Simon Allais
email: simon.allais@imj-prg.fr

Tobias Soethe
email: soethe@mathga.rwth-aachen.de

HOMOLOGICALLY VISIBLE CLOSED GEODESICS ON COMPLETE SURFACES

Keywords: Mathematics Subject Classification. 53C22, 58E10 closed geodesics, cylinder, min-max, Möbius band

In this article, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder, a complete Möbius band or a complete Riemannian plane leads to having infinitely many geometrically distinct closed geodesics. In particular, we prove that any complete cylinder with isolated closed geodesics has zero, one or infinitely many homologically visible closed geodesics; this answers a question of Alberto Abbondandolo.

Introduction

The problem of the existence and multiplicity of closed geodesics plays an important role in both Riemannian geometry and dynamics. Going back to Hadamard and Poincaré [START_REF] Hadamard | Les surfaces à courbures opposées et leurs lignes géodésiques[END_REF][START_REF] Poincaré | Sur les lignes géodésiques des surfaces convexes[END_REF], it is still open for a large class of Riemannian manifolds. Given a complete Riemannian manifold (M, g), a famous question is whether it possesses a closed geodesic for every Riemannian metric g. This is always true if M is closed [10, [START_REF] Lyusternik | Variational problems on closed manifolds[END_REF][START_REF] Fet | Variational problems on closed manifolds[END_REF]. We can then ask whether the number of closed geodesics is infinite or not. It is known that every closed surface has infinitely many geometrically distinct closed geodesics [START_REF] Franks | Geodesics on S 2 and periodic points of annulus homeomorphisms[END_REF][START_REF]On the existence of closed geodesics on two-spheres[END_REF][START_REF] Hingston | On the growth of the number of closed geodesics on the two-sphere[END_REF]. However, this question is still open for spheres of higher dimension. In this article, we are interested in non-compact complete Riemannian surfaces for which even the existence of one closed geodesic fails in general: planes and cylinders (we also study the Möbius band that can have only one closed geodesic). For instance, the Euclidean plane does not possess any closed geodesic. Nevertheless, under specific geometric conditions, interesting results can be stated. In 1980, Bangert proved that any complete Riemannian cylinder, plane or Möbius band of finite area has infinitely many closed geodesics [START_REF] Bangert | Closed geodesics on complete surfaces[END_REF]. For the plane and the cylinder he proved the same result even under the weaker assumption of just the existence of a convex neighborhood of infinity. We will discuss this result in greater depth as it is used extensively in our proofs. The purpose of this article is to give simple conditions under which the existence of one or two distinct closed geodesics implies that a complete Riemannian cylinder, Möbius band or plane contains infinitely many geometrically distinct closed geodesics.

Let S 1 := R/Z and let M S 1 × R be a complete Riemannian cylinder. Let ΛM be its loop space. Two loops α, β ∈ ΛM are said to be geometrically distinct if their images are distinct: α(S 1 ) = β(S 1 ). Throughout the article, by writing that two closed geodesics are distinct we will always mean that they are geometrically distinct. Given a ring R, a closed geodesic γ ∈ ΛM is said to be homologically visible over R if the local homology of the critical circle S 1 •γ ⊂ ΛM of the energy functional is non-zero over the coefficients ring R (see Section 2 for precise definitions). With the exception of the Möbius band, every result is true over any coefficients ring R (once fixed) so the ring R will not be mentioned explicitly.

Theorem 1.1. Let M be a complete Riemannian cylinder where all closed geodesics are isolated and assume one of the following hypotheses: 1. there exists a contractible closed geodesic, 2. there exists a self-intersecting closed geodesic, [START_REF] Ballmann | Closed geodesics on positively curved manifolds[END_REF]. there exist two distinct closed geodesics that intersect, 4. there exists a closed geodesic of non-zero average index, [START_REF]On the existence of closed geodesics on two-spheres[END_REF]. there exist two homologically visible closed geodesics. Then M contains infinitely many homologically visible closed geodesics intersecting some common compact set K ⊂ M and at least one without self-intersection.

Notice that according to Bott iteration theory, a closed geodesic c has a non-zero average index if and only if some iterate c m has a non-zero index. The fact that hypothesis 5 implies that there exists infinitely many homologically visible closed geodesics proves a conjecture of Abbondandolo: Corollary 1.2. Any complete Riemannian cylinder where all closed geodesics are isolated has zero, one or infinitely many homologically visible closed geodesics.

By essentially taking the double cover (see Section 7 for details), one can thus deduce the following counterpart of Theorem 1.1 when M is a complete Möbius band.

Then M contains infinitely many homologically visible closed geodesics intersecting some common compact set K ⊂ M and at least one without self-intersection.

Corollary 1.6. Any complete Riemannian plane where all closed geodesics are isolated has zero or infinitely many homologically visible closed geodesics.

It is easy to give counter-examples to Theorem 1.1 when none of the assumptions 1-5 holds by considering embedded cylinders of revolution (θ, z) → (r(z) cos θ, r(z) sin θ, z), [START_REF] Asselle | On the existence of infinitely many closed geodesics on non-compact manifolds[END_REF] for well-chosen smooth maps r : R → (0, +∞). A complete cylinder may have no closed geodesic at all: take r > 0. It can have an arbitrary large finite number k ∈ N of homologically invisible closed geodesics: take r (z) > 0 for all z ∈ R \ {z 1 , . . . , z k } and r (z i ) = 0. It can also have a unique visible closed geodesic: take r < 0 on (-∞, 0), r (0) = 0 and r > 0 on (0, +∞) (one can as well add to this cylinder an arbitrary large finite number of homologically invisible closed geodesics the same way as before). By taking such an r even and taking the quotient under the involution (θ, z) → (θ + π, -z), one gets Möbius bands with only one homologically visible closed geodesic and as many homologically invisible closed geodesic as wanted.

Remark that in our examples closed geodesics are without self-intersections and not contractible as implied by the theorem. Counter-examples where the theorem fails by lack of completeness can be found as well by choosing embedded cylinders of revolution restricting the domain of the embedding (1) to (θ, z) ∈ R/2πZ × (a, b) for a, b ∈ R. We could proceed as follows: take an even r : [-1, 1] → (0, +∞) with r > 0 on [-1, 0) such that z = 0 is the only closed geodesic of the associated compact embedded cylinder. One can find such an r by slightly modifying a Tannery surface: a sufficient condition is that the metric g on the interior of the cylinder can be written as g = [α + h(cos ρ)] 2 dρ 2 + sin 2 ρ dθ 2 , for a good choice of coordinates (ρ, θ) ∈ (0, π) × S 1 , where α is irrational and h : (-1, 1) → (-α, α) is a smooth odd function (see for instance [START_REF] Arthur | Manifolds all of whose geodesics are closed[END_REF]Theorem 4.13]). Then extend r to a smooth map (-3, 1] → (0, +∞) with r| (-3,-1) < r(-1), r < 0 on (-3, -2) and r > 0 on (-2, -1). Then z = -2 and z = 0 are the only closed geodesic of the cylinder embedded by r| (-3,1) and are both visible.

In a similar way, we can give examples of complete planes with only an arbitrary finite number of homologically invisible closed geodesics by considering surfaces of revolution (1) parametrized by R/2πZ × [0, +∞) with r : [0, +∞) → [0, +∞) being increasing and smooth on (0, +∞) with r(0) = 0 and r (z) → +∞ when z → 0 in a suitable way (i.e. so that the surface is smooth at the origin). Then, as above, we get homologically invisible closed geodesics on the inflexion points of r, and nowhere else.

We say that

C -⊂ M (resp. C + ) is a neighborhood of -∞ (resp. of +∞) if C - contains S 1 × (-∞, a)
for some a ∈ R (resp. S 1 × (b, +∞) for some b ∈ R) for an arbitrarily fixed identification of M with S 1 × R. In these terms, Bangert proved the following theorem (where for the notion of local convexity, we refer to Section 2.2): Theorem 1.7 [START_REF] Bangert | Closed geodesics on complete surfaces[END_REF]Theorem 3,Remark 2]). Let M be a complete Riemannian cylinder where all closed geodesics are isolated and suppose there exist locally convex closed neighborhoods C -and C + with disjoint interior of -∞ and +∞ respectively such that the boundaries ∂C ± are not totally geodesic. Then M contains infinitely many (homologically visible) closed geodesics intersecting a common compact set K and at least one without self-intersections.

In order to prove Theorem 1.1, we will extensively use a variation of the above theorem adapted to our problem. The desired applications require us to work with open neighbourhoods of infinity. This in turn necessitates a slighty different convexity property to be satisfied, which we call Property (C) (and which we introduce properly in Section 2.2, where it is discussed with its connections to local convexity). To avoid technicalities, we impose some slightly stronger assumptions (which will always be satisfied in our applications) to state the following modified version of Bangert's theorem: Theorem 1.8. Let M be a complete Riemannian cylinder where all closed geodesics are isolated and suppose there exist disjoint open neighborhoods C -and C + of -∞ and +∞ respectively satisfying Property (C) and such that the boundaries ∂C ± do not contain a simple closed geodesic. Then M contains infinitely many homologically visible closed geodesics intersecting K = M \ (C -∪ C + ) and at least one without self-intersections.

Our modified statement can be proven in the same way as the original theorem. However, since Bangert did not give the precise proof of Theorem 1.7 (but rather for its analogue in case of the plane), for the sake of completeness we give a comprehensive proof of Theorem 1.8 in the paper. The proof of Theorem 1.5 is quite similar and relies extensively on the analogous theorem of Bangert when M is a plane where all closed geodesics are isolated: if there exists a locally convex open neighborhood of infinity C = M with a boundary ∂C which is not totally geodesic, M contains infinitely many homologically visible closed geodesics [4, Theorem 3]. These two theorems were originally used by Bangert to prove that any complete Riemannian plane of finite area has infinitely many closed geodesics.

In fact, our results extend verbatim to the case where M is a complete reversible Finsler manifold as we will essentially use variational properties of geodesics in our proof with no concern for geometric notion specific to Riemannian manifold. However, nothing can be said concerning the more general case of a complete (asymmetrical) Finsler manifold. The major issue is that, in the asymmetrical case, a closed subset of M which is bounded by a geodesic is not locally convex (and neither does an open such set satisfy Property (C)). In this direction, we point out that the related question of whether or not infinitely many closed geodesics exist on every irreversible Finsler cylinder of finite area is still open [START_REF] Burns | Open problems and questions about geodesics[END_REF]Question 2.3.2].

In order to put these results in perspective, we recall some known results concerning existence of closed geodesics on complete non-compact Riemannian manifolds. In 1978, Thorbergsson proved the existence of closed geodesics on a complete Riemannian manifold M if it contains a convex compact set which is not homotopically trivial or if M has a non-negative sectional curvature outside some compact set [START_REF] Thorbergsson | Closed geodesics on non-compact Riemannian manifolds[END_REF]. In the 1990s, Benci and Giannoni proved that any complete d-dimensional Riemannian manifold such that the limit superior of its sectional curvature at infinity is non-positive and the homology of its free loop space is non-trivial in some degree larger than 2d possesses a closed geodesic [START_REF] Benci | Closed geodesics on noncompact Riemannian manifolds[END_REF][START_REF]On the existence of closed geodesics on noncompact Riemannian manifolds[END_REF]. In 2017, Asselle and Mazzucchelli showed the existence of infinitely many closed geodesics for complete d-dimensional Riemannian manifolds which have no close conjugate points at infinity and a free loop space with unbounded Betti numbers in degrees larger than d [START_REF] Asselle | On the existence of infinitely many closed geodesics on non-compact manifolds[END_REF]. They also improved the result of Benci and Giannoni by replacing the asymptotic curvature assumption by an assumption on the conjugated points at infinity and by improving the bound on the homology of the free loop space. The existence of one closed geodesic in any complete Riemannian manifold of finite volume is a hard open problem (see for instance the following recent review of the subject [START_REF] Burns | Open problems and questions about geodesics[END_REF]).

Organization of the paper. In Section 2, we fix notation and recall results of the variational theory of geodesics that we will need. In Section 3, we give a comprehensive proof of Theorem 1.8. In Section 4, we prove Theorem 1.1 when hypothesis 1, 2 or 3 is assumed. In Section 5, we prove Theorem 1.1 when hypothesis 4 is assumed. In Section 6, we prove the last case of Theorem 1.1. In Section 7, we prove Corollary 1.3. In Section 8, we prove Theorem 1.5.

Acknowledgments. This work started while the first author was visiting the Ruhr-Universität Bochum (Germany) in October 2019. The first author wishes to thank Alberto Abbondandolo and Stefan Suhr for the invitation, and the Ruhr-Universität Bochum for providing an excellent working environment. The first author is also grateful to his advisor Marco Mazzucchelli who introduced him to the result conjectured by Alberto Abbondandolo. To the latter the second author expresses his gratitude for his continuing help and advice. The second author is partially supported by the SFB/TRR 191 'Symplectic Structures in Geometry, Algebra and Dynamics', funded by the DFG (Projektnummer 281071066 -TRR 191). Both authors are thankful to the anonymous referee for his careful reading of the preprint and for providing important comments and suggestions.

Preliminaries

In this section, we recall some results of Riemannian geometry that we will use in our proofs and fix some notation. For the extension of these notions to the Finsler case, the reader may for instance look at [14, Section 2].

2.1. The energy functional. Given a complete Riemannian manifold with boundary W , we denote by ΛW the space of H 1 -maps S 1 → W . In fact, if one wants to avoid analytic questions, we can always reduce our space to a finite-dimensional manifold of broken geodesics. For γ ∈ ΛW and m ∈ N * , the iterated loop γ m ∈ ΛW is defined by t → γ(mt). A geodesic is an immersed path γ : R → W such that

∇ γ γ = 0,
where ∇ denotes the Levi-Civita connection of the metric and γ stands for the derivative of γ. Therefore, in our convention, geodesics have constant speed. A closed geodesic is a geodesic γ which is periodic: γ(t + 1) = γ(t) so that γ ∈ ΛW . Closed geodesics of W are the critical points with non-zero critical value of the energy functional E : ΛW → [0, +∞),

E(γ) := S 1 g γ ( γ, γ)dt, ∀γ ∈ ΛW.
The energy functional E is C 2 . If W is a compact manifold (possibly with boundary), E also satisfies the Palais-Smale condition and the (-∇E)-flow is defined for all time t ≥ 0 (if f is a real-valued on a Riemann-Hilbert manifold, ∇f denotes its gradient). We notice that every closed geodesic lies on a critical circle S 1 • γ, where S 1 acts on ΛW by t • γ := γ(t + •). In our study we assume that E has only isolated critical circles (except for the constant loops which have zero value). Two closed geodesics c 1 and c 2 are said to be geometrically distinct if they do not have the same image in W .

2.2. Finite-dimensional approximation of the loop space. Morse's finitedimensional approximation of the curve space over W , as presented by Bangert in [START_REF] Bangert | Closed geodesics on complete surfaces[END_REF] consists of the following data: an open set O ⊂ W , an energy bound κ > 0 and a parameter j ∈ N satisfying 1 j < ε 2 κ where ε > 0 is smaller than the injectivity radius on O. The positivity of ε will be fulfilled if for instance O has compact closure, as will be the case in our considerations. The finite-dimensional approximation Ω = Ω(O, κ, j) is constructed as follows: it is the set of all curves γ ∈ ΛW such that E(γ) < κ, γ(i/j) ∈ O and such that γ| [i/j,(i+1)/j] is a geodesic of length less than ε for 0

≤ i ≤ j -1.
We call a closed subset C of a manifold locally convex, if every point possesses a neighborhood U ⊂ C such that any two sufficiently close points in U can be joined by a unique geodesic which is entirely contained in U . We only define local convexity for closed sets, as open sets are always satisfying this property. Let Ω be a finite-dimensional approximation of ΛW and C ⊂ W a closed locally convex set with compact boundary such that C ⊂ O. Then there exists an ε > 0 such that for any two points p, q ∈ C with Riemannian distance d(p, q) < ε, if there exists a unique geodesic of length = d(p, q) joining p and q, it is contained entirely in C [4, p. 85]. The negative gradient of the restriction of the energy functional to Ω is given by [20, p. 252]). Now from our choice of j and Cauchy-Schwarz inequality, we get

-∇E| Ω (γ) = -2 γ1 (1/j) -γ2 (1/j), . . . , γj-1 ((j -1)/j) -γj ((j -1)/j) for γ ∈ Ω, where γ i = γ| [(i-1)/j,i/j] for 1 ≤ i ≤ j (see
d(γ((i -1)/j), γ(i/j)) 2 ≤ 1 j E(γ| ((i-1)/j,i/j) ) ≤ ε 2 κ κ = ε 2 .
Consequently, by local convexity of C, the negative gradient flow of the finite-dimensional approximation of the energy functional respects C, i.e. γ(S 1 ) ⊂ C implies φ t (γ)(S 1 ) ⊂ C for all t ≥ 0, where φ t denotes the negative gradient flow (in finitedimensional approximation).

We would like to use this property on slightly more general sets. Therefore we denote B r (p) := {q ∈ W | d(p, q) < r} and state the following property for a subset C ⊂ W .

There exists an ε > 0 such that if p, q are elements in the same connected component of B ε (p) ∩ int(C), then whenever a unique geodesic connecting p and q exists, it is contained in that connected component.

(C)

Notice, that closed locally convex sets with compact boundary satisfy this property. Moreover, so do open sets whose boundary coincides with a broken geodesic with convex vertices. The above discussion carries over and we have that the negative gradient flow of the finite-dimensional approximation of the energy functional respects sets that satisfy Property (C).

Index of a closed geodesic. The index of a closed geodesic γ is the Morse index of E:

ind(γ) := ind(E, γ). It is always finite. The behavior of this index under iteration k → ind(γ k ) was extensively studied by Bott in [START_REF] Bott | On the iteration of closed geodesics and the Sturm intersection theory[END_REF]. We simply recall that

ind(γ k ) ≥ k ind(γ) -dim(W ) + 1, k ∈ N, (2)
where ind(γ) ≥ 0 is the average index of γ defined by

ind(γ) := lim k→∞ ind(γ k ) k .
Let p ∈ W and Ω p W ⊂ ΛW be the set of loops based at p, that is

H 1 -paths γ : [0, 1] → W such that γ(0) = γ(1) = p. Given a closed geodesic γ ∈ ΛW , we denote by ind Ω (γ) ∈ N the Morse index ind Ω (γ) := ind E| Ω γ(0) W , γ .
By inclusion, ind Ω (γ) ≤ ind(γ). In fact, we have the concavity inequality [3, Eq. (1.5)]:

ind(γ) -dim(W ) + 1 ≤ ind Ω (γ) ≤ ind(γ).
(3) A Jacobi field of the geodesic path γ is a smooth map J : R → γ * T W , satisfying

J(t) ∈ T γ(t) W, ∀t ∈ R and ∇ 2 γ J = R( γ, J) γ
, where R denotes the Riemann tensor. Let µ(t) be the number of linearly independent Jacobi fields of γ such that J(0) = J(t) = 0; the Morse index theorem states that ind Ω (γ) = 0<t<1 µ(t).

(4)

The local homology of an isolated critical circle S 1 • γ over a ring R is by definition

C * (S 1 • γ; R) := H * ({E < E(γ)} ∪ S 1 • γ, {E < E(γ)}; R),
where the set

{E < E(γ)} is {δ ∈ ΛW | E(δ) < E(γ)},
C * (S 1 • γ) H * U ∩ {E < E(γ)} ∪ S 1 • γ , U ∩ {E < E(γ)} . ( 5
)
Therefore, every local minimum of E is homologically visible. We will be interested in the properties of the local homology C * (S 1 • γ) especially in the case where γ is a closed geodesic of average index ind(γ) = 0 and whose image γ(S 1 ) lies in the interior of W (ind(γ) = 0 is equivalent to the fact that ind(γ m ) vanishes for all m ≥ 1). Let γ ∈ ΛW be such a closed geodesic. Given m ∈ N, we denote by ψ m : ΛW → ΛW the iteration map ψ m (δ) := δ m . According to a theorem of Gromoll-Meyer [22, Theorem 3], the local homology C d (S 1 • γ) is zero in degrees d ≥ 2 dim W and there exist infinitely many positive integers m such that the induced map in homology

(ψ m ) * : C * (S 1 • γ) → C * (S 1 • γ m ) (6)
is an isomorphism. On the other hand, a theorem of Bangert-Klingenberg [6, Corollary 1] states that there exists m 0 ∈ N above which for all m ≥ m 0 , there exists e m > m 2 E(γ) such that the composition

C * (S 1 • γ) (ψm) * ---→ C * (S 1 • γ m ) inc * --→ H * E < e m , E < m 2 E(γ) ( 7
)
is zero.

Proof of the adapted Bangert theorem

A closed geodesic γ is a mountain pass if, for all neighborhoods U ⊂ ΛM of

S 1 • γ, U ∩ E -1 ([0, E(γ)) is not connected.
For the proof of Theorem 1.8, we need the following statement, which tells us that isolated closed geodesics cannot remain mountain pass critical points of the energy functional when sufficiently iterated. A geometric proof is given by Bangert [START_REF] Bangert | Closed geodesics on complete surfaces[END_REF]. Theorem 2]). Let γ be an isolated closed geodesic on M , where dim M = 2. Then there exists an integer m γ ∈ N such that the following holds: For all integer m ∈ N with m ≥ m γ , there is a neighborhood

Theorem 3.1 ([4,
U of S 1 • γ in ΛM such that U ∩ E -1 ([0, E(γ m ))) is connected.
According to Gromoll-Meyer [START_REF]Periodic geodesics on compact riemannian manifolds[END_REF], given an isolated closed geodesic γ, there exists a connected neighborhood U ⊂ ΛM of the critical circle

S 1 • γ such that C * (S 1 • γ) H * U, U ∩ E -1 [0, E(γ)) .
If γ and all its iterates are homologically invisible, Theorem 3.1 is thus true for m γ = 1.

Proof of Theorem 1.8. Assume there are only finitely many prime closed geodesics γ 1 , . . . , γ k which have homologically visible iterates and which intersect W := M \ (C -∪ C + ). We will now derive a contradiction from this assumption. We will define a suitable finite-dimensional approximation Ω = Ω(O, κ, j). Now as the statement of Theorem 3.1 remains true in a finite-dimensional approximation, we get that there exists m 0 ∈ N such that for all integers m ≥ m 0 and for all i ∈ {1, . . . , k} the following holds: i) There exists a neighborhood

U of S 1 • γ m i in Ω such that U ∩ E -1 ([0, E(γ m i ))) is connected. Set A := max{E(γ m 0 i ) | i ∈ {1, .
. . , k}}, and notice that A is larger than the energy of a closed geodesic of mountain pass type. We fix an identification of π 1 (M ) with Z and denote by [γ] ∈ Z the class of a loop γ ∈ ΛM . We define the following sets of curves:

P ± n := {γ ∈ Ω | γ(S 1 ) ⊂ C ± , [γ] = n}.
In the following for each U, V ⊂ M , we will denote dist(U, V ) := inf x∈U, y∈V

d(x, y).

Choose δ > 0. Then there exists an n ∈ N such that for any curve γ ∈ P ± n satisfying dist(γ(S 1 ), W ) < δ it holds that E(γ) ≥ A. We can now say how exactly the finite-dimensional approximation has to be chosen:

• Choose a κ > 0 large enough such that there exists a homotopy h :

[0, 1] → E -1 ([0, κ)) in Ω from h 0 ∈ P - n to h 1 ∈ P + n with dist h t (S 1 ), W < δ, ∀t ∈ [0, 1]. • Set O := {p ∈ M | dist(p, W ) < R}, where R > 2κ 1 2 +δ such that O contains γ 1 , . . . , γ k .
• Choose j such that the (-∇E)-flow of the finite-dimensional approximation respects C ± , as described above. A technical issue is given by the fact that the gradient flow of -∇E may not be defined for all times as the sublevel sets of E| Ω are not compact. Ultimately we are only going to be interested in curves intersecting the compact set W , i.e. the subset

K := {γ ∈ Ω | γ(S 1 ) ∩ W = ∅}
of Ω. We introduce a smooth function g : Ω → [0, 1] with the property that

   g(γ) = 1 if dist(γ(S 1 ), W ) ≤ 1 2 κ 1 2 , g(γ) = 0 if dist(γ(S 1 ), W ) > 3 2 κ 1 2
. Then the flow φ t of -g∇E is defined for all times t ≥ 0 and coincides with the negative gradient flow for curves in K. Two crucial observations about the set K are the following: firstly, for all κ < κ the set K ∩ E -1 ([0, κ]) is compact. Secondly, if φ t (γ) ∈ K for some γ ∈ Ω and some time t ≥ 0, we already have γ ∈ K as the flow φ t respects the sets C ± (as those satisfy Property (C)). From this it follows:

ii) Let 0 < κ 0 < κ 0 + ε < κ. Let V denote a neighborhood of the closed geodesics in K of energy κ 0 . Suppose there is no closed geodesic in K ∩ E -1 ((κ 0 , κ 0 + ε]).

Then there exists a time τ > 0, such that

φ τ E -1 ([0, κ 0 + ε]) ∩ K ⊂ E -1 ([0, κ 0 )) ∪ V.
This is just the deformation lemma; for a proof see for instance [START_REF] Struwe | Variational methods[END_REF]Lemma 3.4]. We are now set to complete the proof of the theorem. Define the set of homotopies

Π := β : [0, 1] → Ω continuous | β 0 ∈ P - n , β 1 ∈ P + n .
Note that Π is not empty, as h ∈ Π. Furthermore, φ t • β ∈ Π for all β ∈ Π and all t ≥ 0 as the flow respects the sets C ± and therefore φ t (β 0 ) ∈ P - n and φ t (β 1 ) ∈ P + n for all t ≥ 0. Define now

κ 0 := inf β∈Π max t∈[0,1] βt∈K E(β t ) .
By definition of κ, one has κ 0 < κ. For every β ∈ Π for time t 0 := min{t ∈ [0, 1] | β t / ∈ P - n } it holds that β t 0 ∈ K and E(β t 0 ) ≥ A (as E and β are continuous and there exists a sequence (t k ) t 0 such that β t k ∈ P - n and dist(β t k , W ) < δ). Consequently, we get κ 0 ≥ A. Since κ 0 < κ, for ε > 0 small enough, the subset K ∩ E -1 ([0, κ 0 + ε]) is compact and there are only finitely many S 1 -orbits of closed geodesics inside (we assumed every orbit to be isolated). Let {S 1 • d j } 1≤j≤l denote the critical circles of energy κ 0 in K. By definition of A and by using i) when d j is a power of some γ i (otherwise this is true according to the remark just before the beginning of the proof), there exist disjoint neighborhoods U j of the S 1 • d j 's such that U j ∩ E -1 ([0, κ 0 )) is connected for all j. We now want to argue that we can choose U j ⊂ K. We first argue, that d j ∈ ∂K implies d j (S 1 ) ⊂ ∂C ± . To that end suppose that d j ∈ ∂K but it is not contained in ∂C ± . Then d j | [-a,0) is contained in C ± for some a ∈ (0, /2) (where this denotes the constant coming from Property (C)) and d j (0) ∈ ∂C ± . There are two cases:

Either d j (b n ) ∈ C ± for some infinitesimal sequence b n > 0 or d j | [0,b] is contained in ∂C ± for some b ∈ (0, /2).
In the first case: d j (b n ) and d j (-a) belong to the same connected component of C ± ∩ B (δ(-a)) since there is a curve arbitrarily close to d j which is contained in C ± (as d j ∈ ∂K). However, the shortest geodesic connecting d j (b n ) and d j (-a) is not contained in C ± which contradicts Property C. In the second case: let γ be a short geodesic segment which is transverse to d j passing through d j (-a) and contained in C ± . Then since d j is in ∂K there is a curve completely contained in C ± connecting a point x arbitrarily close to d j (b) and a point on γ. This means that x is in the same connected component of C ± ∩ B (x) as are all the points of γ. If x is close enough to d j (b) however, one can find a short geodesic segment from x to a point on γ passing through d j (0). Therefore this geodesic segment is not contained in C ± thus contradicting Property C. If d j is an iterate of a simple closed geodesic, by the above it cannot be contained in ∂K since by assumption ∂C ± do not contain simple closed geodesics. Conversely, if it is not an iterate, it has a (transverse) self-intersection. Then all the curves close to d j will also intersect d j and therefore ∂C ± . Consequently, in this case U j is also contained in K. Now because there are only finitely many closed geodesics in K ∩ E -1 ([0, κ 0 + ε]) for ε > 0 small enough, one can take ε > 0 such that there is no closed geodesic in K ∩ E -1 ((κ 0 , κ 0 + ε]). By the definition of κ 0 there exists a homotopy β ∈ Π satisfying E(β t ) ≤ κ 0 + ε for all t ∈ [0, 1] such that β t ∈ K. Choose neighborhoods V j of S 1 • d j such that V j ⊂ int(U j ) and use property ii) on the neighborhood V := l k=1 V j of closed geodesics of energy κ 0 in K to obtain a τ > 0 with the property that for the homotopy φ τ • β we have that (

φ τ • β) t ∈ K implies E((φ τ • β) t ) < κ 0 or (φ τ • β) t ∈ V . Now (φ τ • β) -1 (V ) = m r=1 (t r , t r )
and by our choice of the V j we have (φ τ • β)([t r , t r ]) ⊂ U j and for the endpoints (φ τ • β) tr , (φ τ • β) t r ∈ U j ∩ E -1 ([0, κ 0 )) for some j ∈ {1, . . . , l} (which is why we applied ii) only to V and not to l j=1 U j directly). Now, by using i) if d j a power of some γ i (otherwise it is true by the remark just before the beginning of the proof), we know that U j ∩ E -1 ([0, κ 0 )) is connected and consequently we can replace (φ τ • β)| [tr,t r ] by a path in E -1 ([0, κ 0 )) with the same endpoints. After m steps we obtain a homotopy β :

[0, 1] → Ω such that E( βt ) < κ 0 when βt ∈ K. Since (φ τ • β) 0 , (φ τ • β) 1 / ∈ K it follows that (φ τ • β) 0 , (φ τ • β) 1 /
∈ l j=1 U j and therefore β0 ∈ P - n , β1 ∈ P + n , hence β ∈ Π. This contradicts the minimality of κ 0 . A closed geodesic without self-intersections can be constructed like in [4, Theorem 1]. One should note that this requires a curve shortening process which takes simple curves into simple curves. Alternatively to the one by Lusternik and Schnirelmann quoted by Bangert, one could also use the more recent Grayson-Oaks curve shortening [START_REF] Grayson | Shortening embedded curves[END_REF][START_REF] Oaks | Singularities and self-intersections of curves evolving on surfaces[END_REF].

Contractible and intersecting closed geodesics

Here M still denotes a complete Riemannian cylinder. We assume that there exists a contractible closed geodesic c ∈ ΛM . Let us consider the unbounded components of M \ c(S 1 ). Since c(S 1 ) is bounded, there are at most two distinct unbounded components. If there are two distinct unbounded components C -and C + , one can assume that C -is a neighborhood of -∞ and C + is a neighborhood of +∞. By C ± we will mean any of these two neighborhoods. Then ∂C ± is a broken geodesic with angles strictly less than π inside C ± since c is a closed geodesic (see Figure 2 for an instance of ∂C + ). Hence C ± is satisfies Property (C). Moreover if the boundary were a simple closed geodesic, then ∂C ± would be parametrised by c which is impossible for c is contractible. We can thus apply Theorem 1.8 in this case.

We now assume that M \ c(S 1 ) has only one unbounded component C. Let us identify M with S 1 × R in the remainder of this proof in order to fix the notation. Let π : R 2 → S 1 ×R be the universal cover of S 1 ×R. By compactness of c(S 1 ), there exists A > 0 such that c(S 1 ) ⊂ S 1 × (-A, A). Let y 0 > A, since S 1 × (-∞, -A) and S 1 × (A, +∞) belong to the same component of M \ c(S 1 ), there exists a smooth path α : [0, 1] → M \ c(S 1 ) such that α(0) = (0, -y 0 ) and α(1) = (0, y 0 ). Let β 0 be the smooth lift of α in R 2 such that β 0 (0) = (0, -y 0 ) and β 0 (1) = (n 0 , y 0 ) for some n 0 ∈ Z that we can take equal to n 0 = 0 by chaining α with t → (tn 0 mod 1, y 0 ). Let δ n,± : [0, 1] → R 2 be the path t → (nt, ±y 0 ) and β n : [0, 1] → R 2 be the family of lifts β n := (n, 0) + β 0 , n ∈ N. We define the family of loops γn ∈ ΛR 2 by γn :

= β 0 • δ n,+ • β -1 n • δ -1 n,-. They project to γ n := π • γn in M \ c(S 1
). Let q 0 ∈ R 2 be a lift of some point of c(S 1 ) and define q n := q 0 + (n, 0). Then the first homology group H 1 (R 2 \ {q n } n∈Z ) is the free abelian group with generators (g n ) n∈Z , and by construction the class of γn is

g 1 + g 2 + • • • + g n . The covering transformations of R 2 \ {q n } n∈Z → S 1 × R \ π(q 0 )
, which form a group isomorphic to Z, act on the first homology group by k •g i = g i+k . Therefore, for natural integers n = m and integers k, l ∈ Z * , the fact that

k(g 1+a + g 2+a + • • • + g n+a ) = l(g 1+b + g 2+b + • • • + g m+b ), ∀a, b ∈ Z,

E(γ).

Let K ⊂ M be a compact set that contains c(S 1 ) such that M \ K has two distinct unbounded component. Since any γ ∈ ΛC that is freely homotopic to γ m must intersect K, one can restrict the domain of the infimum to those γ which image is inside the compact set L ⊂ M of points that are at distance at most √ e m of K (which is compact by completeness of the metric on M ). Indeed, if γ were a loop of length ≥ 2 √ e m then E(γ) ≥ 4e m by Cauchy-Schwarz inequality. By compactness of L, we can use a finite-dimensional approximation to get a closed geodesic c m on C ∪ c(S 1 ) with E(c m ) = e m that is a limit of broken geodesics on C freely homotopic to γ m . By uniqueness of the Cauchy problem, if c m intersect c(S 1 ), the closed geodesic must be a power of c (up to a translation of the parametrisation). This is impossible since the powers of c are not in the closure of {γ ∈ ΛC | [γ] = [γ m ]} for m ≥ 2 (such γ's must intersect every line joining both ends ±∞ of M ). Therefore, the above infimum is reached by the closed geodesic c m ∈ ΛC. We thus get a family of closed geodesics (c m ) such that [c k m ] = [c l n ] for all k, l ∈ Z * and m = n. Therefore the closed geodesics (c m ) are geometrically distinct. They all intersect the compact set K. As a local minimum, every c m is homologically visible. A closed geodesic without self-intersection can be found be choosing a simple closed curve in C which is close to ∂C and by applying Grayson-Oaks curve shortening [START_REF] Grayson | Shortening embedded curves[END_REF][START_REF] Oaks | Singularities and self-intersections of curves evolving on surfaces[END_REF] to this curve. Now that Theorem 1.1 is proved under hypothesis 1, in order to prove it when there is one self-intersecting closed geodesic c or two intersecting ones c 1 and c 2 , one can assume that these geodesics are not contractible. Therefore, in both respective cases, M \c(S 1 ) or M \(c 1 (S 1 )∪c 2 (S 1 )) has exactly two unbounded connected components C -and C + , which satisfy Property (C) by construction. The intersection hypothesis then implies that none of the boundaries ∂C ± is a simple closed geodesic. Hence the conclusion follows by applying Theorem 1.8.

Closed geodesic of non-zero average index

We assume that there exists a closed geodesic c ∈ ΛM of average index ind(c) > 0. If c is contractible or self-intersecting, we already know that there are infinitely many homologically visible closed geodesics. Let us assume that c is an embedded curve generating π 1 (M ) Z. By a slight abuse of notation, we identify the loop c : S 1 → M with its lift R → M . Lemma 5.1. There exist a non-zero Jacobi field J : R → c * T M of c and δ > 0 such that J(s) = 0 for all s ∈ (0, δ) and J(0) = J(δ) = 0.

Proof. Since ind(c) > 0, Bott iteration inequality (2) and the concavity bound (3) imply that there exists k ∈ N * such that ind Ω (c k ) ≥ 1.

Let us fix such a k ≥ 1. The conclusion is now a direct application of the Morse index theorem (4) to the geodesic path c k . In order to fix notation, let us identify the image of the loop c to S 1 × {0}, with c(s) = (s, 0) for s ∈ S 1 , so that M \ c(S 1 ) is the disjoint union of the neighborhood S 1 × (-∞, 0) of -∞ and the neighborhood S 1 × (0, +∞) of +∞ (we only need this identification to be a homeomorphism). Let J : R → c * T M and δ > 0 be the Jacobi field and the positive number given by Lemma 5.1. Let ε > 0 and I := (-ε, δ + ε). Since there exists a smooth family (β s ) s∈(-1,1) of geodesic paths I → M such that J| I = ∂βs ∂s | s=0 , it implies that there exists a geodesic path α : [0, 1] → S 1 × [0, +∞) intersecting c (transversally) only at its endpoints.

By construction, the unbounded component

C + of S 1 × (0, +∞) \ α([0, 1]
) has a boundary which is a broken geodesic with angles strictly less than π. By symmetry, we get two disjoint neighborhoods of +∞ and -∞ respectively which are locally convex and whose boundaries are not totally geodesic, we can thus apply Theorem 1.8.

Two homologically visible closed geodesics

Here M denotes a complete Riemannian cylinder. We fix an identification of π 1 (M ) with Z and denote by [γ] ∈ Z the class of a loop γ ∈ ΛM . We assume that there exist two geometrically distinct and homologically visible closed geodesics. We suppose by contradiction that for any compact set K ⊂ M only a finite number of geometrically distinct homologically visible closed geodesics intersect K. By the previous cases of Theorem 1.1, every prime closed geodesic of M must be embedded, non-contractible, without intersections with another closed geodesic, and of zero average index. Thus the images of closed geodesics of M S 1 × R with a homologically visible iterate are naturally ordered by their smallest intersection with * × R where * denotes any point of S 1 . The order is independent of the choice of * ∈ S 1 . We will say that two closed geodesics are consecutive if they are so with respect to this order. Since only a finite number of geometrically distinct homologically visible closed geodesic intersect a given compact set, one can talk about the next and the previous one with respect to this order. Lemma 6.1. There exist two closed embedded geodesics c 1 and c 2 of M with degree Proof. We first show that two consecutive closed among closed geodesics that possess homologically visible iterates cannot be both local minima of E| ΛC if C is the compact cylinder that they bound. By contradiction, let us assume so and let us call γ 0 and γ 1 these two geodesics. Up to a change of parametrization, one can assume that [γ 0 ] = [γ 1 ] and thus that these two geodesics are homotopic in ΛC . Let Π := {h : [0, 1] → ΛC continuous | h(0) = γ 0 and h(1) = γ 1 } denote the set of homotopy of loops in C starting at γ 0 and ending at γ 1 . We consider the following min-max:

[c 1 ] = [c 2 ] = 1 bounding a compact locally convex cylinder C S 1 × [0, 1] such that (1) c 1 is a local minimum of E| ΛC , (2) c 2 is not a local minimum of E| ΛC , (3)
τ = inf h∈Π max E • h.
By compactness of C , E| ΛC satisfies Palais-Smale (alternatively, one can work in the compact finite-dimensional manifold of k-broken-geodesics of energy ≤ c+ε for a large k ∈ N and ε > 0). Let e := max(E(γ 0 ), E(γ 1 )). Since the critical orbits S 1 • γ 0 and S 1 • γ 1 are isolated local minima of E| ΛC that satisfies Palais-Smale, τ > e. By local convexity of C , the (-∇E)-flow preserves ΛC . By the minimax principle, τ is thus a critical value of E| ΛC and there exists a homologically visible closed geodesic γ ∈ ΛC of energy τ . Hence γ 0 and γ 1 are not consecutive, a contradiction.

By a similar argument, we show that one out of two consecutive closed geodesics among those that possess homologically visible iterates is a local minimum of E| ΛC . Indeed, otherwise one has that inf

γ∈ΛC [γ]=1 E(γ) < min(E(γ 0 ), E(γ 1 )),
and this infimum is reached for some closed geodesic in C by compactness and local convexity of C (and this is not a point since E(γ) ≥ (2r) 2 for all γ ∈ ΛC of degree [γ] = 1 where r > 0 denotes the injectivity radius of the compact Riemannian manifold with boundary C ). This new closed geodesic is a local minimum of E by definition and thus homologically visible.

The requirements of the lemma are thus fulfilled by taking any two consecutive closed geodesics among those with homologically visible iterates.

Proof of Theorem 1.1. Let c 1 and c 2 be closed geodesics of M satisfying Lemma 6.1. We will reach a contradiction by finding a homologically visible geodesic which is not c 1 or c 2 and arbitrarily close to C.

Let x ∈ Int(C) be outside the image of the isolated set of closed geodesics and let γ 1 ∈ ΛC be the loop of degree [γ 1 ] = 1 based at x of minimal length. It exists by local convexity and compactness of C. The loop γ 1 is not a periodic geodesic (this is a geodesic as a path [0, 1] → C but not as a loop S 1 → C) by our specific choice of x. This loop lies inside Int(C) so that either the connected component of C \ γ 1 (S 1 ) containing c 1 or the connected component containing c 2 is locally convex -depending on the angle of γ 1 at γ 1 (0) = γ 1 (1) = x. If the connected component containing c 2 were locally convex, then the infimum of E among loops of degree one lying inside the locally convex compact cylinder bounded by γ 1 and c 2 would give a closed geodesic loop = c 1 which would be a local minimum. Thus the connected component of C \ γ 1 (S 1 ) containing c 1 is a locally convex compact cylinder. Hence the unbounded component of M \ γ 1 (S 1 ) containing c 1 is a locally convex neighborhood of -∞ which is not totally geodesic since γ 1 is not a closed geodesic. ) is thus locally convex, depending on the angle of γ 2 at γ 2 (0) = γ 2 (1) = y. If the neighborhood of +∞ was the locally convex one, by Theorem 1.8 applied to the locally convex neighborhood of -∞ defined above with γ 1 and this neighborhood of +∞, there would be infinitely many homologically visible and geometrically distinct closed geodesics intersecting some compact set of M . Thus the neighborhood of -∞ is the locally convex unbounded component of M \ γ 2 (S 1 ). Restricting this neighborhood to the compact cylinder C ∪ C , one gets a compact locally convex cylinder Z intersecting only two geodesics c 1 and c 2 that possess homologically visible iterates, moreover c 1 (S 1 ) ⊂ ∂Z and c 2 (S 1 ) ⊂ Int(Z).

Let k ∈ N * be such that C * (S 1 • c k 2 ) = 0. Let Λ h ⊂ ΛZ be the connected component of loops γ ∈ ΛZ of degree [γ] = h. For all m ∈ N * , let ψ m : Λ k → Λ km be the iteration map ψ m (γ) := γ m . According to the Bangert-Klingenberg theorem [START_REF] Benci | Closed geodesics on noncompact Riemannian manifolds[END_REF], there exists m 0 ∈ N above which for all m ≥ m 0 there exists e m > m 2 E(c k 2 ) such that the composition

C * (S 1 • c k 2 ) (ψm) * ---→ C * (S 1 • c km 2 ) inc * --→ H * E| Λ km < e m , E| Λ km < m 2 E(c k 2 ) is zero. According to the Gromoll-Meyer theorem (6), since ind(c k 2 ) = k ind(c 2 ) = 0, there exist infinitely many m such that (ψ m ) * : C * (S 1 • c k 2 ) → C * (S 1 • c km 2
) is an isomorphism. Let m ≥ m 0 be such an integer, then the inclusion induces a zero map

C * (S 1 • c km 2 ) inc * --→ H * E| Λ km < e m , E| Λ km < m 2 E(c k 2 )
, which contradicts the fact that c km 2 ) E(c km 1 ) since c 1 is the only local minimum in C). Since Z is locally convex, the (-∇E)-flow preserves ΛZ. Moreover Z is compact and has only isolated closed geodesics, we can thus apply Morse theoretical arguments since E| Λ km has isolated critical circles and satisfies Palais-Smale or, alternatively, one can restrict E to the finite-dimensional subspace of j-broken-geodesics of Λ km of energy less than e m + ε for some large j ∈ N and some ε > 0. Thus, if S 1 • c km 2 were the only homologically visible critical circle of energy ≥ m 2 E(c k 2 ), Morse deformation lemma would imply inc * to be an isomorphism.

The case of the Möbius band

Assuming Theorem 1.1 concerning complete cylinders, we deduce Corollary 1.3. Let (M, g) be a complete Möbius band and let us denote by π : M → M its connected double cover. Hence, ( M , g) with g := π * g is a complete cylinder. Let us denote by E : ΛM → R and E : Λ M → R the respective energy functionals of M and M . Any closed geodesic of M is covered by one or two closed geodesics of M . The proof would be obvious if the homological visibility of one of the iterates of the geodesic on M were equivalent to the homological visibility of one of the iterates of the covering geodesics. However, it is not clear whether this is the case when only one closed geodesic covers the closed geodesic on M . We will see that the Smith inequality (8) will yield equivalence over the field R = Z/2Z.

In the statements of the following two lemmas, we use the above notation M , M , E, E where π : M → M denotes any Riemannian cover of some Riemannian manifold M .

: Λ M → ΛM , γ → π • γ, induces an isomorphism C * (S 1 • c) -→ C * (S 1 • c).

Moreover, ind(c) = ind(c).

Proof. Since π is a covering map, the map π is a diffeomorphism in a small neighborhood U of S 1 • c by the uniqueness of the lift to U of a loop belonging to the neighborhood U Given a group G acting on a space X, let X G ⊂ X be the set of fixed points of G. According to the Smith inequality,

:= π ( U ) of S 1 • c. Since E = E • π ,
dim H * X; Z/pZ ≥ dim H * X Z/pZ ; Z/pZ , ( 8
)
where X is a locally compact space or pair such that H * (X; Z/pZ) is finitely generated, a space on which acts the group Z/pZ with p prime (see for instance [11, Chapter IV, §4.1]). Here dim H * denotes the total dimension k dim H k . The following lemma is a counterpart of a result of Çineli-Ginzburg relating the local homologies of a Hamiltonian orbit and its p-iterate [START_REF] Çineli | On the iterated hamiltonian floer homology[END_REF].

Lemma 7.2. For every isolated closed geodesic c ∈ ΛM and every prime number p,

dim C * (S 1 • c p ; Z/pZ) ≥ dim C * (S 1 • c; Z/pZ).
us notice that this last inequality is an equality when p is large enough according to Gromoll-Meyer theory [START_REF]Periodic geodesics on compact riemannian manifolds[END_REF]Theorem 3] (the coefficients field Z/pZ can be replaced by any ring R in this case).

Proof. Since the local homology of S 1 • c only depends on a small neighborhood of S 1 •c (local property (5)), one can assume that M is a closed manifold. Let X ⊂ ΛM be the topological pair

X := {E < E(c p )} ∪ S 1 • c p , {E < E(c p )} .
This pair retracts on a locally compact pair by using a finite-dimensional approximation. According to the Gromoll-Meyer theory, the homology group H * (X; Z/pZ) = C * (S 1 • c; Z/pZ) is finitely generated (see Section 2). By viewing Z/pZ as the subgroup of p-th roots of unity, Z/pZ ⊂ S 1 acts on ΛM . This action preserves the sublevel sets of E so it preserves X and γ → γ p induces a homeomorphism

{E < E(c)} ∪ S 1 • c, {E < E(c)} -→ X Z/pZ .
This is now a direct consequence of the Smith inequality [START_REF]On the existence of closed geodesics on noncompact Riemannian manifolds[END_REF].

Proof of Corollary 1.3. Let π : M → M be the connected double cover of the complete Möbius band M . Let us identify H 1 (M ; Z) and H 1 ( M ; Z) with Z, so that the induced morphism π * : H 1 ( M ; Z) → H 1 (M ; Z) is the multiplication by 2. Given a closed geodesic γ ∈ ΛM , we denote by [γ] ∈ Z its homology class. By the lifting property of covers, there exists γ ∈ Λ M such that γ = π • γ if and only if [γ] is even (we recall that π 1 (M ) H 1 (M ; Z) for M and for M as well).

A contractible closed geodesic of M is covered by contractible closed geodesics of M thus hypothesis 1 on M implies hypothesis 1 on M . A self-intersecting closed geodesic of M is either covered by a self-intersecting closed geodesic or two intersecting closed geodesics of M thus hypothesis 2 on M either implies hypothesis 2 or hypothesis 3 on M . Two intersecting closed geodesics on M admit intersecting lifts on M thus hypothesis 3 on M implies hypothesis 3 on M . If hypothesis 4 is satisfied on M , let c ∈ ΛM be a closed geodesic with ind(c) > 0. Now [c 2 ] = 2[c] is even so there exists a closed geodesic γ ∈ Λ M such that c 2 = π • γ. According to Lemma 7.1, ind(γ) = ind(c 2 ) = 2 ind(c) > 0 and therefore hypothesis 4 is also satisfied on M . Finally, if hypothesis 5 is satisfied on M and c 1 , c 2 ∈ ΛM denote the two closed geodesics that are homologically visible over Z/2Z, c 2 1 and c 2 2 are also homologically visible over Z/2Z by Lemma 7.2 and one can apply Lemma 7.1 as before to get that hypothesis 5 is satisfied on M over Z/2Z.

According to Theorem 1.1, in any of the above cases M contains infinitely many closed geodesics intersecting some common compact set K that are homologically visible over Z/2Z. By Lemma 7.1, the projection of these closed geodesics gives infinitely many closed geodesics intersecting the compact set π( K) that are homologically visible over Z/2Z.

The case of the plane

Let M R 2 be a complete Riemannian plane with isolated closed geodesics. Using what we have seen in the previous sections, we now give the proof of Theorem 1.5.

Proof of Theorem 1.5. When hypothesis 1, 2 3 is assumed, the conclusion follows from the same argument as in the case of the cylinder: by construction of an open neighborhood C = M of infinity. More precisely, this neighborhood C is the unbounded component of M \ c(S 1 ) or M \ (c 1 (S 1 ) ∪ c 2 (S 1 )) if c is self-intersecting or c 1 and c 2 are intersecting closed geodesics. In the case when there exists a closed geodesic c of non-zero average index, C is constructed by "integrating a Jacobi field" along c as was done in Section 5. Now, let us assume that all the closed geodesics of M are without self-intersection, with zero average index and do not intersect any other closed geodesic. Moreover, let us assume that only finitely many (geometrically distinct) closed geodesics intersect any given compact set among homologically visible closed geodesics. Let us show by contradiction that it cannot occur whenever M possesses at least one homologically visible closed geodesic. Let c be a simple closed geodesic that has a homologically visible iterate and such that there is not any homologically visible closed geodesic inside the disk D bounded by c. Let G = γ γ(S 1 ) ⊂ M be the union of the images of the closed geodesics γ of M . Let U be the connected component of M \ (D ∪ G) that contains c(S 1 ) in its boundary. Since U contains loops that are not contractible in R 2 \ D (by taking loops close to the boundary c(S 1 )), U is not simply connected. Let y ∈ U and let γ ∈ ΛU be a loop minimizing the length among the loop of U based at y that are freely homotopic to c (it exists since U is complete). Since ∂U is a disjoint union of closed geodesics, γ lies in the interior of U and is a geodesic path. Depending on the angle that γ makes at y, either the unbounded component of M \ γ(S 1 ) is locally convex and not totally geodesic or the bounded component containing c is locally convex. In the first case, one can apply Bangert's theorem to get a contradiction.

Let us now apply an argument similar to the one given in [6, Theorem 3] in order to conclude the proof. We can thus assume that c lies in the interior of a compact and locally convex subset K ⊂ M and that some powers of c are the only homologically visible closed geodesics of K. Therefore, by the Morse deformation lemma applied the C 2 function E| ΛK which satisfies the Palais-Smale condition and whose anti-gradient flow preserves ΛK (by compactness and local convexity of K), there must be a closed geodesic γ ∈ ΛK such that C d+1 (S 1 •γ) = 0 (see for instance [START_REF] Chang | Infinite-dimensional Morse theory and multiple solution problems[END_REF] where one can replace isolated critical points by isolated critical S 1 -orbits verbatim). By maximality of d, γ and c are geometrically distinct. But c is the only homologically visible closed geodesic of K, a contradiction.

Figure 1 .

 1 Figure 1. The family of loops (γ n )

  implies that the iterated loops γ k n and γ l m are not freely homotopic in M \ π(q 0 ) and hence in the unbounded component C of M \ c(S 1 ). For γ ∈ ΛC, let us denote by [γ] the free homotopy class of γ. For m ≥ 2, let us consider the infimum e m := inf γ∈ΛC [γ]=[γm]

Figure 2 .

 2 Figure 2. Construction of the neighborhood C + (α could also have self-intersections)

  c 1 and c 2 are the only closed geodesics of M inside C that have homologically visible iterates.

2 Figure 3 .

 23 Figure 3. Construction of cylinder Z

2

  is the homologically visible critical point of E| Λ km of maximal value. Critical points of E| Λ km are closed geodesics of Z of degree km. Thus S 1 • c km 1 and S 1 • c km 2 are the only homologically visible critical circles of E| Λ km (and E(c km

Lemma 7 . 1 .

 71 Let c ∈ Λ M be a closed geodesic and let c := π • c. Then the map π

  the Morse indices ind(c) and ind(c) are equal. The conclusion now follows from the local property (5) of the local homologies of S 1 • c and S 1 • c.

  Since ind(c) = 0, the local homology groups C d (S 1 • c m ) are trivial in degrees d ≥ 4 for all m ∈ N. Let d ∈ {0, 1, 2, 3} be the maximal degree such that C d (S 1 • c m ) = 0 for some m ∈ N * . Let k ∈ N * be such that C d (S 1 • c k ) = 0. According to Gromoll-Meyer theory, there exist infinitely many m ∈ N * such that the map induced by the iteration map(ψ m ) * : C * (S 1 • c k ) → C * (S 1 • c km )is an isomorphism. As above, according to the Bangert-Klingenberg theorem[START_REF] Benci | Closed geodesics on noncompact Riemannian manifolds[END_REF], there exists m 0 ∈ N * such that, for all such m ∈ N * greater than m 0 , the inclusion of sublevel sets of E| ΛKC * (S 1 • c km ) inc * --→ H * E| ΛK < e m , E| ΛK < m 2 E(c k )induces the zero map, for some e m > m 2 E(c k ). Thus, for such an m, the long exact sequence of the tripleE| ΛK < e m , E| ΛK < m 2 E(c k ) ∪ S 1 • c km , E| ΛK < m 2 E(c k ) implies that H d+1 E| ΛK < e m , E| ΛK < m 2 E(c k ) ∪ S 1 • c km = 0.


P C Pinet

D Glenadel-Justaut

Y Daydou

G Ceuleneer

S Gou

P Launeau

S D Chevrel

C Carli

MGM DECONVOLUTION OF COMPLEX MAFIC MINERALOGY ROCK SLAB SPECTRA FROM VISIBLE-NEAR INFRARED IMAGING SPECTROSCOPY : IMPLICATIONS FOR THE CHARACTERIZATION OF THE TERRESTRIAL OCEAN CRUST AND OF THE LUNAR CRUST

Keywords: imaging spectroscopy, crystalline plagioclase, mgm deconvolution, oceanic drilling, lunar crust

Successive studies have shown the interest of the MGM approach for planetary surface characterization. However, these reference studies mainly deal with spectra acquired on controlled laboratory powder samples. Advanced imaging spectroscopic sensors with very high spatial resolution observations, either from orbit or in situ, give access to measurements at the scale of the outcrops down to the microscopic scale. At these scales, the coarse textures of the rock slabs may prevail over the granular nature of the surface. MGM deconvolution is tested here on Stillwater Complex and oceanic drill core slab samples with implications for the study of mafic assemblages in the presence of crystalline plagioclase such as in the lunar highlands crust.

INTRODUCTION

Recent observations from the multiband imager and the spectral profiler onboard the Japanese SELENE spacecraft and from the Moon Mineralogy Mapper imaging spectrometer onboard the Chandrayaan mission identified in a number of lunar regions of the lunar crust (e.g., in crater central peaks) the unambiguous occurrence of a 1.2 -1.25 m spectral feature, indicative of a crystal field absorption consistent with Fe-bearing plagioclase feldspar in anorthosite [1], also confirmed by the Diviner LR Experiment onboard the Lunar Reconnaissance Orbiter [2]. However, given the wealth of the M3 dataset, advanced hyperspectral processing appears needed to fully explore the existing variability involving plagioclase and mafic crystal field absorptions [3,4], and to constrain the crustal lithology and stratigraphy of the lunar Highlands.

BACKGROUND ON EARLIER MGM WORKS

The principle of the Modified Gaussian Model is to deconvolve overlapping absorptions of mafic mineral spectra into their fundamental absorption components. The specific interest of this model is to directly account for electronic transition processes [e.g.,5]. Consequently, the MGM approach is in essence able to achieve a direct detection and quantification of minerals which make up the observed surface.

The work by [5] relied on the fact that spectrum variations in visible and near-infrared are caused by absorption features that can be described by means of Gaussians [6]. [5] used modified Gaussians to obtain a mathematical solution closer to the physical reality. Each Gaussian is parameterized by a band center, band width and band strength. As described in [6], MGM computations are carried out in energy and natural log reflectance space. Thereby, overlapping absorptions are additive and can be modeled using linear inverse theory. Spectra are modeled in the logarithm of reflectance space as a sum of modified Gaussian distributions superimposed on a baseline continuum.The resulting combinations of Gaussians can then be interpreted in terms of mineralogy.

It has been shown that MGM is able to retrieve modal and/or chemical composition from an unknown spectrum in the case of simple mineralogical assemblages. Successive studies have shown the interest of the MGM approach for planetary surface characterization (e.g., [7,8,9]). However, one should note that all the reference studies rely on spectra acquired on laboratory powder samples of controlled composition and granulometry, with grain size between 24 and 250 µm.

So far, situations addressing more complex mineralogies (i.e. olivine and pyroxene(s) and/or different pyroxene composition) and/or actual rock samples (e.g., [10]) have been little explored by the MGM approach at the exception of a few works (e.g., [10,[START_REF] Pinet | LPSC 37th[END_REF][START_REF] Clenet | [END_REF]13,14]) and significant efforts have still to be made for improving our capability of spectroscopic modeling and interpretation when dealing with real world observations of unknown mafic rock lithologies observed under natural conditions.

Concerning the influence of plagioclase in MGM deconvolutions, only few studies have been undertaken [START_REF] Clenet | [END_REF]14].

CRYSTALLINE PLAGIOCLASE MGM MODELING

Our goal here is to improve the capability of the MGM to realistically model complex mafic mineralogies when considering rock slab surfaces with coarse grained textures, involving plagioclase and mafic crystal field absorptions. Two cases are considered hereafter.

Stillwater Complex rock samples

Extensive tests have been performed on a group of slab spectra (spot size ~1cm 2 ) corresponding to Pl-rich rock samples including two anorthosites and one leuco-gabbro. These spectra present a high albedo, a general blue slope typical of slab spectra and a wide absorption band with a minimum close to 1.25 m indicative of Fe 2+ in Pl and are labelled StC18, StC9, StC13. They have been produced from bidirectional measurements in the visible-near infrared of rocks from the Stillwater layered intrusion [14]. Their mineral abundances have been established from careful point counting under optical microscope (StC18 anorthosite sample has 94.1% plagioclase (pl), 4.1% clinopyroxene (cpx), 1.8% zoisite (alteration). StC9 anorthosite sample has 79.0% pl, 5.2% cpx, 15.7% zoisite. StC13 leuco-gabbro sample has 81.3% pl, 15.5% cpx, 3.1% opx, 0.1% zoisite).

Examples from [START_REF] Gou | European Lunar Symp. 3rd[END_REF] are given below (Figs. 1,2,3). For all spectra, the overall blue slope is rather well modelled by the second-order polynomial adjusted on the local maxima along the spectra and used to handle the continuum. The thin hatched line in red corresponds to the initial settings and the solid hatched line shows the MGM modelled continuum after deconvolution. Both lines are quite close and basically overlapping for StC13case. This spectrum also shows much less developed overtone absorption bands, at 1.4, 1.9 and 2.2-2.3m, consistent with the lack or very limited occurrence of secondary minerals (hydrous alteration products). In all cases, a large absorption band with a minimum around 1.25m is identified but its band center position may significantly vary. These cases demonstrate that the MGM defined configuration appears to perform well on crystalline pl-rich rock spectra. However, one notes that a weak px contribution associated with pl (up to 15% for cpx) may be undetected. Further testings are underway on a suite of slab spectra dealing with pl-ol and plpyroxenes assemblages [14,[START_REF] Gou | European Lunar Symp. 3rd[END_REF].

CONCLUSIONS AND PERSPECTIVES

Based on the examples highlighted in this paper, MGM can properly model the typical negative slope of rock slab spectra and discern at microscopic scale the major minerals contribution (ol, cpx, opx), including the detection of crystalline plagioclase, in the spectra across a complex geological sample such as the considered oceanic core. Plagioclase can be detected by an MGM dedicated configuration only when its abundance is greater than 80% (see STC9, STC13, STC18 and pixel X29Y93 on the core). Preliminary results also indicate that it is possible to characterize mixtures involving plagioclase in association with mafic minerals. However, weak contribution of pyroxene in association with plagioclase (up to 15% for clinopyroxene) may be undetected (e.g., STC13).

Though additional work is required and not discussed here [START_REF] Gou | European Lunar Symp. 3rd[END_REF], it seems that MGM modeled band centers can properly predict olivine compositions in acceptable ranges compared with the microprobe results.

We are currently in the process of generating a systematic mapping at microscopic scale (on the order of 1 mm 2 ) of the mineralogical variability found along the core section.

Implications for the future encompass the possibility to make use of microscopic imaging spectroscopy for the characterization of continuous cores from the ocean seafloor acquired with drilling projects. Another direction will be to implement this hyperspectral strategy on M 3 spectra with the objective of documenting the petrology of the lunar crust and its layering through characterization of plagioclase and mafic crystal field absorptions.

Figure 1 .

 1 Figure 1. StC18 MGM modelling: ( rms:0.012). 1.25m band center at 1242nm, bandwidth: 568nm, band depth: -0.34

Figure 2 .Figure 3 . 13 3. 2 .

 23132 Figure 2. StC9 MGM modelling: (rms:0.010). 1.25 m band center at 1280nm, bandwidth: 476nm, band depth: -0.26

Figure 4 .

 4 Figure 4. a) Estimated mineral average abundance as a function of depth; b) Image of the core with the location (triangles) of the 5 spectra discussed hereafter and of the macropicture (black rectangle) shown in fig.5; c) and d) Spectrum in blue (PixelX029_093) is properly modeled as a slightly hydrated plagioclase (rms:0.012), spectrum in purple (Pixel X040_080) is modeled as a Cr-diopside (rms:0.008), spectrum in red (Pixel X037_Y155) is ol-rich (Mg#80) (rms:0.008), spectrum in green (Pixel X040_Y138) is a Cr-diopside/ol mixture rms:0.011). Spectrum in orange (Pixel X038_Y302) is modeled as a mixture of Pl, cpx and ol (see fig.5).

Fig. 5 .

 5 Fig.5. Macropicture revealing mineral textures and assemblages (white color: Pl, dark green-black: Ol, light green-grey: cpx) on a subset of the hyperspectral cube (~6mm wide x 9mm vertical) in the vicinity of the orange square (pixel X038_Y302) (corresponding to the black box on fig.4a) where the MGM deconvolution finds a mixture of Pl and Cpx/Ol in agreement with local point counting estimates. Though not analyzed here, each yellow cell of the grid (scale of analysis on the order of 1 mm 2 ) has a spectrum associated with it, corresponding to 8246 individual spectra over the complete core section.The MGM modeling of the orange spectrum (pixel X038_Y302) (figs.4 and 5) indicates that this pixel has a heterogeneous mineralogy caused by the presence of (variably altered) plagioclase, clinopyroxene and olivine. Despite higher residuals (respective rms: 0.019 and 0.015) than in simple cases, both PLG and OL/CPX configurations appear to provide a quasi correct modeling, the band center positions for Pl and Ol absorptions being however disturbed. It is interesting to note that the point counting under the microscope indicates a 40/40/20 mixture for Pl/Cpx/Ol minerals.


N Amini

F Yang
email: fan.yang@dlr.de

E Pineda

B Ruta

M Sprung

A Meyer

Intrinsic relaxation in a supercooled ZrTiNiCuBe glass forming liquid

We studied structural relaxation in the bulk metallic glass-forming alloy Zr46.8Ti8.2Cu7.5Ni10-Be27.5 on different time and length scales, with emphasis on the supercooled liquid state. Using X-ray photon correlation spectroscopy, we determined the microscopic structural relaxation time covering timescales of more than two decades in the supercooled liquid region, down to sub-second regime. Upon heating across the glass transition, the intermediate scattering function changes from a compressed to a stretched decay, with a smooth transition in the stretching exponent and characteristic relaxation time. In the supercooled liquid state, the macroscopic and microscopic relaxation time, as well as the melt viscosity all exhibit the same temperature dependence. This points to a relaxation mechanism via intrinsic structural relaxation of the majority component Zr, with its microscopic timescale controlling both the stress relaxation and viscous flow of the melt.

I. INTRODUCTION

Metallic glasses (MGs) exhibit out-of-equilibrium disordered structure. As a consequence, their structural configurations permanently change towards a more stable state, a process called structural relaxation or physical aging [1]. The structural relaxation has been subjected to many experimental and theoretical investigations [2][3][START_REF] Donth | Relaxation and Thermodynamics in Polymers: Glass Transition[END_REF][START_REF] Sillescu | [END_REF][6] due to its abundance of phenomenology, which is tightly connected with the structure and dynamics of glasses [7].

Due to the disordered structure, metallic glasses and bulk metallic glasses (BMGs) exhibit extraordinary physical and chemical properties, e.g.: high strength, high hardness, good corrosion and wear resistance, soft magnetic properties, etc [8][9][10]. This makes them attractive candidates for novel applications in many fields. Since the properties of the glass-forming melt are governed by the relaxational dynamics, knowledge of its timescales is not only essential for understanding fundamental processes like glass transition and transport phenomena, but also relevant for a range of technologically important processes related to MGs. For instance, the long range atomic transport, as well as macroscopic thermophysical properties like density and shear viscosity are key parameters during the fabrication of MGs, including additive manufacturing and thermoplastic molding [11,12]. Also, the relaxation of metallic glasses towards a more stable (or aged) glassy state controls the long time evolution of their properties, which are crucial when used as structural materials [1].

BMGs provide a unique possibility for studying the structural relaxation behavior in both the glass and the supercooled liquid, owing to their good thermal stability against crystallization. In addition, compared to molecular or oxide glass formers, MGs are composed of atoms has become feasible only recently with the instrumental developments of the X-ray photon correlation spectroscopy (XPCS) [16,[START_REF] Ruta | [END_REF][20][21][22][23][24]. XPCS directly probes the evolution of the structural relaxation features via monitoring the fluctuation of the scattered speckle pattern from a coherent X-ray beam. With the advances in the high brilliance third generation synchrotron sources, XPCS has emerged as a very powerful technique for studying microscopic dynamics, revealing unexpected phenomena like the intermittent, heterogeneous nature of the aging dynamics in metallic glasses [16]. With the use of a fast X-ray detector, here we determined intrinsic relaxation times in the range of 10 The XPCS measurement was carried out on the P10 beamline at the synchrotron facility PETRA III, DESY.

The Vit4 ribbon was mounted on the standard P10 sample insert, between two copper plates which served both as mechanical support and thermal contact, and then placed into a DN100 vacuum cube evacuated down to better than 1×10 -5 mbar. A coherent X-ray beam of 8.25 keV was used. Speckles patterns were collected by a EIGER X4M detector with a capability of frame rates up to 750 Hz placed at ∼ 1843 mm downstream of the sample, at a scattering angle of 2θ = 36 • . This corresponds to a momentum transfer q ∼ 2.6 Å-1 , i.e., at the position of the first structure factor maximum. A preannealing treatment before the XPCS experiment was performed to release the residual stress, by heating the ribbon up to 600 K using identical heating and cooling rates of 1 K/min, i.e., just above the calorimetric glass transition temperature T g at this heating rate [25], with the ribbons clamped in straight shape. The sample was then measured in isothermal steps during two heating runs, one from 520 K to just above 600 K, and another from 520 K to 673 K [START_REF]A cooling step from 603 K to 520 K was performed in between[END_REF]. A temperature incremental step G(q, t 1 , t 2 ) = I(q, t 1 ), I(q, t 2 ) p I(q, t 1 ) p I(q, t 2 ) p ,

where . . . p denotes an ensemble averaging over equiv-161 alent pixels corresponding to the same scattering vector 162 q. A representative of TTCF of the ribbon in the SCL 163 164

regime at 628 K is shown in Figure 1. For experiments 165 which start at t 1 = t 2 = 0, its averaged g 2 (q, t) function 166 at a mean time of (t 1 + t 2 )/2 and a lag time

t = |t 1 -t 2 |, 167 corresponds to 168 g 2 (q, t) = 1 + γ(q)|f (q, t)| 2 (2)
where γ(q) is the q-dependent speckle contrast in the ex- 175

The common heterogeneous feature of structural relax-176 ation is usually described according to the Kohlrausch-177 Williams-Watts (KWW) function, which gives

178 g 2 (q, t) = 1 + a(q)exp[-2(t/τ (T )) β(T ) ] ( 3
)
where τ is the characteristic relaxation time, β is the 179 shape parameter (β < 1 defines a stretched and β > 1 a 180 compressed exponential decay). a(q) = γ(q)f 2 (q) is the product of the speckle contrast γ(q) and the so-called 182 non-ergodicity parameter f (q) (or Debye-Waller factor) 183 [START_REF] Ruta | [END_REF]. Fig. 2 shows the measured g 2 (t) functions at differ-184 185 ent temperatures. Since all measurements here were done 186 at a single scattering vector q, we dropped the q notation 187 of the correlation function in the following. Without fur-188 ther notice, this refers always to q = 2.6 Å-1 i.e., at the 189 first structural factor maximum. For clarity some tem- in the experiment is capable to be operated at even higher frame rates, we did not explore the g 2 (t) details below 0.1 s.

B. Mechanical relaxation

The stress relaxation is observed by recording the decay of stress change with time. The evolution of the stress, normalized to the initial value, at different temperatures is shown in Fig. 4a. Similar to the XPCS measurement, here the decay of the stress can be described by a stretched exponential

σ(t)/σ 0 = exp[-(t/τ ) β ], ( 4
)
where τ is the timescale of the stress relaxation, β is the stretching KWW exponent, and σ 0 is the initial applied stress. With this we assume that also in the glassy region the applied stresses can be fully relaxed. A similar fitting procedure for determining the stretching exponent was used: below 620 K β was left as a free fitting parameter, and above 620 K a fixed β = 0.525 was used.

The characteristic relaxation timescale is then derived accordingly.

The obtained relaxation time and shape parameter are shown in Fig. 4b region [START_REF] Ruta | [END_REF]30].

281

Moreover, here we are able to show that for Vit4 across the glass transition there is a continuous, monotonic change in both β and f 2 (q): f 2 (q) decreases with increasing temperature until 570 K, and then becomes almost temperature independent within the experimental uncertainty. Also the β value drops continuously from 1.75±0.14 at 523 K to 0.67±0.07 at 618 K. β <1 indicates that the sample has reached the supercooled liquid state [16,[START_REF] Ruta | [END_REF], inline with where the relaxation time τ exhibits a different slope or activation energy (Fig. 3c). The transition occurs during the heating experiment starting around 600 K, at this temperature β falls below 1, and culminates around 618 K, with a relaxation timescale of about 245 s.

On the other hand, below 570 K f 2 (q) decreases continuously upon heating towards an almost constant value in the supercooled liquid region. Such a temperature dependence of f 2 (q) is very similar to the change of the Debye-Waller factor observed in metallic and nonmetallic glasses due to a fast (β) relaxation process, for which the mode coupling theory (MCT) predicts [31]

f (q) = f c (q) + h c (T c -T )/T c (5)
where T c is the critical temperature of the MTC, f c (q)

is the Debye-Waller factor at T c , and h q is an amplitude. However, it is obvious that here this plateau value is reached even below the glass transition temperature T g , and T c = 875 K of the Vit4 alloy is considerably higher [13]. Thus, despite the similar temperature dependence, the standard MCT scenario does not apply.

Certainly a fast relaxation processes upon entering the SCL would be an intuitive interpretation. Yet, the characteristic transition temperature and timescale depends both on the wave number and on the experimental observation time window. Thus, they are not directly comparable with those observed by other experimental techniques. For example, in the neutron back-scattering and the Mössbauer absorption experiments the observed f (q) of Vit4 is completely harmonic in this temperature range [25,32]. Moreover, below T g the sample is a stiffer material and the dynamics observed by XPCS could always be a superposition of possible (residual) stress-driven dynamics and the intrinsic relaxation dynamics. Additional studies, particularly focusing on the short timescale dynamics, are needed to clarify their effect, before the drop of f 2 (q) can be unambiguously attributed to the intrinsic β-relaxation.

The presence of compressed microscopic dynamics has been observed in a variety of out-of-equilibrium systems, jammed soft materials [33][34][35]. In the case of soft glasses, the compressed dynamics has been associated to dipole forces due to a stress field occurring in the outof-equilibrium state. This is related to the densification during aging, which creates some micro-collapses leading to a ballistic-like collective particle motion. Here in Vit4, beside the common β ∼1.5 shape exponent, the observation of a continuous, monotonic decrease of beta across the glass transition appears to be very similar to these soft disordered systems, which has not been re-

E(ω)/E 0 = 1 - 1 [1 + (iωτ ) α ] γ , (6)
with α = 0.836 ± 0.006 and γ = 0.405 ± 0.006, corre-376 sponding to a stretching exponent β of about 0.5 [29],

377
which is rather universal for the α-relaxation in metallic 378 glasses [37,38].

379

Compared to the relaxation of mechanical stress, the of the small Be atoms obtained by previous NMR study 394 [43], and the equilibrium melt viscosity close to T g [44] are instance, such discrepancy is considerably smaller [START_REF] Ruta | [END_REF].

Thus, this is a fundamental point still to be surveyed on

- 1 - 4 (

 14 10 4 s by XPCS, covering a large fraction of the time scales accessible by mechanical spectroscopy. Utilizing both techniques, we investigate structural relaxation processes on different length scales, and how the microscopic processes controls the macroscopic properties of the metallic glasses and supercooled melts. II. EXPERIMENTAL PROCEDURE XPCS and mechanical spectroscopy experiments were carried out on a well-known Zr-based BMG: Zr 46.8 Ti 8.2 -Cu 7.5 Ni 10 Be 27.5 , commercially known as Vitreloy ® Vit4). Its undercooled melt exhibits a remarkable stability against crystallization, allowing the study of the supercooled liquid (SCL) region over a wide temperature range and long annealing times. Glassy ribbons of Vit4 were prepared from pure elements with a purity of at least 99.99% by arc-melting and subsequent melt spinning procedure under high purity argon atmosphere. During the melt spinning process the copper wheel linear tangential velocity was set to 40 m/s. The manufactured ribbons were 30±5 µm thick.

of 5 K157

 5 was used between the individual measurements for 136 the SCL regime. Between the isothermal steps, 2θ scans 137 were performed to confirm the amorphous nature of the 138 ribbon. A temperature offset between the sample tem-139 perature and the furnace temperature of 27 K was cor-140 rected [27]. 141 The mechanical relaxation behavior was investigated 142 with mechanical spectroscopy and static stress relaxation 143 experiments in a TA instruments Q800 Dynamic Me-144 chanical Analyzer. The mechanical spectroscopy mea-145 surements were performed at a heating rate of 1 K/min 146 with driving frequencies from 0.03 to 30 Hz. The static 147 stress relaxation experiments were performed in isother-148 mal steps applying constant strains of 0.1%. Before the 149 application of the strain a 20 min isothermal equilibration 150 was performed at each temperature. Similar to the XPCS 151 measurements, also in both the stress relaxation and me-152 chanical spectroscopy experiments, the ribbon was pre-153 annealed up to T g before the measurements to reduce the 154 In an XPCS experiment the quantitative information 158 of the sample dynamics can be obtained by calculating 159 the so-called two-time correlation function (TTCF)

  160

169

  periment and f (q, t) is the intermediate scattering func-170 tion. The scattering function reflects the time depen-171 dence of the density (electronic) correlation in the sam-172 ple, which is a spatial Fourier transformation of the den-173 sity auto-correlation function, and hence the atomic dy-174 namics [28].

FIG. 1 .FIG. 2 .

 12 FIG. 1. Two-time correlation function (TTCF) measured on the Vit4 ribbon by XPCS at 628 K.

190FIG. 3 .

 3 FIG.3. Evolution of (a) the pre-exponential factor a(q) = γ(q)f 2 (q), (b) the stretching exponent β, and (c) the decay time τ of the obtained g 2 (t) function as a function of temperature. Open symbols correspond to data obtained in the 1 st heating, full symbols correspond to that from the 2 nd heating. Horizontal lines in (a) and (b) represents the average value of a(q) and β in the SCL regime, respectively. The dashed vertical line marks the glass transition region. The calorimetric glass transition temperature at this heating rate is around 596 K[25] (dotted line). Same as in Fig2, the blue, magenta, and red colors represent data obtained in the glassy, glass transition, and SCL regimes, respectively.

  213 214In Fig.3we also distinguished data obtained during 215 the first and second heating and shown fit parameters derived from different exposure times. It can be seen that the parameters obtained from the second heating are a continuation of that obtained from the first heating. The results are in agreement within error bars in the overlapping region. The measured g 2 (t)s agree also between different detector exposure times, with the only difference that at short exposure time the signal-to-noise ratio decreases. Hence, although the Eiger detector used

  and Fig.4cas a function of temperature. In contrast to g 2 (t) in the XPCS experiment, in the glassy region the shape of the stress relaxation curve remains stretched, with a stretching exponent β slightly below 0.5. This increases across the glass transition, here observed around 600 K, towards an average, constant value of β ∼ 0.5. The transition temperature can be also consistently identified from where the obtained relaxation time changes its temperature dependence. In addition to the stress relaxation experiments, the α peak of the loss modulus was obtained from mechanical spectroscopy at different frequencies, and fitted to the Havriliak-Negami function in order to obtain the corresponding relaxation time and shape parameters[29].The relaxation times of the α peak, shown in Fig.4c, are in agreement to that obtained from the static stress relaxation above T g , while below the glass transition they diverge due to the different aging state of the samples used in the two types of probes. The mechanical spectroscopy measurements were performed at a constant heating rate of 1 K/min while the stress relaxations were monitored in long isothermal steps, mimicking the thermal protocol in the XPCS experiments.

FIG. 4 .

 4 FIG. 4. (a): Decay of the normalized stress measured at constant strain steps at different temperatures on the Vit4 sample. (b) and (c): Derived stretching exponent and relaxation times according to eq. 4, respectively. In (c) also the relaxation times of the α-peak obtained from the mechanical spectroscopy measurements are shown. Dotted line represents the temperature of the calorimetric glass transition.

338

  Fig. 5 shows the g 2 (t) function obtained by XPCS and 355

380 g 2 FIG. 5 . 390 C. Relaxation on different length scale 391 Fig. 6

 253903916 FIG. 5. Time-temperature superposition of the g 2 (t) function (a), normalized stress relaxation (b), and the normalized loss modulus (c) in the supercooled liquid regime of Vit4.

395FIG. 6 .

 6 FIG.6. Comparison of the microscopic and macroscopic relaxation time measured by XPCS, stress relaxation, and the peak of the loss modulus (DMA HN) of the Vit4 glass as a function of temperature. The viscosity of the supercooled melt[44] and Be atom relaxation time[43] are also shown. The solid and dashed lines represent VFT-fit of the XPCS relaxation time and the melt viscosity in the SCL, respectively. Star and cross symbols show the relaxation time obtained from heating rate dependent calorimetric Tg and from viscosity relaxation, respectively.

  the decoupled small atom dynamics, 422 the α-relaxation time τ α obtained by mechanical spec-423 troscopy and by XPCS exhibit very similar tempera-424 ture dependence. The fragility parameters obtained from 425 XPCS and stress relaxation times are 21.8±5.7 (solid 426 line in Fig. 6) and 19.8±3.6, respectively. This agrees 427 with that of the melt viscosity within the experimental 428 uncertainty. This confirms that the relaxation time of429 the g 2 (t) function corresponds to the timescale of the α-430 relaxation. It is also known from previous measurements 431 by Busch et al.[44] that the time required for the appar-432 ent viscosity to relax towards its equilibrium value during 433 isothermal three point beam bending viscosity measure-434 ments, and the timescale derived from the heating rate 435 dependent glass transition temperature are proportional 436 to the melt viscosity, as shown in Fig. 6. Thus, in the supercooled liquid regime investigated here, the structural relaxation is governed by a dominant microscopic relaxation timescale, which controls the macroscopic flow, the calorimetric glass transition, and the mechanical behavior. This is further supported by the similar stretching exponents. Beside the similar stretching exponent and temperature dependence, however, the absolute timescale differs by of about a factor of 10 between the microscopic structural relaxation and the decay of the macroscopic stress. The temperature uncertainty in the different experimental techniques has been checked by various methods and cannot explain the difference in timescales. This should be also distinguished from the situation around 600 K observed previously for Zr 44 Ti 11 Ni 10 Cu 10 Be 25 [21], where similar relaxation times were found but, in the case of XPCS, the temperature dependence and the stretching exponent clearly corresponded to that in the glass transition region and hence different from that observed here in the SCL. Such kind of discrepancies of relaxation times in the SCL are known for different physical quantities [18, 47, 48], also here shown for the heating rate dependent glass transition and the equilibrating time of the apparent viscosity in Fig. 6. Concerning the difference between the microscopic structural relaxation and macroscopic stress relaxation times, for Vit4, a slower microscopic relaxation time indicates that the stress can be already relaxed without a complete decay of the atomic correlation at next-nearest neighbor distances. Considering the highly collective nature of the atomic motion in the supercooled liquid, such scenario is not unphysical. The exact ratio between the timescales observed in XPCS and in stress relaxation, however, should reflect how the relaxation of stress is achieved by the mass transport, and depends on the detailed structural dynamics of the melt/glass, e.g.: the dynamics modes involved, and the number of atoms which are participating. In the Mg-Cu-Y glass, for

477a 486 Ni 10

 48610 larger range of alloy systems and wave vectors q[23]. 478 V. CONCLUSION 479 To summarize, in this study we emphasize the ability 480 of measuring microscopic, intrinsic relaxation time with 481 XPCS down to sub-second scale, covering a significant 482 portion of the experimental accessible timescale of me-483 chanical spectroscopy. With this, we compared the mi-484 croscopic and macroscopic structural relaxation behavior 485 of a bulk metallic glass forming alloy Zr 46.8 Ti 8.2 Cu 7.5 -Be 27.5 in the supercooled liquid regime. We show 487 that upon heating the onset of the supercooled liquid ex-488 hibits a continuous decease of the non-ergodicity param-489 eter f 2 (q) and stretching exponent β of the intermediate 490 scattering function. In the supercooled liquid, both the 491 microscopic and macroscopic relaxations exhibit a time 492 temperature superposition, with rather similar stretch-493 ing exponents. In addition, a very similar temperature 494 dependence of the two relaxation times are found, which 495 agree with that of the melt viscosity in the investigated 496 temperature range. This shows that the macroscopic flow 497 and mechanical properties are controlled by the micro-498 scopic structural relaxation, with a dominant intrinsic 499 timescale of the majority component, in our case Zr, com-500 mon for these densely packed systems.

  501

ACKNOWLEDGMENTS

acknowledges the financial support from the German 504 Academic Exchange Service (DAAD) via the DLR-505 DAAD special program. We acknowledge DESY (Ham-506 burg, Germany), a member of the Helmholtz Association