Daniele Alessandrini
email: daniele.alessandrini@gmail.com

Lixin Liu

Athanase Papadopoulos
email: athanase.papadopoulos@math.unistra.fr

Weixu Su
email: suweixu@gmail.com

On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space

Keywords: AMS Mathematics Subject Classification: 32G15, 30F30, 30F60 Length-spectrum metric, quasiconformal metric, quasiconformal map, Teichmüller space, Fenchel-Nielsen coordinates

Given a surface of infinite topological type, there are several Teichmüller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmüller space and the length-spectrum Teichmüller space. We work under this hypothesis that the basepoint is upper-bounded and admits short interior curves. There is a natural inclusion of the quasiconformal space in the length-spectrum space. We prove that, under the above hypothesis, the image of this inclusion is nowhere dense in the length-spectrum space. As a corollary we find an explicit description of the length-spectrum Teichmüller space in terms of Fenchel-Nielsen coordinates and we prove that the length-spectrum Teichmüller space is pathconnected.

Introduction

In this paper, the word surface refers to a connected orientable surface of finite or infinite topological type. To motivate our results, we review some basic preliminary facts about Teichmüller spaces of surfaces of infinite topological type.

1.1. The quasiconformal Teichmüller space. Given a surface S, its Teichmüller space is a parameter space of some homotopy classes of complex structures on S. There are several possible ways for defining what is the set of homotopy classes one wants to parametrize and which topology one wants to put on this set. Usually it is not necessary to worry too much about the details, because in the most common case considered, i.e. the case when S is a closed surface, the set is just the set of all possible homotopy classes of complex structures, and all "reasonable" possible definitions of a topology on that set give the same topology. Therefore, in the case of closed surfaces, this freedom to choose between several possible definitions is not a problem, it is, instead, a very useful tool in the theory: one can choose the definition that best suits the problem studied.

As soon as we leave the setting of closed surfaces, it is necessary to be more careful with the definitions. In this paper we deal with surfaces of infinite topological type, and in this case the different possible definitions do not always agree. First of all, it is necessary to choose a basepoint, i.e. a base complex structure R on the surface S, and then to consider only the set of homotopy classes of complex structures on S that are "comparable" with R in a suitable sense. This notion of comparability usually suggests a good definition of the topology. For example, the most commonly used definition is what we call the quasiconformal Teichmüller Date: January 29, 2012. space, where the set T qc (R) parametrizes all the homotopy classes of complex structures X on S that are quasiconformally equivalent to R, i.e. such that there exists a quasiconformal homeomorphism between R and X that is homotopic to the identity of S. (Note that the space T qc (R) we consider here is the reduced Teichmüller space i.e. homotopies need not fix the ideal boundary point-wise.)

The topology on T qc (R) is given by the quasiconformal distance d qc , also called the Teichmüller distance, defined using quasiconformal dilatations of quasiconformal homeomorphisms: for any two homotopy classes of complex structures X, Y ∈ T qc (R), their quasiconformal distance d qc (X, Y ) is defined as

d qc (X, Y ) = 1 2 log inf f K(f )
where K(f ) is the quasiconformal dilatation of a quasiconformal homeomorphism f : X → Y which is homotopic to the identity.

1.2. Fenchel-Nielsen coordinates. In [START_REF] Alessandrini | On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type[END_REF], we studied the quasiconformal Teichmüller space of a surface of infinite topological type using pair of pants decompositions and Fenchel-Nielsen coordinates. We will use this technique also in this paper, so we recall some of the main facts we need here. The definition of these coordinates depends on the interpretation of every complex structure as a hyperbolic metric on the surface. To do so, we use the intrinsic hyperbolic metric on a complex surface, defined by Bers. For complex structures of the first type (i.e. if the ideal boundary is empty) this metric is just the Poincaré metric, but for complex structures of the second type it is different from this well known metric. For these notions and for other related notions, we refer the reader to the paper [START_REF] Alessandrini | On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type[END_REF] for the details of this definition and for explanations about how the intrinsic metric may differ from the Poincaré metric. This metric has the property that every puncture of the surface shows one of the following behaviors:

(1) It has a neighborhood isometric to a cusp, i.e., the quotient of {z = x+ iy ∈ H 2 | a < y}, for some a > 0, by the group generated by the translation z → z + 1. (2) It is possible to glue to the puncture a boundary component that is a simple closed geodesic for the hyperbolic metric. Punctures of this kind will be called boundary components, and the boundary geodesic will be considered as part of the surface. For infinite-type surfaces we proved in [START_REF] Alessandrini | On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type[END_REF] that given a topological pair of pants decomposition P = {C i } of S and a complex structure X on S, it is always possible to find geodesics {γ i } for the intrinsic metric of X such that each γ i is homotopic to C i and the set {γ i } is again a pair of pants decomposition of S. To every curve C i of the topological decomposition we can associate two numbers (ℓ X (C i ), τ X (C i )), where ℓ X (C i ) is the length in X of the geodesic γ i and τ X (C i ) is the twist parameter between the two pairs of pants (which can be the same) with geodesic boundary adjacent to γ i . In this paper, the twist parameter is a length, and this is a slight change in notation with reference to the paper [START_REF] Alessandrini | On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type[END_REF], where it was an "angle" parameter.

Let R be a complex structure equipped with a geodesic pants decomposition P = {C i }. We say that the pair (R,

P) is upper-bounded if sup Ci ℓ R (C i ) < ∞.
We say that the pair (R, P) is lower-bounded if inf Ci ℓ R (C i ) > 0. If (R, P) is both upper-bounded and lower-bounded, then we say that (R, P) satisfies Shiga's property (this property was first used by Shiga in [START_REF] Shiga | On a distance defined by length spectrum on Teichmüller space[END_REF]). Note that if R is of finite type, then any pants decomposition of R satisfies Shiga's property. Shiga's property was introduced in [START_REF] Shiga | On a distance defined by length spectrum on Teichmüller space[END_REF], and we used it in our papers [START_REF] Alessandrini | On local comparison between various metrics on Teichmüller spaces, to appear in Geometriae Dedicata[END_REF] and [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF]. We note however that this property is used in a weaker form in these papers [START_REF] Alessandrini | On local comparison between various metrics on Teichmüller spaces, to appear in Geometriae Dedicata[END_REF] and [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF], and this is also the form which will be useful in the present paper. In fact, we shall say from now on that Shiga's property holds for the pair (R, P) if the pants decomposition P is upper-bounded and there exists a positive constant δ such that ℓ R (C i ) > δ for any C i ∈ P which is in the interior of the surface.

In this paper, we will often use another condition, we say that (R, P) admits short interior curves if there is a sequence of curves of the pair of pants decomposition α k = C i k such that the curves α k are not boundary components of the surface and such that ℓ R (α i ) tends to zero.

In the paper [START_REF] Alessandrini | On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type[END_REF], we proved that if (R, P) is upper-bounded, then the quasiconformal Teichmüller space (T qc (R), d qc ) is locally bi-Lipschitz equivalent to the sequence space ℓ ∞ , using Fenchel-Nielsen coordinates. An analogous result, in the case of the non-reduced Teichmüller space, is due to Fletcher, see [START_REF] Fletcher | Local rigidity of infinite-dimensional Teichmüller spaces[END_REF] and the survey by Fletcher and Markovic [START_REF] Fletcher | Infinite dimensional Teichmüller spaces[END_REF].

1.3. The length-spectrum Teichmüller space. In this paper we study a different definition of Teichmüller space, which we call the length-spectrum Teichmüller space. The definition of this space and of its distance depend on a measure of how the lengths of essential curves change when we change the complex structure.

A simple closed curve on a surface is said to be essential if it is not homotopic to a point or to a puncture (but it can be homotopic to a boundary component). We denote by S the set of homotopy classes of essential simple closed curves on R.

Given a complex structure X on S and an essential simple closed curve γ on S, we denote by ℓ X (γ) the length, for the intrinsic metric on X, of the unique geodesic that is homotopic to γ. The value ℓ X (γ) does not change if we take another complex structure homotopic to X, hence this function is well defined on homotopy classes of complex structures on S.

Given two homotopy classes X, Y of complex structures on S, we define the functional

L(X, Y ) = sup γ∈S ℓ X (γ) ℓ Y (γ) , ℓ Y (γ) ℓ X (γ) ≤ ∞.
Given a base complex structure R on S, the length-spectrum Teichmüller space T ls (R) is the space of homotopy classes of complex structures X on S satisfying L(R, X) < ∞.

For any two distinct elements X, Y ∈ T ls (R), we have 1 < L(X, Y ) < ∞. We define a metric d ls on T ls (R), called the length-spectrum distance, by setting

d ls (X, Y ) = 1 2 log L(X, Y ).
Historically, the length-spectrum distance was defined before the length-spectrum Teichmüller space. Initially people considered this distance as a distance on the quasiconformal Teichmüller space: they studied the metric space (T qc (R), d ls ). For finite type surfaces this is a perfectly fine distance on T qc (R), it makes this space complete and it induces the ordinary topology, and it is more suitable for studying problems regarding lengths of geodesics. For an example of a paper studying this space, see [START_REF] Choi | Comparison between Teichmüller and Lipschitz metrics[END_REF], some of whose results we will use in the following.

The first paper dealing with the space (T qc (R), d ls ) in the case of surfaces of infinite type was [START_REF] Shiga | On a distance defined by length spectrum on Teichmüller space[END_REF].

We proved in [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF] that the metric space (T qc (R), d ls ) is, in general, not complete. More precisely this happens if there exists a pair of pants decomposition P = {C i } such that the pair (R, P) admits short interior curves, i.e. if there is a sequence of curves of the pair of pants decomposition α k = C i k contained in the interior of R with ℓ R (α i ) → 0. The idea was to construct a sequence of hyperbolic metrics by large twists along short curves. To be more precise, for any X ∈ T qc (R), we have again ℓ X (α i ) = ǫ i → 0. Denote by τ t α (X) the surface obtained from X by a twist of magnitude t along α and let X i = τ ti αi (X), with

t i = log | log ǫ i |. Then we proved that d qc (X, X i ) → ∞, while d ls (X, X i ) → 0. If we set Y n = τ tn αn • • • • • τ t2 α2 • τ t1 α1 (
X) and if we define Y ∞ to be the surface obtained from X by a twist of magnitude t i along α i for every i, then a similar argument shows that Y ∞ ∈ T ls (R) \ T qc (R) and lim n→∞ d ls (Y n , Y ∞ ) = 0; see [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF], [START_REF] Alessandrini | On local comparison between various metrics on Teichmüller spaces, to appear in Geometriae Dedicata[END_REF] for more details. We shall give another proof of the last result in Section 5 below.

The fact that the metric space (T qc (R), d ls ) is, in general, not complete is an indication of the fact that T qc (R) is not the right space for this distance. The length-spectrum Teichmüller space was defined in [START_REF] Liu | Some metrics on Teichmüller spaces of surfaces of infinite type[END_REF], with the idea that it was the most natural space for that distance, and it was studied in [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF], [START_REF] Alessandrini | On local comparison between various metrics on Teichmüller spaces, to appear in Geometriae Dedicata[END_REF]. We proved in [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF] that for every base complex structure R, the metric space (T ls (R), d ls ) is complete. This result answered a question raised in [START_REF] Liu | Some metrics on Teichmüller spaces of surfaces of infinite type[END_REF] (Question 2.22).

Other properties, such as connectedness and contractibility, are unknown in the general case for surfaces of infinite type. If the basepoint R satisfies Shiga's condition, then it follows from the main result of [START_REF] Alessandrini | On local comparison between various metrics on Teichmüller spaces, to appear in Geometriae Dedicata[END_REF] that (T ls (R), d ls ) is homeomorphic to the sequence space ℓ ∞ with an homeomorphism that is locally bi-Lipschitz, and, in particular, the space is contractible. One of the results we prove in the present paper is the following. Theorem 6.3 If (R, P) is upper-bounded and it admits short interior curves, then (T ls (R), d ls ) is path-connected.

To obtain this result, we will need some results about the comparison between the quasiconformal and the length-spectrum spaces (see below), and we will also need the following explicit characterization of the length-spectrum Teichmüller space in terms of Fenchel-Nielsen coordinates, which is interesting in itself: Theorem 6.2 Assume X = (ℓ X (C i ), τ X (C i )). X lies in T ls (R) if and only if there is a constant N > 0 such that for each i,

log ℓ X (C i ) ℓ Y (C i ) < N and |τ X (C i ) -τ R (C i )| < N max{| log ℓ R (C i )|, 1}.
1.4. Comparison between the two spaces. It is interesting to compare the two spaces (T ls (R), d ls ) with (T qc (R), d qc ).

A classical result of Sorvali [START_REF] Sorvali | On the dilatation of isomorphisms between covering groups[END_REF] and Wolpert [START_REF] Wolpert | The length spectra as moduli for compact Riemann surfaces[END_REF] states that for any K-quasiconformal map f : X → Y and any γ ∈ S, we have

1 K ≤ ℓ Y (f (γ)) ℓ X (γ) ≤ K.
It follows from this result that there is a natural inclusion map

I : (T qc (R), d qc ) → (T ls (R), d ls )
and that this map is 1-Lipschitz.

In [START_REF] Liu | Some metrics on Teichmüller spaces of surfaces of infinite type[END_REF] we proved that if R satisfies the Shiga's condition, then this inclusion is surjective, showing that under this hypothesis we have T ls (R) = T qc (R) as sets. In the same paper we also gave an example of a complex structure R such that the inclusion map I is not surjective.

The inverse map of I (defined on the image set T qc (R)) is not always continuous. Shiga gave in [START_REF] Shiga | On a distance defined by length spectrum on Teichmüller space[END_REF] an example of a hyperbolic structure R on a surface of infinite type and a sequence (R n ) of hyperbolic structures in T ls (R) ∩ T qc (R) which satisfy

d ls (R n , R) → 0, while d qc (R n , R) → ∞.
In particular, the metrics d ls and d qc do not induce the same topology on T qc (R). A more general class of surfaces with the same behavior was described in the paper [START_REF] Liu | Topological equivalence of metrics in Teichmüller space[END_REF] by Liu, Sun and Wei.

In the same paper, Shiga showed that if the hyperbolic metric R carries a geodesic pants decomposition that satisfies Shiga's condition, then d ls and d qc induce the same topology on T qc (R). In the paper [START_REF] Alessandrini | On local comparison between various metrics on Teichmüller spaces, to appear in Geometriae Dedicata[END_REF] we strengthened this result by showing that under Shiga's condition the inclusion map is locally bi-Lipschitz.

Several natural questions arise after this, for instance :

(1) Give necessary and sufficient conditions under which the inclusion map I is surjective. ( 2) Under what conditions is the inverse map (defined on the image set) continuous ? Under what conditions is it Lipschitz ? bi-Lipschitz ? (3) Are the two spaces (T qc (R), d qc ) and (T ls (R), d ls ) in the general case locally isometric to the infinite sequence space ℓ ∞ ? If not, are there other "model spaces" to which such a space is locally isometric ? (4) How does the image I ((T qc (R), d qc )) sit in the space (T ls (R), d ls ) ? Is it dense ? Is it nowhere dense ? Some of these problems have been solved in the present paper. Consider T qc (R) as a subset in T ls (R). A natural question which is asked in [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF] is whether the subset T qc (R) is dense in (T ls (R), d ls ). This would tell us that T ls (R) is the metric completion of T qc (R) with reference to the distance d ls , and it would also imply that (T ls (R), d ls ) is connected (since the closure of a connected subset is also connected). In this paper, we will give a negative answer to this question. We prove the following: Theorem 5.8 If R admits a geodesic pants decomposition which is upper-bounded and admits short interior geodesics, then the space (T qc (R), d ls ) is nowhere dense in (T ls (R), d ls ).

The proof of Theorem 5.8 involves some estimates between quasiconformal dilatation and hyperbolic length under the twist deformation. Some of the techniques used in this paper are developed in our papers [START_REF] Alessandrini | On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type[END_REF] and [START_REF] Alessandrini | On local comparison between various metrics on Teichmüller spaces, to appear in Geometriae Dedicata[END_REF]. The upper-boundedness assumption of (R, P) is used here so that we can get a lower bound estimate of the length-spectrum distance under a twist. Without this assumption, the density of

T qc (R) in T ls (R) is unknown.
It would be interesting to study the closure of T qc (R) in T ls (R). This space is the completion of T qc (R) with reference to the length-spectrum metric.

Under the geometric conditions of Theorem 5.8, we showed in [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF] (Example 5.1) that the inverse of the inclusion map I restricted to T qc (R): (T qc (R), d ls ) → (T qc (R), d qc ) is nowhere continuous. We give another proof of this fact in the present paper (Proposition 5.2).

Regarding Item (4) above, we prove (Proposition 5.4 below) that if the surface R admits an upper-bounded pants decomposition P such that (R, P) is upper-bounded and admits short interior curves, then there exists a point in T ls (R) \ T qc (R) which can be approximated by a sequence in T qc (R) with the length-spectrum metric. This gives in particular a new proof of the fact (obtained in [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF]) that the space T qc (R) equipped with the restriction of the metric d ls is not complete.

Preliminaries

Let R be a base topological surface equipped with a hyperbolic structure X and with a geodesic pants decomposition P = {C i }. The pieces of the decomposition (completions of connected components of the complements of the curves C i ) are spheres with three holes equipped with hyperbolic metrics, where a hole is either a cusp or a geodesic boundary component. We call such a piece a generalized pair of pants to stress on the fact that it is not necessarily a hyperbolic pair of pants with three geodesic boundary componentss. To each C i ∈ P, we consider its length parameter ℓ X (C i ) and its twist parameter τ X (C i ). Recall that the latter is only defined if C i is not a boundary component of R, and it is a measure of the relative twist amount along the geodesic C i between the two generalized pairs of pants (which may be the same) that have this geodesic in common. The twist amount per unit time along C i is chosen so that a complete positive Dehn twist along C i changes the twist parameters on C i by addition of ℓ X (C i ). For any hyperbolic metric X, its Fenchel-Nielsen parameters relative to P is the collection of pairs

{(ℓ X (C i ), τ X (C i ))} i=1,2,•••
where it is understood that if C i is a boundary component of R, then there is no twist parameter associated to it, and instead of a pair (ℓ

X (C i ), τ X (C i )) we have a single parameter ℓ X (C i ).
There is an injective mapping from T qc (R) or T ls (R) to an infinite-dimensional real parameter space:

X → log ℓ X (C i ) ℓ R (C i ) , τ X (C i ) -τ R (C i ) i=1,2,••• .
If the image of X belongs to ℓ ∞ , then we say that X is Fenchel-Nielsen bounded (with respect to (R, P)).

For each C i in the interior of X, there is a simple closed curve β i satisfying the following (see Figure 1):

(1) β i and C i intersect minimally, that is, i(C i , β i ) = 1 or 2;

(2) β i does not intersect any C j , j = i. The following result is proved in [START_REF] Alessandrini | On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type[END_REF].

Lemma 2.1. Suppose that the pants decomposition {C i } is upper-bounded by M , that is, sup i {ℓ X (C i )} ≤ M . Then there exists a positive constant ρ depending only on M such that for each i, β i can be chosen so that the intersection angle(s) θ i of C i and β i (there are one or two such angles for each curve C i ) satisfy sin θ i ≥ ρ.

We shall need some background material on the conformal moduli of quadrilaterals. Recall that a quadrilateral Q(z 1 , z 2 , z 3 , z 4 ) consists of a Jordan domain Q in the complex plane and a sequence of ordered vertices z 1 , z 2 , z 3 , z 4 on the boundary of Q. The vertices of a quadrilateral Q(z 1 , z 2 , z 3 , z 4 ) divide its boundary into four Jordan arcs, called the sides of the quadrilateral. The arcs z 1 z 2 and z 3 z 4 are called the a-sides and the other two arcs, the b-sides of Q. Two quadrilaterals Q(z 1 , z 2 , z 3 , z 4 ) and Q ′ (w 1 , w 2 , w 3 , w 4 ) are said to be conformally equivalent if there is a conformal map from Q to Q ′ which carries each z i to w i .

Every quadrilateral

Q(z 1 , z 2 , z 3 , z 4 ) is conformally equivalent to a rectangle R(0, a, a + ib, ib) = {x + iy : 0 < x < a, 0 < y < b}.
It is easy to see that two rectangles R(0, a, a + ib, ib) and R ′ (0, a ′ , a ′ + ib ′ , ib ′ ) are conformally equivalent if and only if there is a similarity transformation between them. Therefore, we can define the (conformal) modulus of the quadrilateral

Q(z 1 , z 2 , z 3 , z 4 ) by mod(Q(z 1 , z 2 , z 3 , z 4 )) = a b .
It follows from the definition that the modulus of a quadrilateral is a conformal invariant and that mod(

Q(z 1 , z 2 , z 3 , z 4 )) = 1/mod(Q(z 2 , z 3 , z 4 , z 1 )).
The modulus of a quadrilateral Q(z 1 , z 2 , z 3 , z 4 ) can be described in terms of extremal length in the following way. Let F = {γ} be the family of curves in Q joining the a-sides. The extremal length of the family F, denoted by Ext(F), is defined by

Ext(F) = sup ρ inf γ∈F ℓ ρ (γ) 2 Area ρ
where the supremum is taken over all conformal metrics ρ on Q of finite positive area. Then it can be shown [START_REF] Ahlfors | Lectures on quasiconformal mappings[END_REF] that mod(Q(z 1 , z 2 , z 3 , z 4 )) = 1 Ext(F) .

A lower bound for the quasiconformal dilatation under a twist

The aim of this section is to prove Corollary 3.3, which is important for Section 5. For a hyperbolic metric X and a simple closed geodesic α on X, we denote by τ t α (X), t ∈ R the hyperbolic metric obtained from X by a Fenchel-Nielsen twist of magnitude t along α. We fix the simple closed curve α and, to simplify the notation, we set X t = τ t α (X). For a small ǫ > 0, let N ǫ be an ǫ-neighborhood of α. We denote by g t ǫ : X → X t a homeomorphism that is the natural isometry outside of N ǫ , and that is homotopic to the identity.

It is convenient to work in the universal cover of the surface. In the case where X has no boundary, its universal cover is the hyperbolic plane H 2 . We let f t ǫ : H 2 → H 2 be a lift of g t ǫ . In the case where X has non-empty boundary, we take the double of X and X t , extend the map g t ǫ to X d → X d t and we then let f t ǫ be the lift of the extended map to H 2 . Thus, in any case, the map f t ǫ is defined on the plane H 2 . Let α be the lift of the closed geodesic α to the universal cover. Then α can be seen as a lamination with discrete leaves.

When ǫ tends to zero, the maps f t ǫ coneverges pointwise on H 2 \ α to a map f t that is an isometry on every connected component of H 2 \ α. We choose an orientation on α. Now every connected component of α divides the plane into two parts, a left part and a right part. We extend f t to α by requiring that this extension is continuous on the left part. We denote the extended map again by f t , a piecewise isometry from H 2 to H 2 .

To make some explicit computations, we work in the upper-half model of the hyperbolic plane H 2 . Up to conjugation in the domain and in the codomain, we can assume that the geodesic iR + is a leaf of α that is fixed pointwise by f t . In particular, f t fixes 0, i, ∞. Lemma 3.1. For any bi-infinite geodesic in the upper-half plane model of H 2 with endpoints x 1 < 0 < x 2 on R and intersecting iR + at i (see Figure 2), we have

f t (x 1 ) < -e 2t + ( x 1 + x 2 2 ) 2 + ( x 1 + x 2 2 ) < 0 and 0 < f t (x 2 ) < x 2 , ∀ t > 0. x y i Figure 2.
Proof. This follows from the construction of the Fenchel-Nielsen twist deformation. See, for example, the proof of Lemma 3.6 in Kerckhoff [START_REF] Kerckhoff | The Nielsen realization problem[END_REF]. We give here the proof for the sake of completeness.

Let γ be the bi-infinite geodesic connecting x 1 and x 2 . By assumption, iR + and γ intersect at the point i. Under the twist deformation, γ is deformed into a sequence of disjoint geodesic arcs {A i }, each coming from γ under the twist deformation. Let γ be the infinite piecewise geodesic arc, which is the union of {A i } and of pieces of leaves of α. See Figure 3 in the case where x 1 = -1 and x 2 = 1. Note that one such arc A 0 passes through the point i. If A 0 is continued to a bi-infinite geodesic, its endpoints will be precisely those of γ. Move along γ in the left direction, running along the leaf iR + (with hyperbolic distance t) until coming to the next arc A 1 . If the arc is continued in the forward direction, one of its

endpoints is -e 2t + ( x1+x2 2 ) 2 + ( x1+x2 2
) (this can be shown by the cosine formula for triangles). Similarly, the forward endpoint of the next arc, A 2 , is strictly to

the left of -e 2t + ( x1+x2 2 ) 2 + ( x1+x2 2
). In fact, the forward endpoint of each arc A i+1 is strictly to the left of A i . Since the forward endpoints of the A , i s converge to f t (x 1 ), we see that f t (x 1 ) is strictly less thane 2t + ( x1+x22

) 2 + ( x1+x2 2
).

An analogous (and simpler) argument shows that 0 < f t (x 2 ) < x 2 .

As before, assume that P = {C i } is a geodesic pants decomposition of the hyperbolic metric R.

Let t = (t 1 , t 2 , . . .) be a sequence of real numbers. Fix X ∈ T qc (R). We say that X t is a multi-twist deformation of X along P if X t is obtained from X by the composition of t i -twists along C i . Let us set t = sup i |t i |. In the following theorem, the simple closed curves (β i ) are chosen as in Section 2. We assume that the intersection angle(s) θ i of C i and β i satisfy sin θ i ≥ ρ i .

Since each β i intersects C i and no C j , j = i, then under the multi-twist deformation, the image of β i depends only on the twist along C i .

We now denote by H(∞, -1, 0, t), t > 0 the quadrilateral where the Jordan domain H is the upper half-plane. Let h(t) = mod(H(∞, -1, 0, t)). Then, h(t) is related to the modulus of the Grötzch ring (the ring domain obtained by deleting the interval [0, r] from the unit disk) µ(r) by the following equality (see Page 60-61 in [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF]):

h(t) = 2 π µ( 1 1 + λ
), where λ = t.

From the known properties of µ(r), it follows that h(t) is a strictly increasing function and lim t→+∞ h(t) = ∞.

Theorem 3.2. For the hyperbolic surface X t defined above, we have

d qc (X, X t ) ≥ 1 2 log sup i h(K i e |ti| ) h( 1+ √ 1-ρ 2 i 1- √ 1-ρ 2 i )
,

where

K i = 1 -1 -ρ 2 i 1 + 1 -ρ 2 i 1 + ( √ 1-ρ 2 i 1- √ 1-ρ 2 i ) 2 + √ 1-ρ 2 i 1- √ 1-ρ 2 i .
Proof. For each C i , consider the twist t i . Without loss of generality, we assume that t i > 0.

Assume that iR + is a lift of C i to the universal cover and that β i has a lift γ which intersects iR + at the point i. Denote by x 1 < 0 < x 2 the two endpoints of γ. The intersection θ of iR + and γ satisfies:

sin θ = 2 |x 1 | + |x 2 | .
Applying Lemma 3.1, we have

(1)

f t (x 1 ) < -e 2ti + ( x 1 + x 2 2 ) 2 + ( x 1 + x 2 2 ) < 0, 0 < f t (x 2 ) < x 2 .
Let us set -e 2ti + ( x1+x22

) 2 + ( x1+x2 2
) = -Ae ti , where

A = 1 1 + e -2ti ( x1+x2 2
) 2 + e -ti ( x1+x2 2 )
.

By the geometric definition of quasiconformal maps,

K(f t ) ≥ mod (H(f t (x 1 ), f t (0), f t (x 2 ), f t (∞))) mod(H(x 1 , 0, x 2 , ∞) .
By (1) and the monotonicity of conformal modulus, we have mod(H(f t (x 1 ), 0, f t (x 2 ), ∞)) ≥ mod(H(-Ae ti , 0, x 2 , ∞)).

Note that mod(H(-Ae

ti , 0, x 2 , ∞)) = mod(H(∞, -x 2 , 0, Ae ti )) = mod(H(∞, -1, 0, Ae ti |x 2 | ). Therefore (2) K(f t ) ≥ mod(H(∞, -1, 0, Ae t i |x2| )) mod(H(∞, -1, 0, | x1 x2 |))
.

We can use the cross ratio to estimate

| x1 x2 | in terms of the lower bound ρ i (X) of sin θ i . Let χ(a, b, c, d) = (a-c)(b-d) (a-d)(b-c) be the cross ratio of a, b, c, d ∈ R ∪ {∞}.
We can map H 2 conformally to the unit disc, and iR + and γ to the geodesics with endpoints ±1 and ±e iθ respectively. It is easy to show that cos 2 (θ/2) = χ(1, e iθ , -e iθ , -1).

By the conformal invariance of the cross ratio, we have

cos 2 (θ/2) = χ(0, x 2 , x 1 , ∞) = |x 1 | |x 1 | + |x 2 | .
Since sin θ ≥ ρ i , we have

(3) 1 -1 -ρ 2 i 1 + 1 -ρ 2 i ≤ |x 1 | |x 2 | ≤ 1 + 1 -ρ 2 i 1 -1 -ρ 2 i .
There are other restrictions on the values x 1 , x 2 . Since the geodesic γ passes through the point i, we can show that |x 2 ||x 1 | = 1.

If

|x 1 | ≥ 1 ≥ |x 2 |, then A |x 2 | ≥ A ≥ 1 1 + ( |x1|/|x2|-1 2 ) 2 + |x1|/|x2|-1 2 .
Combined with the inequality (3), we obtain

A |x 2 | ≥ 1 1 + ( √ 1-ρ 2 i 1- √ 1-ρ 2 i ) 2 + √ 1-ρ 2 i 1- √ 1-ρ 2 i . If |x 2 | ≥ 1 ≥ |x 1 |, then (using again Inequality (3)) we obtain A |x 2 | ≥ A |x 1 | |x 2 | ≥ 1 -1 -ρ 2 i 1 + 1 -ρ 2 i 1 + ( √ 1-ρ 2 i 1- √ 1-ρ 2 i ) 2 + √ 1-ρ 2 i 1- √ 1-ρ 2 i . Denote K i = 1 -1 -ρ 2 i 1 + 1 -ρ 2 i 1 + ( √ 1-ρ 2 i 1- √ 1-ρ 2 i ) 2 + √ 1-ρ 2 i 1- √ 1-ρ 2 i . We conclude that A |x 2 | ≥ K i .
It follows from inequality (2) that

K(f t ) ≥ mod(H(∞, -1, 0, K i e ti ))
mod(H(∞, -1, 0,

1+ √ 1-ρ 2 i 1- √ 1-ρ 2 i ))
.

Using the function h(t) already introduced for moduli of quadrilaterals, we get

K(f t ) ≥ h(K i e ti ) h( 1+ √ 1-ρ 2 i 1- √ 1-ρ 2 i )
.

By Teichmüller's Theorem, there is an extremal quasiconformal map from X to X t that realizes the Teichmüller distance d qc (X t , X). Lifting this map to the universal cover, since it is homotopic to f t , it has the same boundary value as

f t . It follows that 2d T (X, X t ) ≥ log sup i h(K i e ti ) h( 1+ √ 1-ρ 2 i 1- √ 1-ρ 2 i ) . Corollary 3.3. If sup i l X (C i ) < ∞ and if X t is a multi-twist deformation of X along P, then d qc (X, X t ) → ∞ as t → ∞.
Proof. By assumption, there is a positive constant M such that sup i ℓ X (C i ) < M . By Lemma 2.1, there is a positive constant ρ (depending on M ) such that inf i ρ i ≥ ρ.

It is easy to see that

(1- √ 1-ρ 2 ) 2 1+ √ 1-ρ 2 is an increasing function of ρ. For each i, we have (1 + 1 -ρ 2 i ) 2 1 -1 -ρ 2 i ≥ (1 -1 -ρ 2 ) 2 1 + 1 -ρ 2 .
As a result, if follows from Theorem 3.2 that

(4) d qc (X, X t ) ≥ 1 2 log sup i h( (1- √ 1-ρ 2 ) 2 1+ √ 1-ρ 2 e ti ) h( 1+ √ 1-ρ 2 1- √ 1-ρ 2 )
.

As t → ∞, the properties of the function h(t) tell us that d qc (X, X t ) → ∞.

Estimation of hyperbolic length under a twist deformation

The twist deformation is an important tool to understand the difference between the quasiconformal metric and the length-spectrum metric. As in the previous section, we fix a simple closed geodesic α and we set X t = τ t α (X). In this section, we give a upper bound and a lower bound for d ls (X, X t ). Proposition 4.1. For every t in R, we have

d ls (X, X t ) ≤ 1 2 max sup γ,i(α,γ) =0 i(α, γ)|t| ℓ X (γ) , sup γ,i(α,γ) =0 i(α, γ)|t| ℓ Xt (γ) .
Proof. Without loss of generality, we can assume that t > 0. For any simple closed curve γ intersecting α, let ℓ t (γ) denote the hyperbolic length of γ in X t . We have

ℓ X (γ) -i(α, γ)t ≤ ℓ t (γ) ≤ ℓ X (γ) + i(α, γ)t.
Recall that the length-spectrum distance is given by

d ls (X, X t ) = max 1 2 log sup γ ℓ t (γ) ℓ X (γ) , 1 2 log sup γ ℓ X (γ) ℓ t (γ) ,
where the supremum is taken over all essential simple closed curves.

For a simple closed curve γ satisfying i(α, γ) = 0, the hyperbolic length of γ is invariant under the twist along α. As a result, we have

d ls (X, X t ) = max 1 2 log sup γ,i(α,γ) =0 ℓ t (γ) ℓ X (γ) , 1 2 log sup γ,i(α,γ) =0 ℓ X (γ) ℓ t (γ) .
For any simple closed curve γ with i(α, γ) = 0, we have

log ℓ t (γ) ℓ X (γ) ≤ log ℓ X (γ) + i(α, γ)t ℓ X (γ) ≤ i(α, γ)t ℓ X (γ)
(using, for the right-hand side, the inequality log(1 + x) ≤ x for x > 0), and likewise

log ℓ X (γ) ℓ t (γ) ≤ log ℓ t (γ) + i(α, γ)t ℓ t (γ) ≤ i(α, γ)t ℓ X (γ) .
The result is thus proved.

Note that if ℓ X (α) ≤ L, then it follows from the Collar Lemma that there is a constant C depending on L such that for any simple closed geodesic γ with i(α, γ) = 0, we have ℓ

X (γ) ≥ Ci(α, γ)| log ℓ X (α)| and ℓ Xt (γ) ≥ Ci(α, γ)| log ℓ X (α)|.
We deduce from the Proposition 4.1 the following

Corollary 4.2. If ℓ X (α) ≤ L, then there is a constant C depending on L such that d ls (X, X t ) ≤ |t| 2C| log ℓ X (α)| .
Now we need a lower bound. We use the idea of an (ǫ 0 , ǫ 1 )-decomposition of a hyperbolic surface (Minsky [14, sec. 2.4], Choi-Rafi [6, sec. 3.1]).

Consider a hyperbolic metric X on a surface of finite type (we will need only the case when X is homeomorphic to a one-holed torus or to a four-holed sphere).

Choose two numbers ǫ 1 < ǫ 0 less than a Margulis constant of the hyperbolic plane. Assume that α is a closed geodesic in the interior of X with ℓ X (α) ≤ ǫ 1 . Let A be an annular (collar) neighborhood of α such that the two boundary components of A have length ǫ 0 . We can choose ǫ 1 and ǫ 0 small enough such that any simple closed geodesic on X that intersects α is either the core curve of A or crosses A (this upper bound for ǫ 1 , ǫ 0 can be chosen in a way that is independent on the surface X).

Let Q = X -A. For any simple closed geodesic γ on X, its restriction to Q is homotopic (relative to ∂Q ) to a shortest geodesic, which we denote by γ Q .

Lemma 4.3 ( Choi-Rafi [6, prop. 3.1] ).

There is a constant C depending on ǫ 0 , ǫ 1 and on the topology of X, such that

|ℓ X (γ ∩ Q) -ℓ X (γ Q )| ≤ Ci(γ, ∂Q), |ℓ X (γ ∩ A) -[2 log ǫ 0 ℓ X (α) + ℓ X (α)|tw X (γ, α)|]i(γ, α)| ≤ Ci(γ, α).
In the second formula, the quantity tw X (γ, α) is called the twist of γ around α. This quantity is defined in [14, sec. 3]. Its difference with the Fenchel-Nielsen twist coordinate is given by the following estimate (For the proof, see Minsky [START_REF] Minsky | Extremal length estimates and product regions in Teichmüller space[END_REF]Lemma 3.5]). Lemma 4.4 ( Minsky [14, lemma 3.5] ). Suppose that X t is the twist deformation of X by a twist of magnitude t = τ Xt (α) -τ X (α) along α. We normalize the twist coordinate by setting s(X t ) =

τX t (α) ℓX (α) and s(X) = τX (α) ℓX (α) . Then |tw Xt (γ, α) -tw X (γ, α) -(s(X t ) -s(X))| ≤ 4.
Now assume that X is homeomorphic to a one-holed torus or to a four-holed sphere, with the pair of pants decomposition α. Consider the simple closed curve β we constructed in Lemma 2.1. Note that |tw X (β, α)| is less than 4, and i(β, α) is 1 or 2. The following is a direct corollary of Lemma 4.3 and 4.4. Lemma 4.5. If X n is the hyperbolic metric obtained from X by a twist deformation along α, with normalized twist coordinates n =

τX n (α)-τX (α) ℓX (α)
, then

ℓ Xn (β) ℓ X (β) ≥ 2 log ǫ0 ℓX (α) + ℓ X (α)(n -8) + 1 2 ℓ Xn (β Q ) -2C 2 log ǫ0 ℓX (α) + 2ℓ X (α) + ℓ X (β Q ) + 2C
, where C is the same constant as in Lemma 4.3, hence it only depends on ǫ 0 , ǫ 1 and on the topology of X.

Proof. We know that i(β, α) and i(β, ∂Q) are 1 or 2, and that |tw X (β, α)| ≤ 4. By Lemma 4.4 we see that n -8 ≤ |tw X (β, α)| ≤ n + 8. We also know that ℓ Xn (α) = ℓ X (α). We can express ℓ X (β) as ℓ X (β ∩ Q) + ℓ X (β ∩ A), and we do the same for ℓ Xn (β). We then use Lemma 4.3 to estimate separately ℓ X (β∩Q), ℓ X (β∩A), ℓ Xn (β∩ Q), ℓ Xn (β ∩ A). Finally, we estimate the ratio, and we get the formula.

Consider now a hyperbolic metric X with a geodesic pants decomposition P = {C i } satisfying sup Ci ℓ X (C i ) ≤ M . Suppose there exists a geodesic α ∈ P which is in the interior of X and which has length less than ǫ 1 . Let X t = τ t α (X). Here we set t = ℓ R (C i )n, for a sufficiently large number n. With these assumptions, we now apply Lemma 4.5 to give a lower bound for the length-spectrum distance d ls (X, X t ).

As we have done before, choose a simple closed curve β which intersects α once or twice but does not intersect any other curves in P. Under the upper-boundedness assumption on the pants decomposition, there is also a constant K depending on ǫ 0 , ǫ 1 , M such that the length ℓ X (β Q ) is bounded as

1/K ≤ ℓ X (β Q ) ≤ K.
Moreover, we may choose the constant ǫ 0 with upper and lower bounds which only depend on M . Then an analysis of the formula in Lemma 4.5 leads to the following Theorem 4.6.

If sup Ci ℓ X (C i ) ≤ M and ℓ X (α) ≤ ǫ 1 , then there is a constant D depending on ǫ 1 , M such that d ls (X, X t ) ≥ 1 2 log ℓ Xt (β) ℓ X (β) ≥ 1 2 log 2| log ℓ X (α)| + |t| -D 2| log ℓ X (α)| + D .
Proof. We can choose ǫ 0 and ǫ 1 such that they satisfy ǫ 0 = 2ǫ 1 . Then log(ǫ 0 ) can be put in the constant. We remark that the constant ǫ 0 is less than the Margulis constant that is less than 1. In particular ℓ X (α) is less than 1 and it can be put in the constant. The terms ℓ X (β Q ) and ℓ Xn (β Q ) can be estimated with K as above. Moreover, nℓ X (α) = t. Transforming the formula of Lemma 4.5 in this way, we get the conclusion.

In fact, Theorem 4.6 is a particular case of Choi-Rafi's product region formula for the length-spectrum metric. Their proof requires a more detailed and complicated analysis, see Lemma 3.4 and Theorem 3.5 of [START_REF] Choi | Comparison between Teichmüller and Lipschitz metrics[END_REF]. Since we only need to consider a particular curve β to give a lower bound, we don't need the general formula.

Structure on the length-spectrum Teichmüller space

The goal of this section is to show how bad can be the inclusion (T qc , d qc ) → (T ls , d ls ). In Section 6, we will use some of the results of this section to prove connectedness of (T ls , d ls )

Given a surface S, we define a proper subsurface S ′ of S to be an open subset of S such that the frontier ∂S is a union of simple closed essential curves of S. If X is a complex structure on S and if S ′ is a proper subsurface of S, the restriction of X to S ′ is the complex structure that we have on the proper subsurface X ′ that is homotopic to S and such that the boundary curves of X ′ are geodesics for the intrinsic hyperbolic metric of X. The intrinsic metric of X ′ is the restriction of the intrinsic metric of X.

Proposition 5.1. Consider an exhaustion of S by a sequence of subsurfaces with boundary:

S 1 ⊆ S 2 ⊆ • • • ⊆ S n ⊆ • • • and S = ∪ ∞ n=1 S n .
Given two complex structures X, Y on S, let X n and Y n be the complex structures obtained by restriction of X and Y to S n respectively. Then

(5) d ls (X, Y ) = lim n→∞ d ls (X n , Y n ).
Proof. From the definition, for any ǫ > 0, there exists a simple closed curve γ on Σ such that

d ls (X, Y ) < 1 2 log ℓ X (γ) ℓ Y (γ) + ǫ.
Such a curve γ must lie in some subsurface Σ n0 . As a result,

d ls (X, Y ) < d ls (X n0 , Y n0 ) + ǫ.
Since d ls (X n , Y n ) is increasing and ǫ is arbitrary, we conclude that

d ls (X, Y ) ≤ lim n→∞ d ls (X n , Y n ).
The other side of ( 5) is obvious.

Let R be a hyperbolic surface of infinite type with a geodesic pants decomposition P = {C i }. In the rest of this section, we always assume that the pair (R, P) is upper-bounded and that it admits short interior curves, i.e. that there is a sequence of curves of the decomposition α k = C i k contained in the interior of R with ℓ R (α i ) → 0. Note that for any point X ∈ T ls (R), the geodesic representative of P satisfies the same upper-boundedness property and admits short interior curves.

The following result was proved in [START_REF] Liu | Topological equivalence of metrics in Teichmüller space[END_REF] (see also [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF]Example 5.1]). The proof here is simpler. Proposition 5.2. Under the above assumptions, the inverse of the inclusion map I restricted to T qc (R): (T qc (R), d ls ) → (T qc (R), d qc ) is nowhere continuous. More precisely, for any X ∈ T qc (R), there is a sequence

X n ∈ T qc (R) with d qc (X, X n ) → ∞ while d ls (X, X n ) → 0. Proof. Let X ∈ T qc (R). Assume that ℓ X (α n ) = C in = ǫ n → 0. Let X n = τ tn αn (X), with t n = log | log ǫ n | → ∞.
By Corollary 4.2,

d ls (X, X n ) ≤ log | log ǫ n | 2C| log ǫ n | ,
which tends to 0 as n → ∞. On the other hand, Corollary 3.3 shows that d qc (X, X n ) → ∞.

Remark 5.3. There exist hyperbolic surfaces R with no pants decomposition satisfying Shiga's property, but where the space (T qc (R), d qc ) is topologically equivalent to the space (T qc (R), d ls ), see Kinjo [START_REF] Kinjo | On Teichmüller metric and length spectrums of topologically infinite Riemann surfaces[END_REF]. It would be interesting to know that whether the two metrics in Kinjo's examples are locally bi-Lipschitz.

Proposition 5.4 (Boundary point). Under the above assumptions, there exists a point in T ls (R) \ T qc (R) that can be approximated by a sequence in T qc (R) with the length-spectrum metric.

Proof. Let X ∈ T qc (R). By assumption, there exists a sequence of simple closed curves α n such that ℓ

X (α n ) = ǫ n → 0. Let X n = τ tn αn (X n-1 ) = τ tn αn • • • • • τ t2 α2 • τ t1 α1 (X), with t n = log | log ǫ n | → ∞.
Define X ∞ as the surface obtained from X by a twist of magnitude t i along α i for every i.

By Inequality (4), d qc (X, X n ) → ∞ and d qc (X, X ∞ ) = ∞.
For any simple closed curve γ on X, by an argument similar to that of the proof of Proposition 4.1 and Corollary 4.2, we have

d ls (X, X ∞ ) ≤ sup γ∈S ∞ n=1 i(γ, α n ) log | log ǫ n | 2C ∞ n=1 i(γ, α n )| log ǫ n | .
To see that the right hand side is uniformly bounded (independently of γ), we can use the inequality

∞ n=1 x i ∞ n=1 y i ≤ ∞ n=1 x i y i ,
that holds for positive values of x i and y i . To see this just note that all terms of k n=1 x i also appear in the product k n=1

x i y i k n=1 y i . With this inequality we can see that

∞ n=1 i(γ, α n ) log | log ǫ n | ∞ n=1 i(γ, α n )| log ǫ n | ≤ ∞ n=1 log | log ǫ n | | log ǫ n | .
Moreover, the last sum is controlled by

∞ log | log ǫ1| e -x xdx, which is bounded. Note that d ls (X n , X ∞ ) ≤ sup γ∈S ∞ k=n+1 i(γ, α k ) log | log ǫ k | 2C ∞ k=n+1 i(γ, α k )| log ǫ n | .
Since the interval ∞ log | log ǫn| e -x xdx tends to zero as n tends to infinity, we have d ls (X n , X ∞ ) → 0.

The above proposition shows that d ls is not complete when restricted to T qc (R). This result was already obtained in [START_REF] Alessandrini | On various Teichmüller spaces of a surface of infinite topological type[END_REF] and the proof we give here is more direct. Proof. As above, fix t k = log | log ǫ k | → ∞ and assume that ℓ X (α n ) = ǫ n → 0. We let Y n be the surface obtained from X by a twist of magnitude t i along α i for every i ≥ n. Then we can apply the proof of Proposition 5.4. Proposition 5.5 is also a consequence of Theorem 5.8 below.

Theorem 5.6. With the assumptions above, the space (T qc (R), d ls ) is not dense in the space (T ls (R), d ls ).

Proof. We will show that there exists a point in T ls (R) which is not a limit point of T qc (R) with the length-spectrum metric.

Start with a point X ∈ T qc (R). Assume that ℓ

X (α n ) = ǫ n → 0. Let Y n = τ Tn αn (X n-1 ) = τ Tn αn • • • • • τ T2 α2 • τ T1 α1 (X), with T n = N | log ǫ n | → ∞
where N is a fixed positive constant. In this proof, we could have taken N = 1, but taking a general N is important for the proof of Proposition 5.7 below. Define Y ∞ as the surface obtained from X by a twist of magnitude T i along α i for every i. It is not hard to see that Y ∞ ∈ T ls (R) and, in fact, [START_REF] Choi | Comparison between Teichmüller and Lipschitz metrics[END_REF]

d ls (X, Y ∞ ) ≤ N 2C .
Suppose that there is a sequence of X k ∈ T qc (R), such that d ls (Y ∞ , X k ) → 0 as k → ∞. Denote the difference of the twist coordinates of each X k from X (with respect to the pants decomposition P) by (τ k (C i )). Since the pants decomposition is upper-bounded and X k ∈ T qc (R), we have sup

i |τ k (C i )| < ∞, since otherwise, Corollary 3.3 would imply d qc (X, X k ) = ∞.
Note that for each X k , its difference of twist coordinates with Y ∞ is equal to

(1) T n -τ k (α n ), for each α n ; (2) -τ k (C i ), for each C i ∈ P \ {α n }.
For each α n , choose a simple closed curves β n as in the construction before Theorem 4.6; then, by theorem 4.6 we have

d (X k , Y ∞ ) ≥ sup n 1 2 log (N + 2)| log ǫ n | -C -τ k (α n ) 2| log ǫ n | + C .
The right hand side of the above inequality has a positive lower bound that is independent of k (because τ k (α n ) is bounded). As a result, the sequence X k cannot approximate Y ∞ in the length-spectrum metric.

Denote the closure of T qc (R) in T ls (R) (with the length-spectrum metric) by T qc (R). By Theorem 5.6, T ls (R) -T qc (R) is not empty. Proposition 5.7. Under the assumptions above, for any X ∈ T qc (R), there is a sequence of points Z k ∈ T ls (R) -T qc (R), such that d ls (X, Z k ) → 0.

Proof. For any X ∈ T qc (R), we let Z k be the same as Y ∞ , which we constructed in the proof of Theorem 5.6, by setting N = 1 k for each k. As we have shown before, Z k ∈ T ls (R) -T qc (R). Moreover, by inequality (6), we have

d ls (X, Z k ) ≤ 1 2Ck
, which tends to 0 as k tends to ∞.

Using the previous proposition, we can prove now the following result:

Theorem 5.8. Under the assumptions above, the space (T qc (R), d ls ) is nowhere dense in (T ls (R), d ls ).

Proof. This is equivalent to prove that T qc (R) has no interior point. Consider an arbitrary Y ∈ T qc (R) and an arbitrary ǫ > 0. If Y ∈ ∂T qc (R), then we let X be a point in T qc (R) such that d ls (X, Y ) < ǫ 2 . If Y ∈ T qc (R), we just set X = Y . By Proposition 5.7, there is a point Z ∈ T ls (R) -T qc (R) such that d ls (X, Z) < ǫ 2 . By the triangle inequality, d ls (Y, Z) < ǫ.

Connectedness of the length-spectrum Teichmüller space

In this section, we will prove that, if (R, P) is upper-bounded but does not satisfy Shiga's property, then (T ls (R), d ls ) is path-connected.

Assume that sup i ℓ R (C i ) ≤ M . We associate to R the Fenchel-Nielsen coordinates ((ℓ R (C i ), τ R (C i ))) i=1,2,••• , and we choose the twist coordinates in such a way that |τ R (C i )| < ℓ R (C i ).

Consider now an arbitrary X ∈ T ls (R), with Fenchel-Nielsen coordinates

((ℓ X (C i ), τ X (C i ))) i=1,2,••• .
We write X = (ℓ X (C i ), τ X (C i )) for simplicity.

Lemma 6.1. Suppose that d ls (R, X) < 2K. We can find a hyperbolic metric Y ∈ T qc (R), with Fenchel-Nielsen coordinates

Y = (ℓ Y (C i ), τ Y (C i )), such that ℓ Y (C i ) = ℓ X (C i ) and |τ Y (C i ) -τ R (C i )| < 2e K ℓ R (C i ) ≤ 2e K M .
Proof. Using a theorem of Bishop [START_REF] Bishop | Quasiconformal mappings of Y-pieces[END_REF], for any two geodesic pair of pants P and Q, there is a quasiconformal map from P to Q, which satisfies

(1) The quasiconformal dilatation of the map is less than 1 + Cd ls (P, Q), where C > 0 depends on an upper bound of d ls (P, Q) and the boundary length of P . (2) The map is affine on each of the boundary components.

Cut R into pairs of pants along P. We use Bishop's construction [START_REF] Bishop | Quasiconformal mappings of Y-pieces[END_REF] to deform each hyperbolic pair of pants, with boundary lengths ℓ R (•) into a pair of pants with boundary lengths ℓ X (•). Then we patch together the new hyperbolic pairs of pants to get a new hyperbolic metric Y homeomorphic to R, in the following way. Suppose two pairs of pants P 1 , P 2 are joined at α ∈ P and deformed into Q 1 , Q 2 . Then we patch together Q 1 and Q 2 by identifying the common image α by an affine map. In this way, the quasiconformal map between pairs of pants provided by Bishop can be extended to a global quasiconformal map between R and Y , with dilatation bounded by

1 + C sup i | log ℓ X (C i ) ℓ R (C i ) |,
where C is a positive constant depending on K, M . As a result,

2d qc (R, Y ) ≤ log(1 + C sup i | log ℓ X (C i ) ℓ R (C i ) |) ≤ C sup i | log ℓ X (C i ) ℓ R (C i ) |. Let Y = (ℓ Y (C i ), τ Y (C i ))
. Then, it follows from our construction that |τ Y (C i )| ≤ e KℓR(Ci) and |τ Y (C i ) -τ R (C i )| ≤ 2e KℓR(Ci) . Now we construct a continuous path in T ls (R) from Y to X by varying the twist coordinates. First we need the following result. Theorem 6.2. The hyperbolic structure X = (ℓ X (C i ), τ X (C i )) lies in T ls (R) if and only if there is a constant N > 0 such that for each i,

log ℓ X (C i ) ℓ Y (C i ) < N and |τ X (C i ) -τ R (C i )| < N max{| log ℓ R (C i )|, 1}.

Figure 1 .

 1 Figure 1. In each case, we have represented the curve Ci and its dual curve βi.

1 A 2 AFigure 3 .

 123 Figure 3. The image of γ under a twist.

Proposition 5 . 5 (

 55 Nowhere open). With the assumptions above, in the metric space (T ls (R), d ls ), any open neighborhood of a point X ∈ T qc (R) contains a point in T ls (R) \ T qc (R).

Acknowledgements. L. Liu and W. Su were partially supported by NSFC; D. Alessandrini is supported by Schweizerischer Nationalfonds 200021 131967/2.

Proof. By Lemma 6.1, we only need to consider the case where ℓ X (C i ) = ℓ R (C i ) for each i. In this case, X is a multi-twist deformation of R along P. If for each i,

then we can use the proof of Theorem 5.6 to show that d ls (R, X) < ∞.

Conversely, suppose that d ls (R, X) < ∞ and that there is a subsequence {α n } of P, such that

If there is a subsequence of {α n }, still denoted by {α n }, with length ℓ R (α n ) tending to zero, then, by Theorem 4.6, we have d ls (R, X) = ∞, which contradicts our assumption. If {ℓ R (α n )} is bounded below (and bounded above by assumption), then it also follows from Proposition 3.3 of [START_REF] Alessandrini | On local comparison between various metrics on Teichmüller spaces, to appear in Geometriae Dedicata[END_REF] that d ls (R, X) = ∞. As a result, there is a sufficiently large constant N such that for each i,

Now we can prove the following:

if we choose N sufficiently large (depending on K, M ). Now we use the multi-twist deformation to construct a path in T ls (R) connecting Y and X, by letting

shows that Y t lies in T ls (R).

Moreover, we can use the proof of Proposition 5.4 to prove that

If we take the Teichmüller geodesic path from R to Y , and then take the path obtained by the multi-twist deformations from Y to X, then we get a continuous (in fact, Lipschitz) path in T ls (R) connecting R and X.

The following question remains open: Question 6.4. Suppose that (R, P) is upper-bounded but does not satisfy Shiga's condition. Is (T ls (R), d ls ) contractible?


Michael Dalager-Pedersen

Mette Søgaard

Henrik Carl Schønheyder

Staphylococcus aureus skin and soft tissue infections in primary healthcare in 1 Denmark: a 12-year population-based study 2 3

Keywords: Staphylococcus aureus, skin and soft tissue infection, impetigo, wound, community, epidemiology. 21 22

A rise in community-onset Staphylococcus aureus infections has been observed in European countries. To ascertain 3 secular trends of S. aureus infections in primary healthcare in Denmark, we conducted this register-based study in 4 North Denmark Region, 1997Region, -2008. . We identified all skin and mucosa specimens obtained by general practitioners and 5 all prescriptions for the preferred oral anti-staphylococcal antibiotic, dicloxacillin. Repeat observations within a 12-6 month period were excluded prior to calculation of age and gender standardised incidence rates per 100.000 person-7 years.

8

We included 108,758 specimens of which 42,778 (39%) yielded S. aureus. The annual incidence rate of specimens 9 doubled during the study period reaching 2,399 in 2008. The overall rate of S. aureus isolates increased 2-fold to a 10 stable rate at about 850, but for isolates from children and for impetigo specimens the increase was steeper with a peak 11 in 2002. 156,462 dicloxacillin prescriptions had been redeemed and the annual prescription rate increased 2.5-fold, 12 peaking at 3,714 in 2007.

13

In conclusion, annual rates of specimens, S. aureus infections and dicloxacillin prescriptions more than doubled in 14 primary healthcare during the 12-year period.

15

A major impetigo epidemic and calls for antibiotic stewardship with increased utilisation of specimens were 16 contributing factors.

23

Information on specimens was retrieved from the laboratory information system (ADBakt; Autonik, Sköldinge, 24 Sweden). We included specimens from the following sites/lesions: skin, abscess, pus, wound, impetigo, conjunctiva, 25 urethra, vulva or anus, and specimens without specification. We excluded all specimens that indicated previous hospital 26 contact (e.g., incisional wound, tracheostomy, and catheter exit site) and surveillance specimens from the nasal 27 vestibule. Specimens from the respiratory tract including the external auditory meatus were excluded as we could not 28 determine whether S. aureus might represent a colonising flora.

Prescription data 1

In Denmark, antibiotics are available on prescription only and redeemed prescriptions are reimbursed according to 2 changeable administrative rules. Reimbursement had effect during the entire study period for oral prescriptions of the 3 isoxazolylpenicillins dicloxacillin and flucloxacillin (Anatomical Therapeutic Chemical classification J01CF01).

4

Reimbursed prescriptions redeemed at pharmacies within the study area have been recorded in a 5 pharmacoepidemiological register since 1991 [14]. We retrieved information on all prescriptions of dicloxacillin and 6 flucloxacillin, but only the former is included in the study as consumption of the latter was negligible. We defined an incident observation as a person with a specimen, S. aureus isolate or dicloxacillin prescription and no 10 similar record during the preceding 12-month period.

11

Repeat observations within a 12-month period were excluded because we wanted to reduce the impact of recurrent 12 infections.

13

We present annual incidence rates of specimens, S. aureus isolates and redeemed dicloxacillin prescriptions per 14 100,000 person-years at risk (pyr) for the period 1997-2008. Data from 1996 ensured incident observations from 1997.

15

The rate of specimens and S. aureus isolates were stratified into 4 age groups and 5 specimen groups. Age groups were 16 0-14 (children), 15-64, 65-79 and 80+ years (oldest old). Specimen groups were abscess/pus, wound, impetigo, skin 17 (including unspecified site), and mucosa (including conjunctiva, genital area and anus). We calculated the rates by 18 dividing the annual number of patients with a given observation by the number of inhabitants in the study area in the 19 beginning of that same year. To compare the rates among different years, we standardized the rates to the age and sex During the study period, we identified 143,887 specimens of which 108,758 were incident specimens. A total of 42,778 4 (39%) of the incident specimens were positive for S. aureus. Numbers of observations per 4-year period appear from 5 Table 1. There was an approximately 2-fold increase in rates of specimens and S. aureus isolates between 1997 and 6 2008 (Figure 1). The rate of S. aureus isolates paralleled the increase observed for specimens until 2002 and remained 7 stable thereafter. In 2008 the incidence rates of specimens peaked at 2,399 and the rate of S. aureus isolates peaked in 8 2006 at 889 per 100,000 pyr. 9

During the study period there was a steady increase in the incidence rate of S. aureus isolates among adults with the 10 highest rate observed for the oldest old (Figure 2a). For children the overall increase was steeper and peaked at 1,740 11 per 100,000 pyr in 2002. Analysis of incidence rates by specimen group showed a similar peak for impetigo specimens 12 (Figure 2b). A subanalysis of patients with impetigo showed that this rise predominantly was caused by infections 13 among children (Figure 2c). Specimens from wounds accounted for the highest annual incidence rates of S. aureus 14 isolates (Figure 2b), with a particularly high and steady increase among the oldest old (Figure 2d). Rates of S. aureus 15 isolates from other specimen types remained stable during the entire study period (Figure 2b). Incidence rates and time 16 trends were similar for males and females. 17 We identified 246,400 dicloxacillin prescriptions of which 156,642 were incident. The incidence rate of prescriptions 25 more than doubled during the study period reaching ~3,600 per 100,000 pyr in 2008 (Figure 1). In adults, the increase 26 was unremitting and reached almost 7,000 per 100,000 pyr among the oldest old. In children, the time pattern was 27 similar to that seen for S. aureus isolates from impetigo specimens. Notably, the peak incidence of prescriptions 28 (~3,800 per 100,000 pyr) was almost 5-fold the incidence rate of S. aureus isolates.

29

Discussion 1 2

We have documented a continuous rise in the rate of specimens obtained in primary healthcare with a doubling from 3 1997 to 2008. There was a parallel increase of S. aureus isolates up to 2002, thereafter the incidence remained fairly 4 stable. The largest source of S. aureus isolates was wound specimens with the highest incidence rates observed among 5 the oldest old. A remarkable finding was the steep increase in impetigo among children from 1997 to 2003, followed by 6 a rapid decline. The similar trends in the proportion of fusidic acid resistant isolates indicate a prominent role of the 7 EEFIC in this impetigo epidemic as shown in other European countries 8-10]. Dicloxacillin prescriptions followed the 8 trend for specimens but at a higher rate. 9 10 Strengths of the current study include access to data on microbiological specimens and prescriptions and the 11 population-based design that allowed calculation of age and sex adjusted incidence rates. However, this study has 12 limitations and most important is the possibility for surveillance bias. We had little information on GPs decision to 13 obtain specimens and no Danish guideline specifies the indications for obtaining specimens from skin and soft tissue 14 infections. In an effort to reduce surveillance bias we excluded repetitious observations and surveillance specimens. 15 Specimens that indicated hospital or healthcare contact were also excluded but we had no information on previous 16 hospital admissions. Nosocomial infections may therefore have affected the rates of specimens and S. aureus isolates, 17 especially among the elderly. Likewise, dicloxacillin prescriptions from hospital and outpatient clinics may have been 18 included because the prescriber's code was often unavailable. Based on available data, we estimated that less than 15% 19 of prescriptions were related to hospital or ambulatory care. Also, awareness of the EEFIC may have caused a shift 20 from fusidic acid to dicloxacillin, which could have contributed to the steady rise in consumption of dicloxacillin.

21

Fusidic acid is mostly used in a topical formulation that is not reimbursed. Therefore, we lack data on consumption of 22 this drug from the regional database.

24

Another concern is the validity of the clinical information on the request form. This information was given on a 25 voluntary basis and may also have been coded erroneously in the laboratory. We consider the source information 26 reliable for impetigo and abscess/pus. In contrast, information may be less valid for skin and wound specimens; these 27 may include eczematous lesions for which a GP might obtain specimens in order to rule out infection rather than to 28 confirm an infectious aetiology.

29

Our findings are consistent with information available from other countries. Fleming et al. [16] examined the trends in 1 incidence and treatment of skin infections presenting to GPs in England and Wales. From 1999 to 2005, they reported a 2 slight decrease in the incidence of skin infections while the number of flucloxacillin prescriptions increased. The study 3 examined several different skin infections, but it is noteworthy that wound infections were not included. In our study 4 wound specimens was the most important source of S. aureus isolates with a particularly high and steadily increasing 5 rate among the oldest old. The increasing age of the population makes this a worrisome finding [17]. Hospital 6 admission rates for skin infections have also been found to be increasing in the US [18] and the UK [11]. No Danish 7 studies have addressed this issue, which warrants further attention, preferably with inclusion of microbiological data.

9

The increasing rates of specimens and S. aureus isolates in the first half of our study period were driven by an impetigo Resistance to fusidic acid (%)

Denmark

  Staphylococcus aureus infections have attracted increasing attention in recent years. One important 3 reason is the spread of methicillin-resistant S. aureus (MRSA) outside hospitals and healthcare facilities, especially in 4 the United States [1]. Community-onset MRSA infections are also recognized in Europe but the overall prevalence 5 remains low [2-3]. Another possible reason is the increasing incidence of impetigo that has been observed in several 6 Western European countries during the last decade [4-7]. This rise has been linked to the epidemic European fusidic 7 acid resistant impetigo clone (EEFIC), which belongs to clonal complex (CC) 121 and is characterised by low level 8 resistance to fusidic acid [8-10]. 9 10 A recent registry-based study from the United Kingdom reported an increasing rate of hospital admissions for 11 community-onset infections typically caused by S. aureus and a rise of anti-staphylococcal antibiotic prescriptions in 12 primary care settings [11]. The standardised admission rate rose 3-fold for pyogenic skin infections and 6-fold for 13 impetigo from 1989-1990 to 2003-2004. Similarly, flucloxacillin prescriptions increased 1.8-fold and fusidic acid 14 prescriptions 2.5-fold during 1991-2006. Limitations to this study are lack of culture data and the fact that only a 15 fraction of community-onset S. aureus infections lead to hospital admission. Nevertheless, the study highlights the need 16 for better estimates of the burden and time trends of staphylococcal infections in the community. 17 18 Apart from antibiotic resistance surveys, we are not aware of longitudinal studies based on bacteriological specimens 19 taken by general practitioners (GPs). Therefore, we conducted this population-based study of specimens obtained by 20 GPs and practising specialists as well as prescriptions of dicloxacillin which is the preferred oral anti-staphylococcal 21 beta-lactam in Denmark. Our aim was to describe the temporal trends of community-onset S. aureus infections in 22 conducted in North Denmark region within the boundaries of the former North Jutland County. We 4 obtained data for the years 1996-2008; during this period the catchment population was close to 495,000 inhabitants 5 (approximately 9% of the Danish population). The primary care sector included approximately 320 GPs and practising 6 specialists. 7 All Danish residents have free access to tax-supported healthcare provided by GPs and public hospitals. A unique 8 personal identification number is provided at birth or immigration by the Central Person Registry and is used for all 9 healthcare contacts and health administrative databases. Clinical Microbiology, Aalborg Hospital, provided diagnostic bacteriology to all GPs and practising 13 specialists in the study area. Point of care tests undertaken by GPs did not include skin specimens. Specimens were 14 obtained with charcoal-coated cotton wool swabs and transported in Stuart's medium (SSI Diagnostika, Hillerød, France). S. aureus was identified by clumping factor (StaphPlus, bioMérieux) or coagulation of horse citrate plasma. 20 From 2003 onwards, most S. aureus isolates were tested for susceptibility to fusidic acid by tablet diffusion 21 (Neosensitabs, Rosco, Taastrup, Denmark) on Mueller-Hinton II agar (SSI Diagnostika). A break-point of 22 susceptibility >32 mm was confirmed by graphical analysis of recorded diameters [12,13].

  20 distribution of the population in the region in 1997, using direct standardisation. Population statistics for the years 21 1997-2006 was obtained from Statistics Denmark [15]. Data from 2007 and 2008 were unavailable due to a reform of 22 the local government in 2007. Therefore, we used 2006 data instead. 23 To estimate presence of the EEFIC which is distinctive by its low-level resistance to fusidic acid [8-10], we calculated 24 the annual prevalence proportion of fusidic acid resistance overall and per specimen group. 25 26 STATA 9.2 for Windows (Stata Corp., College Station, TX) was used for data analyses. Data were obtained from 27 regional registries which are generally available to researchers and their use does not require informed consent.28Therefore, permission from the Danish Scientific Ethical Committee was not required for this study.

  and S. aureus isolates 3

  S. aureus isolates from impetigo patients were resistant to fusidic acid during 2003-2005, i.e. more than 20 twice the prevalence proportion for isolates from other specimen groups (Figure3). From 2006 there was a 21 considerable fall in the prevalence of fusidic acid resistance among isolates from impetigo specimens.

3Fig. 1 5 Fig. 2 9 Fig. 3

 15293 Figure Captions

  Figure

Table 1

 1 Number of cultures from skin and mucosa and Staphylococcus aureus isolates obtained by general practitioners and practising specialists in North Denmark

	03	04	05	06	07	08
			Year of diagnosis

Denmark). Information on specimen type was given on a request form; further information on type of lesion (e.g. 16 impetigo) and anatomic region were voluntary.

Specimens were processed by standard culture techniques including 5% horse blood agar (SSI Diagnostika). In 2005 a

chromogenic agar medium selective for S. aureus was introduced (chromID S. aureus, bioMérieux, Marcy l'Etoile,

the study. Furthermore, Danish GPs are well informed about antibiotic resistance and antibiotic stewardship through

multiple channels including the annual physician's desk reference (Medicinfortegnelsen) [23], the Institute for Rational

http://www.un.org/esa/population/unpop.htm. Accessed 1 May 2010

Christensen KLY, Holman RC, Steiner CA, Sejvar JJ, Scholl BJ, Schonberger LB. Infectious disease

hospitalizations in the United States. Clin Infect Dis 2009; 49: 1025-35.

Koning S, Mohammedamin RSA, van der Wouden JV, Suiljekom-Smit LW, Schellevis FG, Thomas S. Impetigo:

Vej-Hansen B (Eds). Laegeforeningens medicinfortegnelse. 21. edn. Laegeforeningens Forlag, København 1998, 27 831-55.

Institute for Rational Pharmacotherapy (IRF) [internet]. Copenhagen. http://www.irf.dk. Accessed 1 May 2010


Christophe Prieur

Frédéric Mazenc

ISS Lyapunov functions for time-varying hyperbolic partial differential equations

A family of time-varying hyperbolic systems of balance laws is considered. The partial differential equations of this family can be stabilized by selecting suitable boundary conditions. For the stabilized systems, the classical technique of construction of Lyapunov functions provides with a function which is a weak Lyapunov function in some cases, but is not in others. We transform this function through a strictification approach which gives a time-varying strict Lyapunov function which allows us to establish asymptotic stability in the general case and a robustness property with respect to additive disturbances of Input-to-State Stability (ISS) type.

I. INTRODUCTION

Lyapunov function based techniques are central in the study of dynamical systems. This is especially true for those having an infinite number of dynamics. These systems are usually modelled by time-delay systems or partial differential equations (PDEs). For the latter family of systems, Lyapunov functions are useful for the analysis of many different types of problems, such as the existence of solutions for the heat equation [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF], or the controllability of the semilinear wave equation [START_REF] Coron | Global steady-state stabilization and controllability of 1D semilinear wave equations[END_REF]. The present paper focuses on a class of onedimensional hyperbolic equations like those written as a system of conservation laws. The study of this class of PDEs is crucial when considering a wide range physical networks having an engineering relevance. Among the potential applications we have in mind, there are for instance hydraulic networks (for irrigation or navigation), electric line networks, road traffic networks [START_REF] Haut | A second order model of road junctions in fluid models of traffic networks[END_REF] or gas flow in pipeline networks [START_REF] Bastin | Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks[END_REF], [START_REF] Dick | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF]. The importance of these applications motivates a lot of theoretical questions on hyperbolic systems which for instance pertain to optimal control and controllability as considered in [START_REF] Gugat | Optimal control for traffic flow networks[END_REF], [START_REF] Colombo | Optimal control in networks of pipes and canals[END_REF], [START_REF] Gugat | Global boundary controllability of the Saint-Venant system for sloped canals with friction[END_REF].

The stabilizability of such systems is often proved by means of a Lyapunov function as illustrated by the contributions [START_REF] Dick | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF], [START_REF] Krstic | Backstepping boundary control for first order hyperbolic pdes and application to systems with actuator and sensor delays[END_REF], [START_REF]Backstepping boundary control for first-order hyperbolic pdes and application to systems with actuator and sensor delays[END_REF] where different control problems are solved for particular hyperbolic equations (see also [START_REF] Prieur | Robust boundary control of systems of conservation laws[END_REF]for the use of Riemann coordinates techniques). For more general nonlinear hyperbolic equations, the knowledge of Lyapunov functions can be useful for the stability analysis of a system of conservation laws (see [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]), or even for the design for these systems of stabilizing boundary controls (see the recent work [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF]).

To demonstrate asymptotic stability through the knowledge of a weak Lyapunov function i.e. a Lyapunov function Christophe Prieur is with Department of Automatic Control, Gipsa-lab, 961 rue de la Houille Blanche, BP 46, 38402 Grenoble Cedex, France, christophe.prieur@gipsa-lab.grenoble-inp.fr; Frédéric Mazenc is with Team INRIA DISCO, CNRS-Supelec, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France, Frederic.MAZENC@lss.supelec.fr whose derivative, along the trajectories of the system which is considered, is nonpositive, the celebrated Lasalle invariance principle has to be invoked (see e.g. [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF], [START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF], [START_REF] Luo | Stability and stabilization of infinite dimensional systems and applications[END_REF]). It requires to state a precompactness property for the solutions, which may be difficult to prove (and is not even always satisfied, as illustrated by the hyperbolic systems considered in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]). This technical step is not needed when is available a strict Lyapunov function i.e. a Lyapunov function whose derivative, along the trajectories of the system which is considered, is negative definite. Thus designing such a Lyapunov function is a way to overcome this technical difficulty, as done for example in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]. These remarks motivate the present paper which is devoted to new Lyapunov techniques for the study of stability and robustness properties of Input-to-State-Stable (ISS) type for a family of time-varying linear hyperbolic PDEs with disturbances. By first applying the classical technique of construction of Lyapunov functions available in the literature, we will obtain a function which is a weak Lyapunov function for some of the systems we consider, but not for the others. Next, we will transform this function through a strictification approach, which owes a great deal to the one presented in [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF], (see also [START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF], [START_REF] Mazenc | A simplified design for strict Lyapunov functions under Matrosov conditions[END_REF]) and obtain that way a strict Lyapunov function which allows us to estimate the robustness of the stability of the systems with respect to the presence of uncertainties and or external disturbances. The function we shall construct is given by an explicit expression.

It is worth mentioning that although the ISS notion is very popular in the area of the dynamical systems of finite dimension (see e.g. the recent survey [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]), or for systems with delay (see for instance [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]), the present work is, to the best of our knowledge, the first one which uses it to characterize a robustness property for hyperbolic PDEs. This work parallels what has been done in [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] where ISS-Lyapunov functions for semilinear time-invariant parabolic PDEs are derived using strictification techniques (see also [START_REF] Dashkovskiy | On the uniform input-to-state stability of reaction-diffusion systems[END_REF] where ISS properties are compared for a reaction-diffusion system with its finite dimensional counterpart without diffusion). On the other hand, the construction we shall present is significantly disconnected from the one in [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] because the family of systems we will study is very different from the one considered in [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF]. In particular the systems studied in [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] are time-invariant whereas the systems considered in the present work are time-varying.

The paper is organized as follows. Basic definitions and notations are introduced in Section II. In Section III the analysis of the robustness of a linear time-varying hyperbolic PDE with uncertainties is carried out by means of the design of an ISS Lyapunov function. In Section IV, the main result is illustrated by an example. Concluding remarks in Section V end the work.

Notation. Throughout the paper, the argument of the functions will be omitted or simplified when no confusion can arise from the context. A function α is said to be of class K ∞ if it is continuous, zero at zero, increasing and unbounded. For any integer n, we let Id denote the identity matrix of dimension n. Given a continuously differentiable function A : R n → R, ∂A ∂Ξ (Ξ) stands for the vector ( ∂A ∂ξ1 (Ξ)..... ∂A ∂ξn (Ξ)) ∈ R n . The norm induced from the Euclidean inner product of two vectors will be denoted by | • |. Given a matrix A, its induced matrix norm will be denoted by ||A||, and Sym(A) = 1 2 (A + A ) stands for the symmetric part of A. We denote the set of diagonal positive definite matrices by D n,+ . The norm |•| L 2 (0,L) is defined by:

|φ| L 2 (0,L) = L 0 |φ(z)| 2 dz.
Finally, given two topological spaces X and Y , we denote by C 0 (X; Y ) (resp. C 1 (X; Y )) the set of the continuous (resp. continuously differentiable) functions from X to Y . Following [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], we introduce the notation, for all matrices M ∈ R n×n ,

ρ 1 (M ) = inf{ ∆M ∆ -1 , ∆ ∈ D n,+ } . (1)

II. BASIC DEFINITIONS AND NOTIONS

Throughout our work, we will consider partial differential equations of the form

∂X ∂t (z, t)+Λ(z, t) ∂X ∂z (z, t) = F (z, t)X(z, t)+δ(z, t) , (2)
where z ∈ [0, L], t ∈ [0, +∞), and Λ(z, t) is a diagonal matrix in R n×n whose m first diagonal terms are nonnegative and the n -m last terms are nonpositive. We assume that the function δ(z, t) is a disturbance of class C 1 , F (z, t) is a periodic function with respect to t of class C 1 , Λ(z, t) is of class C 2 , periodic with respect to t of period T . The boundary conditions are written as

X + (0, t) X -(L, t) = K X + (L, t) X -(0, t) (3)
where

X = X + X - , X + ∈ R m , X -∈ R n-m , and K ∈ R n×n is a constant matrix. The initial condition is X(z, 0) = X 0 (z) , ∀z ∈ (0, L) , (4)
where X 0 is a function in L 2 (0, L)).

When the functions Λ, and F do not depend on z and t, and when Λ is invertible, the Cauchy problem (2), ( 3) and ( 4) is well-posed (see e.g. [23, Section 2.1 and Section 2.3]). This means that for any initial condition X 0 in L 2 ((0, L), R n ), for any T > 0, and any δ in L 2 ((0, L) × (0, T ); R n ), there exists C(T ) > 0 such that a solution X ∈ C 0 ([0, +∞); L 2 ((0, L); R n )) exists, is unique and satisfies, for all t ∈ [0, T ],

|X(., t)| L 2 ((0,L);R n ) ≤ C(T )|X 0 | L 2 ((0,L);R n ) .
Now we introduce the notions of weak, strict and ISS Lyapunov functions that we consider in this paper (see for instance [START_REF] Luo | Stability and stabilization of infinite dimensional systems and applications[END_REF]Def. 3.62] for the notion of Lyapunov functions and [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] for the notion of ISS Lyapunov functions in an infinite dimensional context).

Definition 2.1: Let ν : L 2 (0, L) × R → R be a continuously differentiable function, periodic with respect to t. The function ν is said to be a weak Lyapunov function for (2) with (3), if there are two functions κ S and κ M of class K ∞ such that, for all functions φ ∈ L 2 (0, L) and for all t ∈ [0, +∞),

κ S |φ| L 2 (0,L) ≤ ν(φ, t) ≤ L 0 κ M (|φ(z)|) dz (5)
and, in the absence of δ, for all solutions of (2) satisfying (3), and all t ≥ 0, dν(X(., t), t) dt ≤ 0 .

The function ν is said to be a strict Lyapunov function for (2) with (3) if, in the absence of δ, there exits a real number λ 1 > 0 such that, for all solutions of (2) satisfying (3), and for all t ≥ 0, dν(X(., t), t) dt ≤ -λ 1 ν(X(., t), t) .

The function ν is said to be an ISS Lyapunov function for (2) with (3) if there exit a positive real number λ 1 > 0 and a function λ 2 of class K such that, for all continuous functions δ, for all solutions of (2) satisfying (3), and for all t ≥ 0, dν(X(., t), t) dt ≤ -λ 1 ν(X(., t), t)

+ L 0 λ 2 (|δ(z, t)|)dz .
Remark 1: 1. For conciseness, we will often use the notation ν instead of dν(X(.,t),t) dt .

2.

Let us recall that, when is known a weak Lyapunov function, asymptotic stability can be often established via the celebrated LaSalle invariance principle applies (see [START_REF] Luo | Stability and stabilization of infinite dimensional systems and applications[END_REF]Theorem 3.64] among other references).

3. When a strict Lyapunov function ν exists for (2) with (3) and δ is not present, then the value of ν along the solutions of (2) satisfying (3) exponentially decays to zero and therefore each solution X(z, t) satisfies lim t→+∞ |X(., t)| L 2 (0,L) = 0. When in addition, there exists a function κ L of class K ∞ , such that, for all functions φ ∈ L 2 (0, L) and for all t ≥ 0,

ν(φ, t) ≤ κ L |φ| L 2 (0,L) , (6)
then the system (2) is globally asymptotically stable for the topology of the norm L 2 . 4. When the system (2) with (3) admits an ISS Lyapunov function ν, then, one can check through elementary calculations 1 that, for all solutions of (2) satisfying (3) and for all 1 For instance the next inequality follows from the fact that we have, for all κ of class K, and for all positive values a and b, κ(a + b) ≤ κ(2a) + κ(2b) , and from the fact that the function κ -1

S is zero at zero and nondecreasing.

instants t ≥ t 0 , the inequality

|X(., t)| L 2 (0,L) ≤ κ -1 S (ϕ 1 (t, t 0 , X)) + κ -1 S (ϕ 2 (t, t 0 )) with ϕ 1 (t, t 0 , X) = 2e -λ1(t-t0) L 0 κ M (|X(z, t 0 )|)dz and ϕ 2 (t, t 0 ) = 2 λ 1 sup ∈[t0,t] L 0 λ 2 (|δ(z, )|)dz
holds. This inequality is the analogue for the PDEs (2) with (3) of the ISS inequalities for ordinary differential equations.

It gives an estimation of the influence of the disturbance δ on the solutions of the system (2) with (3). •

III. ISS LYAPUNOV FUNCTIONS FOR LINEAR HYPERBOLIC SYSTEMS

Before stating the main theoretical result of the work, some comments are needed. Since, in the case where the system ( 2) is such that m < n, one can replace X(z, t) by

X + (z, t) X -(L -z, t)
and obtain that way a new PDE whose corresponding diagonal matrix function Λ has only nonnegative entries, we may assume without loss of generality that Λ is diagonal with nonnegative entries and then the boundary conditions (3) are the boundary condition

X(0, t) = KX(L, t) . (7)
Next, we recall an important result given in [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF] because it sheds light on the problem we consider and, more precisely, on the assumptions we introduce below. If Λ is constant, positive, and ρ 1 (K) < 1, where the ρ 1 is the function defined in (1), then the system (2), when F (z, t) = 0 and δ(z, t) = 0 for all z ∈ [0, L], t ≥ 0, with the boundary conditions ( 3) is exponential stable in H 2 -norm. Moreover there exist a diagonal positive definite matrix 2 Q ∈ R n×n and a positive constant ε > 0 such that

Sym(QΛ -K QΛK) ≥ εId . (8)
Furthermore, following what has been assumed for the parabolic equation in [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF], it might seem natural to consider the case where F (z, t) possesses some stability properties. On the other hand, there is no reason to believe that this property is always needed. These remarks lead us to introduce the following assumption:

Assumption 1: For all t ≥ 0 and for all z ∈ [0, L], all the entries of Λ(z, t) are nonnegative. There exist a symmetric positive definite matrix Q, a real number α ∈ (0, 1), a continuous real-valued function r, periodic of period T > 0 with a positive mean value, i.e. such that the constant

B = T 0 r(m)dm (9)
2 such a matrix Q may be obtained by selecting a diagonal positive matrix ∆ such that ∆K∆ -1 < I. Then selecting Q = ∆ 2 Λ -1 and ε > 0 sufficiently small we have [START_REF] Gugat | Global boundary controllability of the Saint-Venant system for sloped canals with friction[END_REF].

is positive, satisfying, for all t ≥ 0 and for all z ∈ [0, L], the following inequalities:

Sym αQΛ(L, t) -K QΛ(L, t)K ≥ 0 , (10)
Sym (QΛ(z, t)) ≥ r(t)Id , (11)
Sym Q ∂Λ ∂z (z, t) + 2QF (z, t) ≤ 0 . ( 12
)
We are ready to state and prove the main result of the paper.

Theorem 1: Assume the system (2) with the boundary conditions ( 7) satisfies Assumption 1. Let µ be any real number such that

0 < µ ≤ - 1 L ln(α) . ( 13
)
Then the function U :

L 2 (0, L) × R → R defined, for all φ ∈ L 2 (0, L) and t ∈ R, by U (φ, t) = e 1 T R t t-T R t q(m)dmd L 0 φ(z) Qφ(z)e -µz dz , (14)
with

q(t) = µ ||Q|| r(t) - B 2T , ( 15
)
where Q and r are the matrix and the function in Assumption 1, is an ISS Lyapunov function for the system ( 2) with ( 7). Remark 2: 1. Assumption 1 does not imply that for all fixed z ∈ [0, L], the ordinary differential equation Ẋ = F (z, t)X is stable. In Section IV, we will study an example where this system is exponentially unstable.

2. The fact that Q is symmetric positive definite and all the entries of Λ(z, t) are nonnegative does not imply that Sym (QΛ(z, t)) is positive definite. That is the reason why we do not assume that r is a nonnegative function.

3. Assumption 1 holds when, in the system (2), Λ(z, t) is constant, F (z, t) is constant and diagonal, δ(z, t) = 0 for all z ∈ [0, L] and t ≥ 0 and the boundary condition (3) satisfies Sym QΛ -K QΛK ≥ 0 , Sym (QF ) ≤ 0 for a suitable symmetric positive definite matrix Q. Therefore Theorem 1 generalizes the sufficient conditions of [START_REF] Diagne | Lyapunov exponential stability of linear hyperbolic systems of balance laws[END_REF] for the exponential stability of linear hyperbolic systems of balance laws to the time-varying case and to the perturbed case.

4. The Lyapunov function U defined in ( 14) is a timevarying function, periodic of period T . In the case where the system is time-invariant, one can chose a constant function q(t), which results with a time-invariant function [START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF]. This function is a quite usual Lyapunov function candidate in the context of the stability analysis of PDEs (see e.g., [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], [START_REF] Coron | On the null asymptotic stabilization of the twodimensional incompressible Euler equations in a simply connected domain[END_REF], [START_REF] Xu | Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems[END_REF]).

• Proof. We begin the proof by showing that the function

V (φ) = L 0 φ(z) Qφ(z)e -µz dz ,
is a weak Lyapunov function for the system (2) with ( 7) when Assumption 1 and ( 13) are satisfied and δ is identically equal to zero.

We note for later use that, for all φ ∈ L 2 (0, L),

1 β L 0 |φ(z)| 2 dz ≤ V (φ) ≤ β L 0 |φ(z)| 2 dz (16)
with β = max ||Q||, e µL λ Q , where λ Q is the smallest eigenvalue of Q. Now, we compute the time-derivative of V along the solutions of ( 2) with ( 7):

V = 2 L 0 X(z, t) Q ∂X ∂t (z, t)e -µz dz = 2 L 0 X(z, t) Q × -Λ(z, t) ∂X ∂z (z, t) + F (z, t)X(z, t) + δ(z, t) ×e -µz dz = -R 1 (X(., t), t) + R 2 (X(., t), t) +R 3 (X(., t), t) , (17)
with

R 1 (φ, t) = 2 L 0 φ(z) QΛ(z, t) ∂φ ∂z (z)e -µz dz , R 2 (φ, t) = 2 L 0 φ(z) QF (z, t)φ(z)e -µz dz , R 3 (φ, t) = 2 L 0 φ(z) Qδ(z, t)e -µz dz . Now, observe that R 1 (φ, t) = L 0 ∂(φ(z) QΛ(z, t)φ(z)) ∂z e -µz dz - L 0 φ(z) Q ∂Λ ∂z (z, t)φ(z)e -µz dz .
Performing an integration by part on the first integral and using [START_REF] Colombo | Optimal control in networks of pipes and canals[END_REF], we get

R 1 (φ, t) = φ(L) QΛ(L, t)φ(L)e -µL -φ(0) QΛ(0, t)φ(0) +µ L 0 φ(z) QΛ(z, t)φ(z)e -µz dz - L 0 φ(z) Q ∂Λ ∂z (z, t)φ(z)e -µz dz . (18)
Combining ( 17) and ( 18), we obtain V = -X(L, t) QΛ(L, t)X(L, t)e -µL +X(0, t) QΛ(0, t)X(0, t) +R 4 (X(., t), t) Using [START_REF] Colombo | Optimal control in networks of pipes and canals[END_REF], we obtain V = -X(L, t) QΛ(L, t)X(L, t)e -µL +X(L, t) K QΛ(L, t)KX(L, t) +R 4 (X(., t), t) +2 L 0 X(z, t) Qδ(z, t)e -µz dz .

By grouping the terms and using the notation

N (t) = K QΛ(L, t)K
we obtain V = X(L, t) N (t) -e -µL QΛ(L, t) X(L, t) +R 4 (X(., t), t)

+2 L 0 X(z, t) Qδ(z, t)e -µz dz .
Grouping the terms in R 4 (X(., t), t), we obtain V = X(L, t) N (t) -e -µL QΛ(L, t) X(L, t)

- L 0 X(z, t) QM (z, t)X(z, t)e -µz dz +2 L 0 X(z, t) Qδ(z, t)e -µz dz , with M (z, t) = µΛ(z, t) - ∂Λ ∂z (z, t) -2F (z, t) .
The inequalities [START_REF]Backstepping boundary control for first-order hyperbolic pdes and application to systems with actuator and sensor delays[END_REF] and ( 13) imply that

V ≤ - L 0 X(z, t) QM (z, t)X(z, t)e -µz dz +2 L 0
X(z, t) Qδ(z, t)e -µz dz .

It follows from ( 12) that V ≤ -µ L 0 X(z, t) QΛ(z, t)X(z, t)e -µz dz +2 L 0 X(z, t) Qδ(z, t)e -µz dz .

Using [START_REF] Prieur | Robust boundary control of systems of conservation laws[END_REF], we deduce that

V ≤ -µr(t) L 0 |X(z, t)| 2 e -µz dz +2 L 0 X(z, t) Qδ(z, t)e -µz dz .
From ( 16), we deduce that V ≤ -µ ||Q|| r(t)V (X(., t))

+2||Q|| L 0 |X(z, t)||δ(z, t)|e -µz dz . (19)
It follows that, for all κ > 0, V ≤ -µ ||Q|| r(t)V (X(., t)) [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF] implies that when r(t) is nonnegative and δ is identically equal to zero, the function V is a weak Lyapunov function for the system (2) with the initial conditions [START_REF] Colombo | Optimal control in networks of pipes and canals[END_REF]. However, we did not assume that the function r is nonnegative and we aim at establishing that the system is ISS with respect to δ. This leads us to apply a strictification technique which transforms V into a strict Lyapunov function. The technique of [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF]Chapter 11] leads us to consider the time-varying candidate Lyapunov function

+2||Q||κ L 0 |X(z, t)| 2 e -µz dz + ||Q|| 2κ L 0 |δ(z, t)| 2 e -µz dz ≤ -q κ (t)V (X(., t)) + ||Q|| 2κ L 0 |δ(z, t)| 2 dz , with q κ (t) = µ ||Q|| r(t) -2||Q||κ λ Q . The inequality
U κ (t, φ) = e sκ(t) V (φ) , with s κ (t) = 1 T t t-T t q κ (
m)dmd . Through elementary calculations, one can prove that the time-derivative of U κ along the solutions of (2) with the initial conditions [START_REF] Colombo | Optimal control in networks of pipes and canals[END_REF] satisfies:

Uκ ≤ -e sκ(t) q κ (t)V (X(., t))

+ ||Q|| 2κ e sκ(t) L 0 |δ(z, t)| 2 dz +e sκ(t) q κ (t) -1 T t t-T q κ (m)dm V (X(., t)) ≤ ||Q|| 2κ e sκ(t) L 0 |δ(z, t)| 2 dz -e sκ(t) 1 T t t-T q κ (m)dmV (X(., t)) .
Since r is periodic of period T , we have

t t-T q κ (m)dm = t t-T µ ||Q|| r(m)dm - 2T ||Q||κ λ Q = µ ||Q|| B -2T ||Q||κ λ Q
, where B is the constant defined in [START_REF] Krstic | Backstepping boundary control for first order hyperbolic pdes and application to systems with actuator and sensor delays[END_REF]. We deduce that the value

κ = µBλ Q 4T ||Q|| 2 , (20)
which is positive because B is positive, gives U ≤ -µ 2||Q|| BU (X(., t), t)

+ ||Q|| 2κ e sκ(t) L 0 |δ(z, t)| 2 dz ≤ -µ 2||Q|| BU (X(., t), t) + c 1 L 0 |δ(z, t)| 2 dz , with c 1 = T ||Q|| 3 µBλ Q e T µr M ||Q|| , r M = sup {m∈[0,T ]}
{r(m)} and U = U κ for κ defined in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]. Moreover, [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF] ensures that, for all t ∈ R and φ ∈ L 2 (0, L), This concludes the proof of Theorem 1.

c 2 L 0 |φ(z)| 2 dz ≤ U (t, φ) ≤ c 3 L 0 |φ(z)| 2 dz , with c 2 = 1 β e µ 4||Q|| (
•

IV. BENCHMARK EXAMPLE

In this section we consider the system (2) and the boundary condition (3) with the following data, for all z in (0, 1) and for all t ≥ 0, X(z, t) ∈ R , L = 1 , Λ(z, t) = cos 2 (t) + 1 -z, F (z, t) = sin 2 (t) 5 , K = 1 2 . A remarkable feature of this system if that the system ξ = F (z, t)ξ, which rewrites as ξ = sin 2 (t) 5 ξ is exponentially unstable. Now, we show that Theorem 1 applies to it. Let Q be any positive real number. Then we have, for all t ≥ 0, N (t) = Q 4 cos 2 (t) and QΛ(L, t) = Q cos 2 (t) and thus ( 10) is satisfied with α = 1 4 . It is clear that [START_REF] Prieur | Robust boundary control of systems of conservation laws[END_REF] holds with r(t) = cos 2 (t). Finally, for all z ∈ [0, L], t ≥ 0, for all z in (0, 1) and for all t ≥ 0, with the boundary condition X(0, t) = 1 2 X(1, t) is asymptotically stable and the function ( 14) is an ISS Lyapunov function (with respect to δ).

Sym Q ∂Λ ∂z (z, t) + 2QF (z, t) = -Q + 2Q

V. CONCLUSIONS

For time-varying hyperbolic PDEs, we have designed ISS-Lyapunov functions. These functions are time-varying and periodic. They make it possible to derive robustness properties of the ISS type.

This work leaves many questions open. The problem of designing ISS Lyapunov functions for nonlinear hyperbolic equations will be considered in future works, possibly with the help of Lyapunov functions considered in [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF]. In addition, it would be of interest to use an experimental channel to validate experimentally the prediction of the offset that is inferred from the ISS-Lyapunov function in a similar way as what is done in [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF].

+2 L 0 X

 0 (z, t) Qδ(z, t)e -µz dz , withR 4 (φ, t) = -µ L 0 φ(z) QΛ(z, t)φ(z)e -µz dz + L 0 φ(z) Q ∂Λ ∂z (z, t)φ(z)e -µz dz+2L 0 φ(z) QF (z, t)φ(z)e -µz dz .

  2T r S -B) , r S = inf {m∈[0,T ]} {r(m)} and c 3 = exp T µr M ||Q|| β. Therefore inequalities of the type (5) are satisfied. We deduce that U is an ISS Lyapunov function, as introduced in Definition 2.1.

  Therefore[START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] holds and Assumption 1 is satisfied. We conclude that Theorem 1 applies. It follows that the system

						sin 2 (t)
				≤ -3Q 5 .	5
	∂X ∂t	(z, t) + (cos 2 (t) + 1 -z)	∂X ∂z	(z, t) =	sin 2 (t) 5 +δ(z, t) X(z, t)