such that the following holds: denoting by h 1 1 , h 1 2 the lifts of h 1 , h 2 , there exists a free homotopy class of simple closed curves α 1 P CpΣ 1 q on Σ 1 such that π : γ h 1 1 pα 1 q Ñ γ h 1 pαq is 1-to-1, and for every free homotopy class of simple closed curves β P CpΣ 1 q such that β ‰ α 1 , h 1 1 pβq ą K (see [ABMP22, Appendix A, Theorem C] for the existence of such a covering). In particular, choosing K ą 0 such that K ą h 1 pαq, we can guarantee that γ h 1 1 pα 1 q is the systole of pΣ 1 , h 1 1 q with length h 1 pαq. Now, as Σ is compact, there exists a constant C ą 1 such that for all c P C (not necessarily simple), 1{C ď h 1 pcq h 2 pcq ď C.

We fix K ą 0 in the construction above such that K{C ą maxp h 1 pαq, h 2 pαqq. We claim that γ h 1 2 pα 1 q is also the systole of pΣ 1 , h 1 2 q with length h 2 pαq. Indeed, observe first that π : γ h 1 2 pα 1 q Ñ γ h 2 pαq is also 1-to-1 (so h 1 2 pα 1 q " h 2 pαq), and for every free homotopy class of simple closed curves β P CpΣ 1 q, β ‰ α 1 , one has:

h 1 2 pβq " h 1 1 pβq ¨ h 1 2 pβq h 1 1 pβq " h 1 1 pβq ¨
h 2 pπpβqq h 1 pπpβqq ą K{C ą maxp h 1 pαq, h 2 pαqq ě h 1 2 pα 1 q. As a consequence, the systoles of pΣ 1 , h 1 1 q and pΣ 1 , h 1 2 q have different lengths, so rh 1 1 s and rh 1 2 s are not in the same MCGpΣ 1 q-orbit. Finally, we complete the proof of Theorem 1.1.

Proof of Theorem 1.1. If L g 1 " L g 2 , Proposition 3.10 yields that there exists ψ : Σ Ñ Σ isotopic to the identity such that ψ ˚g1 " e 2f g 2 . But then, applying Katok's argument [START_REF] Katok | Four applications of conformal equivalence to geometry and dynamics[END_REF] for conformal metrics with same marked length spectrum, we get that f " 0. This concludes the proof.

We conclude with a remark on Theorem 1.1 for metrics with low regularity: Remark 3.12. Theorem 1.1 still holds for metrics with C 4 -regularity. Indeed, if g 1 and g 2 are C k -regular, the unit tangent bundles SΣ 1,2 are also C kregular and the geodesic vector fields X 1,2 are C k´1 . The conjugacy φ in Proposition 3.2 is then C k´1 by [GR22, Theorem 1.1, item 1]. By [PSU14, Theorem 5.5], the right-inverse e 1 defined in §3.2 can be constructed with values in H ´1pSΣq flow-invariant distributions, and the proof of [PSU14, Theorem 5.5], based on the Pestov identity, applies with metrics which are only C 2 -regular.

In order to prove that φ ˚maps A `pSΣ 1 q to A ˘pSΣ 2 q in Proposition 3.2, one can use the notion of H s wave-front set WF s of distributions (for s ě ´1) in the Sobolev space H ´1pSΣ 1,2 q, see [Hör97, Section 8.2] for a definition. Indeed, the action by pullback by φ ˚of distributions in H ´1pSΣ 1 q is welldefined and maps to distributions with regularity H ´1pSΣ 2 q if φ is at least C 2 , using that C 2 embeds into H 1 , the dual of H ´1. For u P H ´1pSΣ 1 q, the pull-back φ ˚u P H ´1pSΣ 2 q is such that dφ J sends WF s puq to WF s pφ ˚uq if φ is C k´1 and ´1 ď s ď k ´2. The tensor tomography result of Theorem 2.4 applies to functions and 1-forms in H 2 (and for C 2 -regular metrics in order to apply [PSU14, Theorem 5.5] based on the Pestov identity). Hence, we need to work with the WF s puq wave-front set with s ě 2 and this forces k ě s `2 ě 4.

Finally, in the pairing formula of Lemma 3.8, S 1 γ is C k´1 (since γ Ă SΣ is itself C k´1 as an integral curve of the C k´1 -vector field X), and the restriction of an H ´1pSΣq distribution to S 1 γ is H ´1´1{2´ε -regular (for every ε ą 0); moreover, the contact form λ is C k´1 , thus dλ is C k´2 . The pairing pS 1 γ, f dλq is then well-defined as long as k ´2 ą 1 `1{2 `ε, that is k ě 4. The end of the proof goes through as in the smooth case. Overall, one can run the arguments with C 4 -regular metrics.

  2.3.2. Tensor tomography. The tensor tomography problem consists in studying the transport equation Xu " f , where u, f P C 8 pSΣq are smooth, f has finite Fourier degree equal to n, and showing that u has finite Fourier degree bounded by maxpn ´1, 0q.

żS 1

 1 γg 2 pcq φ ˚pe 1 ωq dλ 2 "


N Tranchant

M.-P Ginot

A Besland

T Caillard

A Minea

P.-Y Rhallabi

Jouan

J Zgheib

L Berthelot

J Tranchant

N Ginot

M-P Besland

A Caillard

T Minea

A Rhallabi

P-Y Jouan

Electron-enhanced

come

that adds two distinguished voltage-pulse sequences, one between 5-15 µs and the last between 20-40 µs. It comes out that this e-HiPIMS sequence significantly increases the electron density.

I. Introduction

Magnetron sputtering is a widespread process for thin film deposition, allowing to tune in a wide range structural, electrical and mechanical properties of deposited films. Commonly known as Physical Vapor Deposition (PVD), the physical sputtering of a target under energetic ion bombardment generated by a plasma discharge is a process widely used in many technological domains such as microelectronics [START_REF] Assouar | Piezoelectric Aluminum Nitride Films Deposited by Triode Sputtering for Surface Acoustic Wave Devices[END_REF] , photovoltaics [START_REF] He | Microstructural and chemical investigation of PVD-CdS/PVD-CuIn1-xGaxSe2 heterojunctions[END_REF][START_REF] Siegrist | Triple-cation perovskite solar cells fabricated by a hybrid PVD/blade coating process using green solvents[END_REF] , biology, [START_REF] Bahi | Surface performances of Ti-6Al-4V substrates coated PVD multilayered films in biological environments[END_REF][START_REF] Hovsepian | Microstructure and load bearing capacity of TiN/NbN superlattice coatings deposited on medical grade CoCrMo alloy by HIPIMS[END_REF] or mechanics, [START_REF] Tricoteaux | Fretting wear properties of CrN and Cr2N coatings[END_REF] among other fields. Moreover, adding a reactive gas such as oxygen or nitrogen 7-11, allows to enlarge the panel of deposited material and open the way to a wider range of materials and applications.

Conventional direct current (DC) or radio frequency (RF) plasma-based magnetron sputtering have shown some limitations. Hence, nowadays, the request for thin film properties has increased in a wide extend of targeted electrical and structural properties. To obtain denser layers with enhanced properties associated with a good surface conformity, the generation of a discharge with a high ionization degree and very energetic ions is needed. Moreover, a high density of ionized species extracted from the target are highly needed to reach dense coatings [START_REF] Sarakinos | High power pulsed magnetron sputtering: A review on scientific and engineering state of the art[END_REF] . In the meantime, such energetic conditions can damage the substrate and promote an important increase of the substrate temperature, which might be unsuitable for sensitive substrates like polymers or specific materials and devices.

To circumvent such drawback, an alternative way emerged is the beginning of this 2000's 13- 15 with high voltage and high-current magnetron discharge, so called High Power Impulse Magnetron Sputtering (HiPIMS) that has been classified in the ionized-PVD plasma processing technologies. Such power supply allows to generate a very dense plasma with a very high degree of ionization degree of sputtered atoms from the target thanks to a few tens microseconds pulse duration (typically in the 30 to 100 µs range) at a high frequency up to a few kHz. The generation of such a dense plasma in a very short period requires instantaneous powers of some tens of kW. The interest of HiPIMS discharges has been demonstrated in many studies [START_REF] Zheng | A global plasma model for reactive deposition of compound films by modulated pulsed power magnetron sputtering discharges[END_REF][START_REF] Raman | A high power impulse magnetron sputtering model to explain high deposition rate magnetic field configurations[END_REF][START_REF] Zheng | On the pressure effect in energetic deposition of Cu thin films by modulated pulsed power magnetron sputtering: A global plasma model and experiments[END_REF] highlighting how the enhancement of the ionization degree of the sputtered particles can promote a huge effect on the microstructure and functional properties of the deposited films.

To get more understanding on the electrical characteristics of the discharge and a higher control of the neutral and ion transport phenomena as well as their effects on the thin film growth mechanisms, a large panel of experimental investigations has been performed using Langmuir probes [START_REF] Čada | Time-resolved plasma parameters in the HiPIMS discharge with Ti target in Ar/O2 atmosphere[END_REF][START_REF] Gudmundsson | Spatial and temporal behavior of the plasma parameters in a pulsed magnetron discharge[END_REF][START_REF] Hippler | Time-resolved Langmuir probe diagnostics of a bipolar high power impulse magnetron sputtering discharge[END_REF] , Optical Emission Spectroscopy, [START_REF] Hippler | Time-resolved optical emission spectroscopy of a unipolar and a bipolar pulsed magnetron sputtering discharge in an argon/oxygen gas mixture with a cobalt target[END_REF] and Mass Spectroscopy [START_REF] Ferrec | Mass spectrometry analyzes to highlight differences between short and long HiPIMS discharges[END_REF] , among many others [START_REF] Lundin | High Power Impulse Magnetron Sputtering: Fundamentals, Technologies, Challenges and Applications[END_REF] . These different studies demonstrated the high ionization degree of sputtered species about two orders of magnitude as compared to the typical values of conventional magnetron discharges. The first effect is thus the huge ionization fraction of the sputtered species that can, in particular experimental conditions, exceed 90% of the discharge. [START_REF] Ferrec | Mass spectrometry analyzes to highlight differences between short and long HiPIMS discharges[END_REF][START_REF] Kubart | Investigation of ionized metal flux fraction in HiPIMS discharges with Ti and Ni targets[END_REF] Such high ionization degree of sputtered species leads to a very directional ionized discharge leading to a highly oriented film with a typical columnar structures perpendicular to the substrate. Such process allows to enhance the deposition conformity of the deposited film on 3D patterned surfaces such as trenches or via structures with a high aspect ratio. [START_REF] Bobzin | High Power Impulse Magnetron Sputtering -HIPIMS[END_REF] In most cases, publishes results revealed that the dominant population is the sputtered ionized metal species (M + ) as compared to the argon ion population (Ar + ). Indeed, the pulse shape applied to the target directly affects the electrical behavior of the HiPIMS discharge and the time evolution of ions and neutrals densities. Ina previous study, the effects of the duty cycle, i.e. time on (t on ) and time off (t off ), on the temporal evolution of the electron density and electron temperature, as well as the ion and neutral densities, have been analyzed. [START_REF] Ferrec | Mass spectrometry analyzes to highlight differences between short and long HiPIMS discharges[END_REF] Moreover, the authors demonstrated that the densities of double-charged ions (Ar 2+ and M 2+ ) could be more significant. [START_REF] Ferrec | Mass spectrometry analyzes to highlight differences between short and long HiPIMS discharges[END_REF] In addition, in the previous decade, other configurations of the instantaneous voltage applied to the target have been investigated. The effect of HiPIMS multi-pulse (m-HiPIMS) was studied [START_REF] Antonin | On the HiPIMS benefits of multi-pulse operating mode[END_REF][START_REF] Fekete | Ti atom and Ti ion number density evolution in standard and multi-pulse HiPIMS[END_REF] and brought more information on the mechanisms involved in the deposition process. The first advantage of m-HiPIMS is to provide a better control of the discharge parameters than a standard HiPIMS (s-HiPIMS) process. [START_REF] Antonin | On the HiPIMS benefits of multi-pulse operating mode[END_REF][START_REF] Souček | Microstructure of titanium coatings controlled by pulse sequence in multipulse HiPIMS[END_REF] O. Antonin at al. [START_REF] Antonin | On the HiPIMS benefits of multi-pulse operating mode[END_REF] investigated the effect of five high voltage micro-pulses of 3μs separated by varying delays time (t off ) at different frequencies to guarantee the same average power. As a result, they observed an increase of the deposition rate by 50%, and an enhancement of the ionization rate within the discharge, leading to a higher content of sputtered metallic species. Moreover, in further work of M. Fekete et al. [START_REF] Fekete | Ti atom and Ti ion number density evolution in standard and multi-pulse HiPIMS[END_REF] , it is demonstrated that the ion flux significantly increases compared to s-HiPIMS with tightly dependency on the delay between pulses (t off ) . In addition, at high pressure, remote probe measurements revealed a peak density after the power pulse, significant of a slow and diffusive ion transport.

Other HiPIMS configuration including the association of different pulses, so-called bipolar pulse, has been further investigated. [START_REF] Keraudy | Bipolar HiPIMS for tailoring ion energies in thin film deposition[END_REF][START_REF] Hippler | Time-resolved optical emission spectroscopy of a unipolar and a bipolar pulsed magnetron sputtering discharge in an argon/oxygen gas mixture with a cobalt target[END_REF][START_REF] Britun | Ion density evolution in a highpower sputtering discharge with bipolar pulsing[END_REF][START_REF] Velicu | Energy-enhanced deposition of copper thin films by bipolar high power impulse magnetron sputtering[END_REF] : a small positive pulse was added after the highpower initial negative pulse on the target. Such HiPIMS configuration allows the increase of the sputtered ions energy.

On the basis of these previous works, we developed a new HiPIMS power supply, so-called e-HiPIMS for electron enhanced HiPIMS that associate a standard HiPIMS (s-HiPIMS) and the multi-pulse HiPIMS (m-HiPIMS). Such power supply allows to deliver multi-voltage pulses separated by various and tuned t 0ff values during the t on period. After a first stage of discharge studies, we investigated the effect of such supply on a pure Ar discharge with a chromium target. Our objective has been to tune the pulse shape in order to amplify only a specific ion population compared to the other plasma species. Indeed, previous studies [START_REF] Yushkov | Origin of the Delayed Current Onset in High-Power Impulse Magnetron Sputtering[END_REF] have shown that in HiPIMS discharges, the peak density associated to each ion population can be shifted at different time occurrence during the main voltage pulse. As shown in previous work [START_REF] Ferrec | Mass spectrometry analyzes to highlight differences between short and long HiPIMS discharges[END_REF] , during the pulse time-on t on in a HiPIMS pure Ar discharge involved with a chromium (Cr) target, the Ar + peak density appears first followed by Cr + peak. Our hypothesis is thus to add a pulse at the moment of maximum density of an ion species assuming that this species density might be amplified. As an illustration, apply an over-pulse synchronized either with the maximum of Ar + density or with the Cr + maximum density, and then observe if any effects on the structural and chemical properties of the deposited films.

It is well known and established that an increase of Ar + ion density promotes denser films denser thanks to enhancement of ion bombardment. However, most of the time, the deposition rate is weakly affected and the crystalline quality of the film tends to decrease and even change to an amorphous structure [START_REF] Ji | Effect of ion bombardment on in-plane texture, surface morphology, and microstructure of vapor deposited Nb thin films[END_REF][START_REF] Mattox | Particle bombardment effects on thin-film deposition: A review[END_REF] . On the contrary, the increase of Cr + ion density usually improves the conformity and crystalline quality of the deposited films [START_REF] Cemin | Low-energy ion irradiation in HiPIMS to enable anatase TiO2 selective growth[END_REF][START_REF] Cemin | Tuning high power impulse magnetron sputtering discharge and substrate bias conditions to reduce the intrinsic stress of TiN thin films[END_REF] . In addition, the metallic ions directly contribute to the film growth.

In the present paper, we present the effect of synchronized additional pulses applied at different time of the main HiPIMS pulse on both the discharge composition and the quality of deposited films in the case of pure Cr films.

II. Experimental details

The HiPIMS measurements and films depositions were performed in a cylindrical stainlesssteel chamber where the cathode is located vertically above the sample holder, which can be polarized to control the energy of ions impinging the substrate surface. Thanks to a turbo molecular pumping unit from Leybold, a background pressure close to 3.10 -4 Pa can be achieved. Two capacitance and ionization gauges allow to follow the deposition pressure and the background pressure, respectively. The size of magnetron target is a disc of 7.6 cm diameter.

Since the rotational speed of the turbo molecular pump remains constant, the working pressure is manually controlled by a regulation valve between the deposition chamber and the pumping system (Leybold). The gas is introduced via a mass flowmeter (Brooks®) and and the gas flux is monitored by a computer interface.

To analyze the discharge, and in particular the evolution of electrons population versus time, we performed electrical characterization of the discharge by a commercial Langmuir probe measurements (using a commercial probe from Impedans 35 ). A single tungsten tip (10 mm length and 0.195 mm radius) in the standard configuration was used to probe the plasma out of the ionization region (magnetized plasma).

The probe tip is connected to RF and DC compensation electrodes to increase the capacitive coupling between the probe tip and the plasma and to compensate for the perturbation of the plasma potential caused by the probe biasing. Temporal analysis of plasma is possible because the Langmuir probe operates at an 80 MHz sampling rate, which allows a plasma diagnostic every 12.5 ns. This enables a very good time resolution for HiPIMS pulses as short as a few tens of s.

This probe has very good performances in terms of the plasma potential range (-100 V to +145 V), the ion current density (1 µA/cm 2 to 300 mA/cm 2 ), and the electron temperature (0.1 eV to 15 eV). Out of the magnetic trap, the probe was placed 5 cm away from the cathode.

The radial displacement of the probe permits the radial measurements of current-voltage (I-V) characteristics. Here we fixed the radial position in front of the racetrack. Plasma parameters such as the electron temperature, electron density, plasma potential and floating potential can be easily extracted from the I-V characteristics. The ion mean free path is large for low pressures, and the sheath around the probe is assumed to be collisionless. Thus, the probe current is determined by the Laframboise orbital motion theory. [START_REF] Rousseau | Comparison between Langmuir probe and microwave autointerferometry measurements at intermediate pressure in an argon surface wave discharge[END_REF] Therefore, all these plasma parameters are automatically evaluated using the Impedans software. 35 A typical HiPIMS pulse comprises two phases: the time-on and the time-off. The time-on, t on , corresponds to applying a high negative voltage (about 500-700 V in absolute value) to the cathode during tens of microseconds. The time-off, t off , corresponds to the cut-off of the high voltage applied to the cathode between two successive pulses. In the latter phase, a progressive extinction of the plasma is observed. These two phases are repeated with a period T.

Figure 1. Simplified scheme of the e-HiPIMS power supply that displays LC circuits.

In our previous study, [START_REF] Ferrec | Mass spectrometry analyzes to highlight differences between short and long HiPIMS discharges[END_REF] , we showed that two regimes exist during the same HiPIMS Ar/Cr discharge pulse. The first one characterizes the beginning of the pulse, in which Ar + dominates. This is the period when the current rises and the gas atoms get massively ionized, but the Cr atom sputtering stays deficient. The second regime is Cr + dominated, due to the argon rarefaction that increases the relative number of Cr atoms in the ionization region.

Therefore, their ionization is enhanced due to the higher direct collision probability with the trapped electrons and their lower ionization energy than argon.

In this study, we propose adding an extra short-voltage pulse at the beginning of the regime dominated by the Cr + ions. Hence, it should boost the ionization process of Cr by electron impact and amplify the Cr + production.

Therefore, the experiments were carried out using the e-HiPIMS power supply. It comprises three LC circuits (three stages, Fig. 1) operating repetitively and synchronized. In addition, each stage can be turned on or off, and can deliver a negative voltage between 150 and 300 V. This means that during a pulse sequence, each stage can deliver a square voltage signal with a specific duration t on , and with a specific delay with respect to the beginning of the sequence.

Hence, the operation mode can be standard HiPIMS (s-HiPIMS) when all the stages deliver the voltage simultaneously (no delay between the stages) or multi-voltage pulse sequence m-HiPIMS when the stages are shifted in time. [START_REF] Antonin | On the HiPIMS benefits of multi-pulse operating mode[END_REF][START_REF] Souček | Microstructure of titanium coatings controlled by pulse sequence in multipulse HiPIMS[END_REF] A homemade software manages each voltage stage in the desired sequence (amplitude, width, delay, etc.). The voltage pulse value is a set point, meaning the cathode applied value can differ slightly from the imposed reference. For example, if the set point value is 700 V, the applied value would be 700 ± 20 V.

Figure 2 shows, for the sake of example, three multi-voltage pulse sequences produced by the new power supply. Each configuration comprises a main voltage stage of 500 V between 0 and 30 µs (the orange section) and an additional voltage stage of 200 V (the blue section).

Standard HiPIMS (D1, Fig. 2) is the addition of these two stages (700 V). For D 2 , the first stage is 500 V between 0 and 25 µs, 700 V between 25 and 30 µs, and 200 V between 30 and 40 µs. This sequence aims to favor the Cr + production. For D3, two additional voltage stages are added at different intervals. The first is between 5 and 15 µs to favor Ar + production and help Cr sputtering. The second one is added between 20 and 40 µs to favor Cr + production. Table 1 summarizes the three configurations with associated discharge voltages. These three configurations D i (i=1-3) are discussed in the present study. For different D i (i = 1-3), a delay between the current and the voltage is observed at the beginning of the pulse. This delay corresponds to the ionization time to build up the ion (Ar + ) space charge, mainly in the ionization region (IR), and the gas ions' flight time across the sheath between the IR and the cathode. Once the Ar + hits the target, a significant amount of metal atoms leaves the target. After a few µs (typical flight time), they reach the IR. After that, the ionization of sputtered metal (Cr) leads to a significant production of Cr + and Cr ++ , contributing to the increase of the total discharge current. [START_REF] Gudmundsson | An ionization region model of the reactive Ar/O2high power impulse magnetron sputtering discharge[END_REF] Notice that the amount of metal ions increases, and the self-sputtering can also increase.

Voltage -D 1 ≈ 700 V 700 V 700 V 700 V 700 V 0 V Voltage -D 2 ≈ 500 V 500 V 500 V 500 V 700 V 200 V Voltage -D 3 ≈ 500 V 700 V 500 V 700 V 700 V 200 V Figures 3, 4
For D 1 , the voltage and the current behavior during t on , from 0 to 30 µs, are as expected, typical for the s-HiPIMS. The voltage slightly decreases at the beginning of the pulse due to the current rise and the voltage drop on the internal exit resistance of the pulsing unit. When the current stabilizes, the discharge voltage reaches a stable value of 600 V. During t on ; the current reaches a maximum of 50 A (1.13 A/cm 2 ). After this maximum, the total current slightly decreases, which is the typical behavior of the HiPIMS plasma operated with a chromium target.

One explanation is the replacement of a significant fraction of the gas ions by metal ions (single and multiple-charged). After 30 µs, the voltage switches off to zero. Consequently, the discharge current abruptly decreases.

For configuration D 2 , the sequence starts with a high voltage pulse, like in the previous sequence, with the same duration of 30 µs. In D 2 , a voltage over-pulse of 200V adds power to the discharge starting at 25µs. This additional pulse ends beyond the end of the starting pulse at 40 µs. As Figure 4 illustrates, the current and the voltage evolution shows three different levels: L 1 , L 2 and L 3 , since the voltage changes during all three phases (see Table 1 and Fig.

2).

At the beginning of the pulse from 0 to 25 µs (level L 1 ), the initial voltage (~ 900 V, Fig. 4) Let us now analyze the voltage and current evolutions for the D 3 sequence (Fig. 5) with respect to D 2 . At the beginning of the pulse from 0 to 5 µs (level L 1 ), the voltage should normally reach 500V. This voltage level is close to the transient value observed in the previous sequences D1 or D2 exactly after 5 µs from the beginning of the pulse. However, since the time interval is very short, the steady state is not reached, and the voltage is higher than the expected set point.

decreases
Then at L 2, after adding the first over-pulse (~5 µs), the voltage slightly increases, followed by a decay around 700V. The voltage level starts to stabilize in the second half of this first over-pulse (between 10 at 15 µs). However, the steady state does not survive because phase L 3 begins. This level of L 3 is very short (~5 µs length), between 15 to 20 µs. During L 3, the voltage drops from 700 V to 500 V. Then, the second over-pulse starts at 20 µs. Up to 30 µs, a new level of L 4 is attained. The voltage reaches 700 V again, very fast. Further, during the transition from L 4 to L 5 , the voltage drops to a low value close to zero (like in sequence D 2 ) before entering the L 5 phase. In L 5, which lies from 30 to 40 µs, the voltage increases to around 200 V without reaching a steady state.

Let us comment on the current evolution (Fig. 5, right Y-axis). The discharge current keeps increasing during levels L 1 and L 2 , from 0 to 15 µs, without observing any transition until reaching a maximum value of 50 A. Notice this peak value is the same as for s-HiPIMS (D 1 ), but in D 3, the maximum is reached faster, after only 15µs, while in D 1, the current peaks after about 20 µs (Fig. 3). In the following, the current decays because of the voltage set point decreases from 700 (L 2 ) to 500 V (L 3 ). At L 4, the second over-pulse brings additional power to the discharge, and the current increases again until around 45 A, below the peak value (recorded in L 2 ). As in D 2 , for level L 5 , the current decreases continuously and smoothly due to the abrupt voltage drop from 700 to 200 V.

Figure 6 shows the influence of the pressure, for 0.2, 0.6, and 1 Pa, on the current waveform for the three pulse sequences D 1 , D 2, and D 3 . The current increases with the pressure independent of how the power is delivered to the plasma. The fingerprint of the current stays similar for the three studied pressures, and the signature of specific levels discussed above characterizes each pulse sequence. However, at the pulse's beginning, the delay between the voltage and the current decreases while the pressure increases, as expected. The higher the pressure, the faster the space charge builds up.

For all studied configurations, the observed delay of around 1 to 5µs decreases by increasing the pressure, at constant applied voltage. As previously studied30, this delay is also linked to the applied voltage.

Indeed, the delay is related to the ionization time necessary to the discharge to create enough electrons and ions and the time for the primary electrons (ejected from the target) to get accelerated in the sheath electric field, between IR and the cathode. [START_REF] Cemin | Tuning high power impulse magnetron sputtering discharge and substrate bias conditions to reduce the intrinsic stress of TiN thin films[END_REF]35 Let us consider the following approximation to evaluate the ionization time's dependence on the pressure. The ionization time ( ) is related to the argon density (n Ar ) and the argon ionization rate, which depends on the electron temperature T e . This can be expressed as:

(1)

At the beginning of the voltage sequence, time-on T e is around 6 -8 eV (previously calculated in our HiPIMS model [START_REF] Zgheib | A high-power impulse magnetron sputtering global model for argon plasma-chromium target interactions[END_REF] ), n Ar is proportional to the pressure:

(2) K B is the Boltzmann constant and T is the gas temperature (300 K). This is particularly true in the early stages of the pulse when the gas rarefaction is not yet active. Moreover, K ionization (T e ) of argon is calculated as 38 :

(3)

Therefore, for T e equal to 6.5 eV and pressure of 0.2 Pa, 0.6 Pa and 1 Pa ionization equals 5 µs, 1.7 µs, and 1 µs, respectively. P. Poolcharuansin and J.W. Bradley 39 have measured between 0 and 4μs, three distinct groups of electrons with effective temperatures of 70-100 eV, 5-7 eV and 0.8-1 eV. After ~ 4 μs, it remains only a single distribution with an electron temperature of about 5 eV. It means that our mean electron temperature estimated at 6.5 eV is consistent with these measurements.

As explained by G.Y. Yushkov and A. Anders, [START_REF] Yushkov | Origin of the Delayed Current Onset in High-Power Impulse Magnetron Sputtering[END_REF] the delay time is close to the time required for the discharge to develop using the primary electrons, corresponding to the ionization time (Eq. ( 1), see also Fig. 6-a). Hence, by increasing the pressure, the ionization time of atoms decreases. For low pressure, electron-atom collisions are scarcer, reducing the frequency of the ionization events. Consequently, the delay between the applied voltage and current growth becomes more significant. Remember that all previously reported results have been obtained

for an average power of 200 W on the cathode. Hereafter the power is 100 W for the following experiments. First, the current and voltage were measured for different conditions.

For configuration D 1 , the current peaks to 18 A (at 0.2 Pa) when the voltage pulse is on and decreases immediately at 30 µs when the discharge is off.

Therefore, the current delay with respect to the voltage is ~5 µs for 0.2 Pa. Meanwhile, it occurs at 2 and 1 µs for 0.6 Pa and 1 Pa, respectively. Table 2 shows the voltage set point and peak current values for each pulse sequence at different periods during the sequence for the three studied argon pressures, namely 0.2, 0.6 and 1 Pa. practically the same path, independent of the working pressure. However, at 1 Pa, there is a slight overshot (increase) of the current before starting the last level of this sequence, absent for the other pressures. This could be attributed to the enhanced ionization of Ar. Similar behavior appears for D 3 . For low pressure of 0.2 Pa, the delay between the current and voltage slightly exceeds 5 s. When the first voltage over-pulse starts, during L 2 between 5 and 15 µs, the current peaks to its maximum for the three pressures. These maxima slightly increase with the pressure. Then the current decreases by 2/3 between 15 and 20 µs (level L 3 )

when the voltage also decreases. After that, the second voltage over-pulse starts (corresponding to L 4 from 20 to 30 µs). The voltage has the same level as in L 2 , but the current increases less and reaches a value below the maximum (in L 2 ). This current limitation could be attributed to the intensification of the gas rarefaction phenomenon in the ionization region, inducing a decrease in the ionization rate and, therefore, a decrease in the discharge current. [START_REF] Zgheib | A high-power impulse magnetron sputtering global model for argon plasma-chromium target interactions[END_REF] Then, at L 5, between 30 and 40 µs, a sharp current decay occurs.

In this part, it is established that using an over-pulse enhances the total current. The overpulse creates hot electrons that ionize the species present in the IR. Naturally, when the pressure increases, the density of these species increases as well, and consequently, the total charge carried by the discharge (table 3).

III. Langmuir probe results

Langmuir probe measurements were carried out to obtain time resolved electron density and temperature. The Langmuir probe was outside the ionization region (IR), characterized by a very high electron density n e . Note that the IR is located a few millimeters from the cathode and typically expands about 2 cm. The probe location (8 cm from the target) was far enough from the magnetic trap, so one can assume there is no magnetic field effect on the probe signal. The results presented hereafter do not consider any correction due to the magnetic field. The measured values of n e and T e outside of the IR are obviously different from those in the IR. [START_REF] Gudmundsson | An ionization region model of the reactive Ar/O2high power impulse magnetron sputtering discharge[END_REF] Figure 7 shows the electron density and temperature evolution during the pulse and the beginning of the afterglow of all three studied pulse sequences D i (i=1-3) at low pressure, 0.2 Pa. When the discharge is 'on', the electron density increases until it reaches a maximum value around (2.8 ± 0.5)×10 16 m -3 , (1.2 ± 0.5)×10 17 m -3 , and (1.3 ± 0.5)×10 17 m -3 for D 1 , D 2 and D 3 , respectively. Noting that at the beginning of the discharge, D 1 starts with a voltage of 700V, while D 2 and D 3 start with 500V each. During the pulse sequence, the electron density increases continuously until reaching a maximum, about 35 µs after the start of the sequence, independent of the way the power is delivered to the discharge. Then the electron density decreases, surviving even 100µs after the end of the pulse when it is typically one decade lower. In addition, the electron density correlates well with the pulse sequence voltage level. D 3 has the highest electron density, followed by D 2 and D 1 . Notice, however, that the difference between D 3 and D 2 is in the margin of the error bar. This means that the effect of the over-pulse on the diffusion plasma keeps less the memory of the moment when it is applied but depends significantly on the voltage level.

In D 3 , the first voltage over-pulse starts only 5 µs after the beginning of the sequence (up to 15 µs), reaching around 700 V. The electron density rises fast and reaches a first maximum of 9.4 ± 0.5×10 16 m -3 at the end of this over-pulse (~15 µs), then decreases when the voltage also decreases (down to 500V in L3). The second voltage over-pulse starts at 20 µs (up to 40 µs).

The electron density peaks at 1.3 ± 0.5×10 17 m -3 at around 35 µs. Notice that at this moment, the voltage is only 200V since the original pulse lies 30µs. However, the probe's position is out of the IR, and it captures the density value in the diffusion plasma. Is expected a delay between the maximum value in the IR (not measured, but expected to be in phase with the injected power, which is maximum in L 4 ) and at the location of the probe.

Comparing the electron density augmentation between D 2 and D 3 with respect to reference D 1 unveils the influence of the over-pulses on plasma. For D 3 , the electron density increases between 5 and 15 µs, followed by a second boost between 27 and 37 µs.

Hence, at least in front of the substrate, the electron density is higher when a second voltage over-pulse is added around the middle of the discharge. The first observed electron density peak is assigned mainly to the Ar ionization process. The second one mainly produces metal ionization, enhancing the density of Cr + and Cr ++ . They were also observed and discussed by

Poolcharuansin and Bradley. [START_REF] Poolcharuansin | Shortand long-term plasma phenomena in a HiPIMS discharge[END_REF] The effect of the second peak on the electron density is higher than the first one due to the lower ionization energy threshold of Cr (7.68 eV) compared to Ar (15.76 eV). After reaching the maximum, the electron density decreases because the power decreases.

For similar conditions, other studies observed an electron density of 10 18 -10 19 m -3 . On the other hand, in our study, the electron density is around 10 17 m -3 . This is due to the placement of the Langmuir probe, which is not in the ionization region (magnetized trap) where the density is expected to be at least one order of magnitude higher; it is placed at 8cm (close to the substrate position) and perpendicular to the target. Thus, some of the electrons could eventually not be detected, which could affect the electron density 40,[START_REF] Gudmundsson | On the electron energy in the high power impulse magnetron sputtering discharge[END_REF] . Therefore, the apparent electron density is lower than what has been observed for the case of HiPIMS in other studies. Furthermore, one can see a striking difference between the n e peaks' shape (Fig. 7) and the currents' evolution (Fig. 6). The current is measured on the cathode, which is very close to the IR. In contrast, the n e is measured by the Langmuir probe, which is located 5 cm away from the cathode. For example, the two current peaks are associated with the two voltage overpulses of the D 3 sequence and registered in the time intervals 5-15 s and 20-30 µs. They are less obvious but still present in the time profile of n e . The overlapping of the peaks in the case

of n e could be attributed to the electron diffusion phenomenon from IR towards the Langmuir probe.

The diffusion results in an enlargement (dispersion) of peaks in time, inducing an overlapping of the two (close) successive peaks.

The electron density increases when the power injected in the plasma increases by applying the additional over-pulse. For the D2 sequence, the voltage over-pulse starts at 25 µs and finishes at 40 µs. Hence, the electron population is less spread than in D3 at the discharge's beginning, but the density level is also lower. The time shift between the maxima values of recorded electron density and current (Fig. 6) is about 8 µs, which could correspond to the ionization enhancement and electron diffusion out of the IR to the Langmuir probe.

In fact, in first approximation, the average diffusion time of electrons writes:

Where L is the distance between IR and Langmuir probe, and D e is the Bohm diffusion coefficient of electrons. The probe is placed outside the magnetized zone. Hence, the Bohm diffusion is important because inside IR, it delays the electron's diffusion, and outside IR, it neglects the diffusion time. Notice also that most ionization events occur within a few millimeters in front of the racetrack. 40 So, it is reasonable to assume . For , . This value for D e is in good agreement with the value reported in REF [START_REF] Zgheib | A high-power impulse magnetron sputtering global model for argon plasma-chromium target interactions[END_REF] for our pressure range.

Let us look at the evolution of the electron energy, namely electron temperature (Fig. 7(b)),

given by Langmuir probe measurements. By disregarding the strong noise, the evolution of T e as a function of time shows a sharp increase at the beginning of time-on, followed by a decay (relaxation) until the end of the sequence at 40 µs. In the afterglow, between 40 and 80 µs, T e increases again for D 1, while for D 2 and D 3, only a weak increase is observed. Beyond 80 µs, T e unconditionally decreases.

The electron temperature evolution in the afterglow is related to the recombination kinetics.

Indeed, low-energy electrons attach first, letting free only a fraction of energetic electrons. On the other hand, non-magnetized electrons diffuse faster to the wall, and the positive space charge dominates the afterglow plasma. Hence, the surviving free electrons have higher energy than during the plasma pulse, but their density is drastically reduced. . It is about 4 ± 0.5 eV for all sequences at the beginning of time-on and continuously decays to about 1 eV until the end of the sequence. In the afterglow, it rises again (see the explanations above). However, for both D 2 and D 3 , the rise of the electron temperature at regular intervals is less pronounced at higher pressure, but it does occur.

Figure 9 shows the electron density of each configuration D i for 0.2 Pa and 0.6 Pa pressures.

The maximum of n e is outside of the time-on. As said, this shift is due to the diffusion time of the electrons from IR towards the Langmuir probe (see Eq. 4).

For D 1 , at first, n e decreases with the pressure increase. This decrease could be due to the slight cooling of the electrons induced by the higher collision frequency (higher pressure).

About the afterglow, n e survives longer at higher pressure beyond the end of the pulse because the density at the end of the pulse is higher. This could be explained by the decrease of the rarefaction phenomenon of Ar gas, which tends to increase the partial pressure of Ar and consequently increase the electron production rate by the ionization process. Also, as discussed in Gudmundsson et al. study [START_REF] Gudmundsson | Are the argon metastables important in high power impulse magnetron sputtering discharges?[END_REF] , the argon metastable can also play an important role with the pressure increase. However, the influence of the pressure less affects sequence D 2 than D 3 . 2.

For all sequences, the electron density is higher at the end of the pulse at higher pressure.

Even when the second voltage over-pulse is applied, the electron density stays higher for low pressure (0.2 Pa). This is because the ionizations occur in the IR, but their diffusion is highly limited with the pressure increase. For D3, when the first over-pulse is added between 5 and 15 µs, the electron density rises less by increasing the pressure to 0.6 Pa. 2.

IV. CONCLUSION

(c)

which drives the total ion flux, by choosing a reduced pulse duration. Boosting the discharge at a specific moment of the discharge, it could favor the preferential ionization of one species over the others. Finally, this will certainly modify the physical and structural properties of the deposited film.

Figure 2 .

 2 Figure 2. The three configurations of pulse sequences (D 1 , D 2, and D 3 ) obtained with the new pulsed power supply. They are compared hereafter.

  Figures 3, 4 and 5 present the measured voltage and current waveforms of the three pulse sequences D 1 , D 2 and D 3 (detailed in Fig.2and Table1), respectively. In all cases, a transient regime is observed before reaching a constant voltage (steady state). The experiments were performed for a pressure of 0.6 Pa and an average power of around 200 W.

Figure 3 .

 3 Figure 3. Typical current and voltage waveforms for D 1 (s-HiPIMS).

  to about 500V, towards a plateau. Then the voltage jumps to the L 2 level, around 700 V, due to the over-pulse (additional ~200V). Due to this short added voltage on the cathode, the steady state of nearly constant voltage does not occur. Finally, when the first pulse of the sequence is cut off (30 µs), the transition from L 2 to L 3 marks a significant voltage drop, close to zero, before entering phase L 3 . During L 3 , from 30 to 40 µs, the voltage gradually increases to reach the reference operation point of about 200 V. Still, no steady state can be reached since the plasma evolves in a transitory regime.

Figure 4 . 3 ) indicate the time intervals with different voltage levels summarized in Table 1 .

 431 Figure 4. Current and voltage waveforms for the pulse sequence D 2 . Labels L i (i = 1,3) indicate the time intervals with different voltage levels summarized in Table1.

Figure 5 .For configuration D 3 (

 53 Figure 5. Current and voltage waveforms for the pulse sequence D 3 . Labels L i (i = 1,5) indicate the time intervals with different voltage levels summarized in Table 1.

Figure 6 . 2 and (c) D 3 (

 623 Figure 6. The evolution of the discharge current for three pulse sequences (a) D 1 , (b) D 2 and (c) D 3 (detailed in Table 1 and Fig. 2) for three argon pressures of 0.2, 0.6, and 1 Pa.

Figure 7 .

 7 Figure 7. Time evolution of the electron density (a) and electron temperature (b) for the three pulse sequences in Table 2 and argon pressure of 0.2 Pa.

Figure 8 .

 8 Figure 8. Time evolution of the electron density (a) and electron temperature (b) for the three pulse sequences in Table 2 and argon pressure of 0.6 Pa.

Figure 8 2 (

 82 Figure8presents the electron density and temperature of all configurations D i (i=1,3) for 0.6 Pa pressure. The average power was fixed at 100 W. The electron density behavior is similar to the evolution observed at lower pressure, 0.2 Pa (Fig.7). At the beginning of the pulse, the electron density increases. Then it reaches a maximum value of (3.5 ± 0.5)10 16 m -3 , (9.6 ± 0.5) 10 16 m -3 , and (1.2 ± 0.5) 10 17 m -3 for D 1 , D 2 and D 3 , respectively. During time-on, n e is higher for D 3 than for D 1 and D 2 . Adding the voltage over-pulses in D 2 and D 3 increases the electron density compared to D 1 . This could be explained by the fact that at high pressure, the Cr thermalization is faster; hence, the ionization during the additional pulse becomes more efficient for D 2 and D 3 . In the afterglow (time-off phase), the electron density smoothly decays for all of the configurations.The electron temperature has a very similar behavior when the pressure increases from 0.2 (Fig.7(b)) to 0.6 Pa (Fig. 8(b)). It is about 4 ± 0.5 eV for all sequences at the beginning of

16 Figure 9 .

 169 Figure 9. Influence of the pressure on the electron density for different pulse sequences (a) D 1 , (b) D 2 and (c) D 3 given in Table2.

Figure 10 shows

 10 Figure10shows the electron temperature of each configuration for the same two pressures, 0.2 Pa and 0.6 Pa. Disregarding the noise generated during the measurement of T e ; globally, T e increases with the pressure whatever the HiPIMS sequence D i (i=1,3) is.

Figure 10 .

 10 Figure 10. Influence of the pressure on the electron temperature for different pulse sequences (a) D 1 , (b) D 2 and (c) D 3 given in Table 2.

Table 1 : Voltage values for each configuration during t on . Notice that all the voltage values are negative but indicated in absolute value for convenience. Red values indicate the addition of the voltage delivered by at least two stages of the power supply, while blue values the remaining voltage from one single stage after the cut-off of the main stage. The black values indicate the voltage of the main stage.

 1

	t on (µs)	0 -5	5 -15	15 -20	20 -25	25 -30	30 -40

Table 2 : The current peak and voltage values for each pulse sequence during different time intervals of the t on for the same average power of 100 W and three working pressures of 0.2, 0.6, and 1 Pa of argon.

 2

		D 1 10-30 µs	D 2 10-15 µs	D 2 25-30 µs	D 3 10-15 µs	D 3 25-30 µs
		700 V	500 V	700 V	700 V	700 V
	0.2 Pa	17 A	15 A	22 A	20 A	18 A
	0.6 Pa	18 A	17 A	22 A	22 A	20 A
	1 Pa	18 A	17 A	25 A	23 A	20 A

Table 3 : Integration of the current for each pulse in Coulomb sequence (same average power of 100 W) and three working pressures of 0.2, 0.6, and 1 Pa of argon.

 3

		D 1	D 2	D 3
	0.2 Pa	381 C	415 C	416 C
	0.6 Pa	503 C	544 C	538 C
	1 Pa	537 C	563 C	547 C

Acknowledgment

We thank Jordan Champlon, Institute of Technology of Nantes, for his expertise in building the new HiPIMS power supply.

Data Availability

The data supporting this study's findings are available within the article.

In this paper, we investigated a prototype of e-HiPIMS power supply, neither studied nor published before. We characterized the effect of such power supply on the generated plasma characteristics. The advantage of such supply lies in the possibility to generate independently multi-pulses and to synchronize a main pulse with one or several over-pulses at chosen moments during the time-on of the main pulse with an independent delay and duration. Thus, specific ion populations can be amplified by well-chosen over-pulses.

Win the present study, we compared three different pulse sequences; thereafter called D 1 , D 2 and D 3 . The first sequence D 1 stands as the reference and corresponds to a conventional HiPIMS power supply behaving like the standard HiPIMS (s-HiPIMS).

The second sequence, D 2 , has an additional voltage over-pulse between 25 and 40 µs of the main pulse, aiming at amplifying the chromium ionization by the end of the pulse. For the third sequence, D 3 , two voltage over-pulses have been added: a the first one between 5 and 15 µs of the main pulse, aiming at amplifying the argon ion density and thus increase the target sputtering, going along with bringing more Cr atoms into the plasma phase, while the second one applied between 20 and 30 µs is expected to boosts the chromium ionization. Timeresolved measurements by the Langmuir probe gave allowed to collect supplementary information on electron density and temperature evolution.

The pressure and power changes revealed their influence on each pulse sequence. As a result, we found that sequence D 3 provides the highest electron density during the discharge. Then, D 2 has the second highest electron density. Note that when the voltage over-pulse is added, the sequence has the same total voltage as in D 1 (s-HiPIMS). For D 3 , the increase of n e with the pressure is remarkable because of the addition of two voltage over-pulses. However, when the pressure increases, the electron density decreases for an average power fixed at 100 W.

Meanwhile, the electron temperature presents different behaviors: at the beginning of the pulse, the temperature is high and decreases in a second time to further to increase at the beginning of the afterglow (time-off). The electron temperature increases again when the following over-pulse is added.

Such results have thus shown their interest with respect to the standard HiPIMS. The e-HiPIMS offers a higher flexibility by adding one or several over-pulses at well-chosen moments in the main pulse with tunable duration in addition to the typical s-HiPIMS parameters, i.e. the pulse width and the frequency in s-HiPIMS.

Thus, this novel e-HiPIMS power supply allows a better control of the HiPIMS deposition process. Our study has shown the possibility of amplifying the HiPIMS discharge current,

Conflict of interest

The authors have no conflicts to disclose.

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