For completeness, one should apply some classical fibration theorems also due to Cerf [1, Appendice].

1.1. Every non-vanishing closed one-form ω on S2 × [0, 1] that is tangent to the boundary reads φ * dz for some diffeomorphism φ that is equal to identity on S 2 × {0}. This follows from the fact that every orbit of a vector field X that everywhere fulfills ω(X) > 0 goes from S 2 × {0} to S 2 × {1}. Moreover, if φ * dz = dz then φ is isotopic to the identity among the diffeomorphisms of the same type. [START_REF] Eliashberg | Contact 3-manifolds twenty years since J. Martinet's work[END_REF] 1.2. Moser's method. Given two non-vanishing closed 1-forms ω 0 and ω 1 on S 2 × [0, 1], tangent to the boundary, a point a ∈ S 2 × [0, 1] is said to be a negative contact (resp. a positive contact) if there exists µ > 0 (resp. µ < 0) such that ω 0 (a) ± µ ω 1 (a) = 0. In other words, the kernels of ω 0 (a) and ω 1 (a) coincide but their co-orientations are opposite (resp. equal). The locus of negative (resp. positive) contacts will be denoted by C -(resp. C + .) Lemma 1.3. If the two non-vanishing closed one-forms are cohomologous and their mutual negative contact locus C -is empty, then the two forms are isotopic.

Indeed, the barycentric combination ω t := (1 -t)ω 0 + tω 1 provides a path (ω t ) t∈[0,1] of closed 1-forms, from ω 0 to ω 1 , that are cohomologous, tangent to the boundary and nowhere singular. In that case, Moser's isotopy theorem applies, as in the case of volume forms [7]. Namely, there is an autonomous flow (ψ t ) t∈[0,1] that conjugates ω t to ω 0 for every t ∈ [0, 1]. [START_REF] Hatcher | Pseudo-Isotopies of Compact Manifolds[END_REF] Therefore, our method to prove the isotopy Theorem (A') on S 2 × [0, 1] will be to cancel the locus C -of negative contacts between ω 0 := dz and ω 1 := φ * dz.

1.4. Reduction to a more special case. Let f be the function which is equal to the projection onto the factor [0, 1]; we set ℓ := φ * f = f • φ. From now on, the two forms ω 0 and ω 1 will be respectively denoted by ω F and ω L ; and the foliations they define will be respectively named F and L. The leaves of ω F will be named the level sets (understood of f ) while the level sets of ℓ will continue to be named the leaves (understood of L.)

Without loss of generality, to prove Theorem (A') we may assume the following in the rest of the paper:

(1. 1) The restriction of f to every leaf of ℓ, close enough to the boundary S 2 × {0, 1}, has only two critical points, a maximum and a minimum, and these belong to C + .

1.5. Plan of the proof of Theorem (A'). Generically, the contact points (positive or negative) have a Morse index: 0, 1 or 2. The contact points of index 1 are named saddles.

They have also a type: Y, λ, or X. A saddle s on a leaf L ∈ L is said to be of type λ (resp. Y ) if the connected component of L ∩ [f (s) -ε, f (s) + ε] which contains s looks like a pair-of-pants (resp. a reversed one.) A saddle is said to be of type X if it is the common limit of two sequences of saddles, one of λ-saddles and the other of Y -saddles, the pair (f, ℓ) being kept fixed (Definition 3.12.) This necessarily involves a pair of saddles both of type X in the same connected component of level curves (Lemma 3.13.) One will speak of λ-, Y -, and X-saddle respectively.

Section 2 is devoted to basics on generic properties. Section 3 reviews elementary isotopies whose effect is to simplify C -. One of them will allow us to kill all the finitely many X-saddles. Typically, this phenomenon occurs in the setting saddle-center-saddle (Definition 3.4.)

The next operation will consist of pushing the saddles of type Y to levels higher than all saddles of type λ. Unfortunately, this operation is obstrued by connecting orbits Y → λ of the L-gradient of f , a phenomenon that generically appears finitely many times. Here, the L-gradient of f is the orthogonal projection of ∇f = ∂ z onto the leaves of ℓ with respect to an understood Riemaniann metric; it will be denoted by ∇ L f . These obstructions are destroyed thanks to the main operation of the paper, namely the Dehn modification. Section 4 will be devoted to this type of 3-dimensional surgery considered in our specific setting. Section 5 makes use of all these tools to prove the main result, namely Theorem (A').

Basics on pairs of non-vanishing closed one-forms

2.1. Regularity. We set C = C -∪ C + . With local coordinates (x, y) on S 2 , the contact locus is defined by the following system:

(2.1) C = ∂ℓ ∂x (x, y, z) = 0, ∂ℓ ∂y (x, y, z) = 0 .
The subsets C + (resp. C -) are defined by adding the inequation ∂ℓ ∂z (x, y, z) > 0 (resp. < 0).

By the assumption (1.1), C -does not approach the boundary and hence is compact. By the transversality theorem of Thom in jet spaces, some approximation of φ makes maximal the rank of the linearized system associated with system (2.1) at every point of S 2 ×]0, 1[. In this case the pair (ω F , ω L )-or (f, ℓ)-is said to be regular and C is a smooth curve. The open subset of points in C where

(2.2) ∆ := ∂ 2 ℓ ∂x 2 ∂ 2 ℓ ∂y 2 - ∂ 2 ℓ ∂x∂y 2
is non-zero is the locus where C is transverse to both foliations F and L; overthere, their contact is quadratic. If a is such a point and L is the leaf of a, this point has a Morse index in {0, 1, 2} as a critical point of the function f L , that is the restriction of f to L. The critical points of f L are named minimum, saddle, maximum, depending on their Morse index. The remaining points of C are called inflection points. Generically, the equation ∆(x, y, z) = 0 is regular and hence, the inflections are isolated in C and do not approach the boundary. So, there are finitely many of them. At an inflection, the tangency of both F and L with C is quadratic while the mutual tangency of F and L is cubic.

By the versal unfolding theory [6] [8], there are coordinates in a neighborhood of an inflection where the two functions f and ℓ read

(2.3) f (x, y, z) = z and ℓ(x, y, z) = x 3 ± y 2 + λ(z) x + µ(z).
The sign in the above formula depends on the Morse indices of contact points nearby. The Hessian of f L at an inflection I ∈ L has index 0 (resp. 1), and is said to be a saddle-min (resp. saddle-max ) inflection. If the restriction f C of f to C is locally minimal (resp. maximal) at p ∈ C, p is said to be a birth (resp. cancellation) inflection.

Since C -is generically a closed-in general non-connex-curve, every of its connected component carries at least two inflections, one maximum and one minimum of the restriction of f to C -.

The Morse index of a quadratic contact is constant on each arc of C ending at an inflection or boundary points. When two arcs in C of quadratic contact have a common inflection in their closure, their indices differ by 1. So, one of these two arcs has a constant index 1; every contact point of this arc is a saddle, and the other arc has the index of an f L -extremum. (E2) There are neither three contact points in the same leaf and in the same level set nor two pairs of contact points in the same leaf at two distinct levels. (E3) Let σ be an inflection point. Then the leaf of σ (resp. its level set) contains no contact point at the same level (resp. in the same leaf) as σ.

Lemma 2.3. In the space of smooth functions ℓ : S 2 × [0, 1] → [0, 1] such that the pair (f, ℓ) fulfils the assumption (1.1), the subspace such that the pair (f, ℓ) is regular and excellent is an open dense set.

Proof. Such a statement is well known to the topologists when one speaks of a generic path of functions, the variable z being the time. But every path of functions is not the primitive of a non-vanishing one-form ω L tangent to the boundary. The main difference comes from the points where ∂ℓ ∂z = 0. In our setting, ∂ℓ ∂z is not vanishing at every point where the two other partial derivatives vanish. That means that we work in an open set of the general space of smooth real functions on S 2 . So, Thom's transversality theorems in bi-one-jet4 spaces of real functions apply. □

Elementary isotopies and application.

Here is the list of the isotopies of ℓ that we consider as elementary: cancellation-or creationof a simple loop in the contact locus C; bypass of the cusp singularity x 4 ; exchange (that is another way of bypassing a cusp.) To this list, we add the isotopies along a satured set, though they are not related to singularity theory.

As announced, these techniques will allow us to cancel the saddles of type X (in the sense of subsection 1.5.) Before starting with the description of these isotopies, some notation is needed.

3.1. Given a leaf L ∈ L and an extremum m of f L , the cone of m, denoted by C L (m), is the largest open disc in L-if it exists-that contains m and is bounded by a singular simple closed curve ∂C L (m) in a level set of f such that:

(i) ∂C L (m) contains a saddle (or an inflection) and bounds a disc ∆(m) in its level set;

(ii) ∆(m) ∪ C L (m) bounds a (topological) ball B L (m) in the ambient manifold.

Note that if L is close to the boundary of S 2 × [0, 1], by (1.1) the cone of the minimum (resp. maximum) is not defined since the leaf contains no saddles. Definition 3.2. (Simple loop.) A simple closed curve Γ in the contact locus C of the pair (f, ℓ) of smooth functions on S 2 × [0, 1] is said to be a simple loop if the following conditions are fulfilled.

(1) Γ is made of two open arcs, α and β, of quadratic contact points of the same signs and two inflection points, I 0 and I 1 , which make the topological closure of each of the arcs α and β; say α is of Morse index 1; the index of β, 0 or 2, depends on the sign in front of y 2 in formula (2.3). (2) Every leaf L that crosses α at a saddle s L5 also crosses β at a unique extremum m L and conversely. Moreover, s L belongs to ∂C L (m L ).

(3) There is a unique L-gradient line6 γ L joining s L to m L on every leaf L meeting α ∪ β. (4) The only arc of definite quadratic contacts located in the ball

B L (m L ) is β ∩ B L (m L ).
By formula (2.3), if s L goes to I j (j = 0 or 1) then γ L with its field of unit tangent vectors goes to I j with ±∂ x . Note that, by permuting f and ℓ, there are F-gradient lines connecting pair of points in α × β lying in the same level set of f . Proposition 3.3. Let Γ be a simple loop in the contact locus of the pair (f, ℓ). Then, there exists a smooth closed 3-ball N which is an arbitrary small neighbourhood of the union m L ∈β B L (m L ) satisfying the following two properties.

(1) The boundary is foliated by smooth closed curves drawn by L ∩ ∂N , with exactly two singularities at max ℓ ∂N and min ℓ ∂N .7 (2) The germ of ℓ along ∂N extends to N as a smooth function ℓ ′ without contact point with f in N . Moreover, ℓ ′ is isotopic to ℓ by an isotopy supported in the interior of N .

Proof. We may only consider the case where β is an arc of maxima. Let ε be a small positive number. For every m L ∈ β, consider the stable manifold W s (m L , ∇ L f ), truncated at the level f = f (s L ) -ε. Then slightly enlarge this surface to a disc D m L ⊂ L in order to have a smooth boundary and to contain an ε-neighbourhood of the truncated stable manifold of s L . If ε is small enough, D m L avoids all other connected components of the contact locus. If this construction is performed smoothly with respect to m L with a fixed ε, the desired

N is ∪ m L ∈β D m L .
There exists a proper disc S ⊂ N , transverse both to the leaves of ℓ and the level sets of f , that contains all segments γ L (item (3) in Definition 3.2.) This S can be endowed with coordinates (x, z) where f (x, z) = z and the pair (x, z) allows us to use the normal form α arc of saddles of inflections. Finally, one finds the y-coordinate by slicing N transversely to S so that the maximum of f |{x=x 0 ,ℓ(x 0 ,y,z)=ℓ 0 } is quadratic non-degenerate and located in {y = 0}. In Morse theory, such extension of coordinates is named a suspension [3].

I 1 I 0 β arc of max.
We are able to solve the problem of isotopy for a pair of functions like (f |S , ℓ |S ) since the two-dimensional problem is known by the contractibility of Diff(D 2 rel. S 1 ) [10]. An isotopy of S extends to N , relatively to a small neighbourhood of ∂N by suspension. Nothing is changed in (S 2 × [0, 1]) ∖ N . The proof is now complete. [START_REF] Poenaru | Singularities of analytical spaces[END_REF] □ On the opposite, creating a simple loop requires no condition. It can be realized in any open set of the ambient manifold where the two functions f and ℓ have no contact points. The details are left to the reader.

The next elementary isotopy will follow the same idea of suspension. This deals with the configuration of contacts named saddle-center-saddle. Center stands for minimum or maximum; here, we only consider the case of a minimum. The case where the center is a maximum is similar.

Definition 3.4. The saddle-center-saddle configuration is the following. The three contact points are in the same leaf L of L; the center is a minimum m L whose cone C L (m L ) contains two saddles s L and s ′ L in its boundary; there is exactly one L-gradient line from m L to s L , and

similarly from m L to s ′ L . So, f (s L ) = f (s ′ L ). Let α, β, α ′ denote the contact arcs containing s L , m L , s ′ L respectively.
Here are the main requirements:

(1) Let L ′ be a leaf close to L crossing β at m L ′ with f (m L ′ ) < f (m L ). Then f (s L ′ ) < f (s ′ L ′ ). (2) Let L ′′ be a leaf close to L crossing β at m L ′′ with f (m L ′′ ) > f (m L ). Then f (s L ′′ ) > f (s ′ L ′′ ). (3)
The interior of the ball B L (m L ) meets no other contact arc than an arc of β. [START_REF] Quê | Sur l'isotopie des formes fermées en dimension 3[END_REF] Here, it is meant that {s

L ′ , s L" } ⊂ α and {s ′ L ′ , s ′ L" } ⊂ α ′ . α β α L L L Figure 2.
The leaves are denoted on the left. Vertically are the three contact arcs.

Note that, up permuting α and α ′ , ( 1) and ( 2) above-mentioned follow from the excellence of the pair (f, ℓ); moreover, it is easily seen by ( 3) that s L , m L , and s ′ L have the same sign as contact point. The planar figure of this configuration is represented in Figure 2. One recognizes-up to sign-the singularity x 4 , named cusp, whose versal unfolding-up to conjugation sourcextarget-reads

(3.1) (x, λ, µ) → -(x 4 + λx 2 + µx)
where λ and µ are two real parameters. As m L is a minimum, the suspension consists of adding a positive square in the added variable y. Hence, it reads

(3.2) (x, y, λ, µ) → -(x 4 + λx 2 + µx) + y 2 .
The critical points of f restricted to a leaf close to L are given by y = 0, 4x 3 + 2λx + µ = 0. Up to positive coefficients, the equation of this discriminant locus reads

(3.3) 4λ 3 + 27µ 2 = 0.
The (λ, µ)-space, denoted by R (λ,µ) , is stratified, apart from the origin, by the discriminant locus D of the fold, and the half axis A = {λ < 0}, the set of parameters for which the corresponding function has two critical points with the same value. This is exactly the case of f L . The deformation from

L ′ to L ′′ corresponds to a path in R (λ,µ) crossing A transversely (Figure 3.) λ µ D A bypass f L f L f L Figure 3.
Discriminant locus of the singular function x 4 or x 4 -y 2 .

In the setting saddle-center-saddle, we have the 3-ball B L (m L ) that we can extend up to the level {f = f (s L ) + ε}. The closure of this extension contains a piece of the unstable manifolds

W u (s L , ∇ L f ) and W u (s ′ L , ∇ L f ).
Then, we take a neighbourhood N of this closure. And hence, there are coordinates (x, y, z) on N such that the surface S := {y = 0} is the locus of the minimum of f on the curve in N defined by (x, ℓ) = (x 0 , ℓ 0 ) where the pair (x 0 , ℓ 0 ) ranges in a convenient 2-dimensional domain.

Definition 3.5. A bypass of the cusp singularity is any path of functions as in Figure 3.

Figure 4 represents the modification of the contact locus along such a path. Proposition 3.6. In the above-mentioned setting, there is an isotopy supported in N that, at time 1, carries ℓ to a function ℓ ′ whose one-parameter family of leaves realizes a bypass of the cusp singularity. After these first two elementary isotopies, we present more shortly one useful variation of the latter.

Definition 3.7. The configuration min-saddle-min is the following. A quadratic contact m L of index 0, located in the leaf L, has a cone C L (m L ) whose boundary has an inflection I L of the same sign as m L . Let β be the contact arc containing m L . It is assumed:

1) The ball B L (m L ) contains no other contact arcs but a sub-arc of β.

2) I L is the cancellation point of a pair saddle-min, that is the common upper bound of an arc α of saddles and an arc β ′ ̸ = β of minima (Figure 6, left-hand side.)

Therefore, there is an L-gradient line from m L to I L . So, if m L ′ is just below m L on β, then there is a saddle s L ′ connected to m L ′ by an L-gradient line, and hence, the other branch of the stable manifold

W s (s L ′ , ∇ L f ) comes from a minimum m ′ L ′ distinct from m L ′ .
If one forgets the so-called suspension by +y 2 , it is clear that the new configuration deals with the cusp singularity x 4 . Instead of crossing the equality of two critical values, one crosses the cancellation of a saddle with a center.

In this setting, bypassing the cusp singularity has the following effect on the contact curves, as shown in Figure 6. We now turn to isotopies along saturated sets. Definition 3.9. Let x ∈ S 2 × [0, 1] and z 0 > f (x). The ascending saturated set of x up to level z 0 , denoted by S z 0 (x), is the minimal closed subset of {f ≤ z 0 } that contains x and satisfies:

(i) For every y ∈ S z 0 (x), the positive orbit of y under ∇ L f , truncated at level z 0 , is included in S z 0 (x). (ii) Let C denote the contact locus of (F, L). If y ∈ (C ∩ S z 0 (x)), every arc of C ascending from y and truncated at level z 0 is contained in S z 0 (x).

Of course, there is an analogous definition of a descending satured set. Such a subset is stratified in an obvious way. By excellence assumption, S z 0 (x) contains finitely many accidents, meaning level sets where the stratified type changes. From one accident to the next one, one proves that S z 0 (x) is surrounded by a fundamental system of collapsible domains that, in our setting, are defined as follows.

Definition 3.10. A collapsible domain K is a 3-ball whose boundary ∂K is the angular union of two discs, the lower boundary ∂ lo K and the horizontal boundary ∂ h K, fulfilling the next conditions:

(i)

∂ h K is contained in {f = z 0 }. (ii) ∂ lo K (resp. its interior) is contained in {f ≤ z 0 } (resp. {f < z 0 }).
(iii) There exists a field of directions on K transverse to the f -level sets and also transverse to ∂ lo K.

Lemma 3.11. Let K be a collapsible domain which contains S z 0 (x) and whose lower boundary is disjoint from S z 0 (x). For every ε > 0, there exists an isotopy (Φ t ) t∈[0,1] , supported in the interior of K, such that:

(1) Φ 0 = Id and Φ 1 (S z 0 (x)) is contained in {z 0 -ε < f ≤ z 0 };
(2) for every t ∈ [0, 1] and every contact point y of the pair (f, ℓ), its image Φ t (y) is a contact point of the pair (f, ℓ • (Φ t ) -1 ), with the same sign and Morse index; (3) if γ is an L-gradient arc, then Φ t (γ) is transverse to the f -level sets. [START_REF] Smale | Diffeomorphisms of the 2-sphere[END_REF] Such an isotopy is said to be an isotopy along a saturated set.

3.12. Application. Here we give an application of Proposition 3.6, namely, the cancellation of a pair of saddles, that are of the type X. We specify the definition of an X-saddle when the pair (f, ℓ) is excellent.

Definition. A saddle s in a contact arc α is said to be of type X, or an X-saddle, if s is the upper bound on α of an interval of saddles of type λ (resp. Y ) and the lower bound of an interval of saddles of type Y (resp. λ).

The following facts are easily checkable when the pair (F, L) is excellent and the leaves are 2-spheres. Lemma 3.13. (1) If s is an X-saddle there is another X-saddle s ′ in some contact arc α ′ interacting with s, meaning that the two saddles are located in the same connected component Ĉ0 of the intersection L 0 ∩ F 0 of a leaf L 0 := {ℓ = u 0 } and a level set F 0 := {f = z 0 }. Proof. It almost clear that a second saddle in Ĉ0 is necessary. Indeed, if not, the germ of L ∩ F 0 Ĉ0 is stable in the sense that, up to isotopy, it does not depend on small variations of z 0 . Therefore, s could not be of type X. By excellence, Ĉ0 cannot have three saddles (or double points.) Then, it contains exactly two saddles. Up to isotopy of F 0 , there are very few configurations for Ĉ0 : (a) The one represented in Figure 7. (b) A second one that is the image-named a chain-of a smooth closed cuve through an immersion with two double points (Figure 8.) The rule for the co-oriention of Ĉ0 in F 0 = S 2 near the double points leaves only these two cases. In the same way, one checks that in case (a) the two contact points have the same sign.

In case (b), whatever the signs of contacts the saddles cannot be X-saddles. □

C - C - T T S z 0 -(Γ) S z 0 -(I 1 )
Figure 9. The gray rectangle represents the domain marked by the ascending saturated set of Γ in the level set z 0 -ε .

Denote by α ′ the arc of saddles passing through s ′ . Let s(t) and s ′ (t) be two local parametrizations of α and α ′ , with s(0) = s, s ′ (0) = s ′ , such that ℓ(s(t)) = ℓ(s ′ (t)) for every t close to 0. Say f (s(t)) and f (s ′ (t)) are increasing with t. As the pair (f, ℓ) is excellent, the two f -velocities at t = 0 are different. We continue this analysis by specifying that the two contacts are positive.

For ε small enough, let C -and C ′ -be the two smooth closed curves in the level set {f = z 0 -ε} corresponding to C 0 and C ′ 0 respectively by the L-gradient flow lines on L 0 . Let T and T ′ be the lower boundaries of the saturated sets descending from s(0) and s ′ (0) respectively (Figure 9). Proposition 3.14. There is an isotopy supported in {z 0 -2ε ≤ f ≤ z 0 + ε} whose effect on ℓ is to decrease by one the number of pairs of contact points with F that are of type X.

Proof. The idea is to create a configuration saddle-center-saddle which contains s(0) and s ′ (0). Since the center is missing, one starts by introducing a simple loop Γ of type saddle-min whose inflection points I 0 and I 1 satisfy z 0 -2ε < f (I 0 ) < f (I 1 ) < z 0 -ε and so that its upper boundary in {f = z 0 -ε} is a position as shown in Figure 9.

The next operation is an isotopy along the satured set S z 0 +ε (I 1 ). By application of Lemma 3.11, one can collapse this domain above z 0 . After this isotopy, still denoting by ℓ and Γ the carried function and simple loop, one sees a configuration saddle-center-saddle as shown in Figure 10. Then Proposition 3.6 is available that makes the desired cancellation of a pair of X-saddles.

There is a new arc α ′′ of positive saddles which crosses {f = z 0 -ε}-this is the arc of saddles in the simple loop Γ after the just above mentioned isotopy; the arc of minima is denoted by β. One checks that α ′′ contains only saddles of type λ. Now we can apply Proposition 3.6. This cancels the pair (s, s ′ ) of X-saddles without creating a new one and Proposition 3.14 is proved. □

The outcome of Proposition 3.14 is the cancellation of every X-saddle.

Dehn modification

After Thom's transversality arguments that generate generic properties, the Dehn modification, also called Dehn surgery, is the main tool to prove Theorem (A), and already regarding its 1-connected version Theorem (A'). We are going to speak of this notion in our setting only: the ambient manifold is M := S 2 × [0, 1], provided with two foliations, respectively as the level sets of two functions, f and ℓ, that are constant on each component boundary components and whose critical sets are empty.

As an application, we prove that a suitable series of Dehn modifications allows us to move the Y -saddles above the λ-saddles. where φ is a smooth non-decreasing function from the value φ(

[0, ε]) = 0 to φ([1 -ε, 1]) = 2π.
The right Dehn twist is the inverse τ -1 of the left one. 2) Let A be an annulus embedded in M and parametrized by A; for i = 1, 2, ∂ i A is parametrized by ∂ i A := S 1 × {i}. Let M be the manifold obtained by cutting M along the interior of A; it is provided with a singular boundary made of two lips A + and A -corresponding to the coorientation of A. The Dehn modification of M along A is the manifold M τ obtained from M by gluing x ∈ A + to τ (x) ∈ A -.

A + A - N ∂ 2 A ∂ 2 A
Figure 11. The two dotted vertical arrows indicate the gluing that makes M τ .

In our setting, A is an annulus, embedded in a level set F 0 of f transversely to L; the leaves of L∩A are closed curves isotopic to each component of ∂A. The identification with A is chosen so that these curves are parametrized by the circles of A. In this setting we shall say that the Dehn modification is adapted to the pair (f, ℓ) or (F, L). By construction, the functions f and ℓ are carried to some uniquely defined functions f τ and ℓ τ .

Important remark 4.2. In the previous setting, a Dehn modification along A keeps the contact locus of the pair (f, ℓ) invariant. Indeed, the support of this modification is away from the contact locus. Definition 4.3. Let A be an annulus, parametrized by A, in a level set F 0 = {f = z 0 }. Let p A ∈ S 2 such that the vertical arc {p A } × [0, 1] ⊂ S 2 × [0, 1] avoids A and intersects the unique disc in F 0 that contains A and is bounded by ∂ 2 A.

Let D A denote the space of smooth oriented 2-discs embedded in M that contain A as oriented annulus, have algebraic intersection +1 with {p A } × [0, 1] and have ∂ 2 A as a boundary. Proof. 1) Since A is an imposed collar of ∂D for every D ∈ D A , it is natural to say that, abstractly, D is the disc of radius 2 in the Euclidean plane. This disc may be considered as embedded in M or in M τ as well. Firstly, we consider D in M τ . One endows D with a 3dimensional collar N ∼ = D × [0, 1] on the side of A + , that is A + ⊂ D × {0}. So, N already existed in M and M .

One defines a diffeomorphism ρ : D → D by ρ(x) = τ (x) if ∥x∥ ∈ [1, 2] and ρ(x) = x if ∥x∥ ≤ 1. One chooses an isotopy (h t ) t∈[0,1] among the diffeomorphisms of D from ρ for t = 0 to the identity of D for t = 1, with an extra requirement:

(4.2) ∥h t (x)∥ = ∥x∥ for every t ∈ [0, 1].
Then, on defines

ψ D : M τ → M by (4.3) ψ D (y) = y if y ∈ M ∖ N ψ D (x, t) = (h t (x), t) if (x, t) ∈ N.
Since all choices are made in contractible spaces, ψ D is uniquely defined up to isotopy. 2) Usual tools of 3-dimensional topology yield the following topological result. 12 Each 2-disc D ∈ D A has a winding number around A, w A (D) ∈ Z, that is the algebraic intersection of D with a vertical arc V in S 2 × [0, 1] starting from the interior of A and ending in S 2 × {1}. Moreover, w A provides a bijection from π 0 (D A ) onto Z.

Here, the linking condition imposed to D for being in D A plays an important role. This statement also holds in M τ . Moreover, one can think of A as a flat annulus in a solid torus T A ∼ = S 1 × D 2 , each ray of A being a diameter of {θ} × D 2 . An exterior collar of ∂T A has coordinates (θ, θ ′ , t) with (θ ′ , t) ∈ ∂D 2 × [0, 1]. This allows one to define a meridional Dehn twist in the coordinates (θ ′ , t) independant of θ. 13 Let us denote it by µ. For D ∈ D A , one has

(4.4) w A (µ(D)) = w A (D) + 1.
This holds true on M and M τ as well because the support of τ and µ are disjoint.

Using that translations on the torus are commuting, one has for every D ∈ D A :

(4.5) ψ µ(D) = µ • ψ D • µ -1
Moreover, µ is isotopic to the identity if A is let free to rotate in T A around its core. rotating A does not change M τ . Therefore, ψ µ(D) is isotopic to ψ D which proves what was desired.

3) The disc D 0 is the unique disc in the level set F 0 fulfilling the requirement ∂D 0 = ∂ 2 A. One can choose its collar N ∼ = D 0 × [0, 1] to be foliated by level sets of f . Then formula (4.3) shows that (ψ D 0 ) * f is sensitive neither to τ nor to the isotopy (h t ). So, (ψ D 0 ) * f = f is clear.

12 Unfortunately, I have no reference for this exercise. One has to use the classical method of the innermost intersection curve for two discs in D A with the same winding number w A .

13 Use formula (4.1) replacing (θ, r) with (θ ′ , t).

For D 1 , this is slightly more subtle. There is a disc ∆ which is bounded by ∂ 1 A in some L-leaf and intersects {p A } × [0, 1] positively. The union A ∪ ∆ is angular along ∂ 1 A. But its smoothing is easy: Take a collar N ∆ of ∆ in the direction outward to A along ∂ 1 A and foliated by L. Then, cap ∂ 1 A in N ∆ with a disc ∆ tangent to A along ∂ 1 A and so that L induces a foliation by circles with one center. Now, we take

D 1 = A ∪ ∆.
The last step consists of choosing a collar

N 1 = D 1 × [0, 1] of D 1 in M ,
that exists in M (on the side of A + ) so that L induces on each D 1 × {t} a foliation by circles. Therefore, by formula (4.3) the isotopy is compatible with L. And hence,

ψ D 1 (L τ ) = L.
4) This is a formal consequence of items 2) and 3). □

We have seen that a Dehn modification adapted to (F, L) allows us to carry some structures, like f or ℓ, from M to M τ . But an isotopy, for instance along a saturated set, cannot be carried. Fortunately, the last item of Proposition 4.4 allows us to carry the property that the pair (F, L) is made of isotopic foliations. This is the reason why we introduce the notion of weak isotopy. Definition 4.5. A weak isotopy of L to F in M is an alternating finite sequence of Dehn modifications-right or left-and isotopies, M 1 , (ψ t 1 ) , . . . , M k , (ψ t k ) , t ∈ [0, 1], and a sequence of pairs of foliations, (F, L), (F 1 , L 1 ), . . . , (F k , L k ), fulfilling the following recursive conditions for every integer j ∈ [1, k].

(i) M j is a Dehn modification adapted to the pair of foliations (F j-1 , ψ 1 j-1 (L j-1 ));14 (ii) M j carries F j-1 to F j and ψ 1 j-1 (L j-1 ) to L j ; (iii) F k = ψ 1 k (L k ). In this language, Theorem (A') reduces to: F and L are weakly isotopic. Now is the announced application of Dehn modification dealing with connecting orbits of saddles. One should first mention that, as for a one-parameter family of functions on a Riemannian manifold, generically on the metric the vector field ∇ L f has finitely many orbits connecting two saddles; moreover there are connecting orbits neither from a max-saddle inflection to a saddle nor from a saddle to a min-saddle inflection. We now include this property in the definition of excellence and assume it in the remainder of this paper. We recall that, by Proposition 3.14, we may assume that no saddles are of type X. Proposition 4.6. In this settting there exists a weak isotopy of ℓ to a new function ℓ ′ in M such that all of its Y -saddles are located above its λ-saddles with respect to the order of their f -values.

Proof. Without X-saddles the type of saddles is constant on every saddle arc. The two ends of such an arc are inflections of the same Morse index:15 if this index is 0 (resp. 1) the saddles are of type λ (resp. Y ) as it can be seen near the inflection.

The matter is to destroy the L-gradient connecting orbits Y → λ. Indeed, assume the metric is chosen so that there are no such connections. Consider a level z 0 above all λ-saddles and a contact arc α of Y -saddles. By assumption on the metric, the ascending saturated set S z 0 (α) avoids every closure of λ-saddle arcs. Of course, S z 0 (α) could meet another Y -saddles not belonging to α or maxima and inflections of Morse index 1. This does not matter, all thses contacts have to be incorporated to S z 0 (α). An isotopy along S z 0 (α) realizes what is wished for α, and so on for the other Y -saddle arcs.

To bypass the connecting orbits we use Dehn modifications. At the beginning, we have finitely many connecting orbits Y → λ lying in distinct leaves L 1 , L 2 , . . . , L n where each L i = ℓ -1 (y i ) is a generic leaf. There exists ε > 0 such that Li := ℓ -1 ([y i -ε, y i + ε]) is still made of generic leaves.

The connecting orbit on L i goes from Y -saddle s i to λ-saddle s ′ i . Denote by α i (resp. α ′ i ) the arc of contacts that contains s i (resp. s ′ i ). Set αi := Li ∩ α i and α′ i := Li ∩ α ′ i ; all are named saddle intervals. If ε is small enough there exists η such that every interval in the collection f (α i ), f (α ′ j ) i,j∈ [1,n] has a length less than η and every two of them have a mutual distance larger than η. In addition, one may require, for every j ∈ [1, n] and every pair (a, b) of distinct contact points in a same leaf from Lj , Say f (s 1 ) > f (s i ) for every i > 1. Cut L1 at the level z 1 := f (s ′ 1 ) -η. By condition (4.6), this level set of L1 is a finite collection of annuli. Among them, call A 1 the one which meets the connecting orbit s 1 → s ′ 1 . The foliation L induces on A 1 a foliation by circles; the central circle lies in the leaf L 1 . Knowing that s 1 is of type Y and s ′ 1 of type λ one sees in A 1 two arcs T and T ′ joining the two boundary circles:

T = A 1 ∩ S z 1 (α 1 ) and T ′ = A 1 ∩ S z 1 (α ′ 1 )
. Here, the first satured set is ascending and the second one is descending. Moreover, T ∩ T ′ is the point where the connecting orbit crosses A 1 -because the connecting orbits are on isolated leaves.

Applying a Dehn twist τ to T ′ -left or right depending on the sign of intersection ⟨T, T ′ ⟩together with the corresponding Dehn modification to M kills the connecting orbit since τ (T ′ )∩ T = ∅.

However, new connecting orbits could have been created in the foliated domain L1 by this first Dehn modification (this is the case in Figure 13 due to τ (T ′′ ) ∩ T ̸ = ∅.) One solves this new difficulty as follows.

Look at all saddle intervals of type λ in L1 at level higher than z 1 . Name them β1 , ..., βk such that f ( β1 16 The number of them is not affected by forthcoming Dehn modifications, it is kept invariant (Remark 4.2.) Say that β1 ⊂ L1 is the lowest saddle interval of type λ being L τ -gradient connected to α1 ; if there is no such a connecting orbit, consider β2 , and so on. Take a proper annulus A ′ 1 ⊂ L1 , at the level z ′ 1 := inf f ( β1 ) -η, that crosses a connecting orbit from α1 to β1 . The figure is similar to Figure 13, but S z ′ 1 ( β1 ) marks a short arc T ′ ( β1 ) joining the two boundary components of A ′ 1 while S z ′ 1 (α 1 ) marks a long arc like τ -1 (T ). These two arcs are transverse to the foliation L τ ∩A ′ 1 and, up to isotopy, there is only one intersection point between them. One can repeat the trick of Dehn modification to cancel the connection α1 → β1 . Of course, new connecting orbits are possibly created, but only from α1 to βj , j > 1. So, the process goes on, without increasing the complexity.

) < f ( β2 ) < • • • < f ( βk ).
One can destroy all connecting orbits from α1 to β1 ∪ • • • ∪ βk . As a result, after this long series of Dehn modifications, the ascending saturated set S z 0 (α 1 ) avoids every λ-saddle, where z 0 has been chosen at the very beginning of the proof. If the saddles (s i ) n i=1 , from which one connecting orbit goes up in the leaf L i , are ranked in the decreasing order of their f -values (f (s i )) n 1 , the process continues with the interval α2 and so on. □

Proof of Theorem (A')

5.1. So far, we got that every saddle is of type λ or Y -that is, no X-saddle-and, up to a weak isotopy applied to the function ℓ, every Y -saddle is located at a level higher to all λ-saddles. Recall that an inflection adhering to a contact arc of λ-saddles (resp. Y -saddles) is of type saddle-min (resp. saddle-max). Say that the λ-saddles (resp. le Y -saddles) are located in {0 < f < 1/2} (resp. {1/2 < f < 1}.) A way to achieve the proof of Theorem (A') is, by Lemma 1.3, to kill every negative contact of the pair (f, ℓ). This requires some algorithm since the leaves of L could have a very tricky topology as embedded surface in S 2 × [0, 1]. In [5], this algorithm was based on words in an alphabet with four letters. Here, it is much simpler; it is based on two topological concepts: basin and co-basin that we are going to define. 5.4. Lower complexity. Given a pair (f, ℓ) of functions without critical points, we define its lower complexity κ -(f, ℓ) as the pair (µ, ν) of non-negative integers arranged in lexicographical order and defined in the following way. 18 The entry µ is the number of contact arcs of negative minima. Let I 0 be the upper, unique by excellence, negative inflection which is the birth of an arc β of negative minima. Every minimum m ∈ β defines a unique basin. A minimum m ∈ β, distinct from a cancellation inflection, is said to be accidental if one of the following cases happens:

-the boundary of ∂ F B(m) contains either two saddles or a negative inflection-necessarily a cancellation inflection-by definition of I 0 , -∂ L B(m) contains a positive inflection. By definition, the entry ν of the lower complexity is the number of accidental minima in β.

If µ = 0 then there are neither negative minima nor negative λ-saddles anymore and, by convention, κ -(f, ℓ) is equal to (0, 0); then the set of negative contacts is empty in {f < 1/2}. the level is close to z I ′ ? Let s ′ denote the threshold of the basin B(m ′ ) and let s ′ (z) denote that nearby threshold at level z close to z I ′ . Since the crossing can happen, certainly s ′ (z) has no connecting orbits with points of Σ ′ if ε is small enough and z ranges in (z(I ′ ) -ε, z(I ′ ) + ε). 20 So, for such a z, the negative saddle s ′ (z) in not in the closure of Σ ′ . □ By Lemma 3.11, there is an isotopy along Σ ′ that pushes it above the level z(I ′ ). Its support is located in a neighborhood of Σ ′ . The time one of this isotopy fulfills the demand. About the last claim, it is sufficient to recall that there are only finitely many such positive birth inflections. Let I ′′ be one of them. 21 One can consider its saturated set Σ ′′ with respect to Φ 1 (L) up to a level which is determined as previously by the crossing argument. An isotopy along Σ ′′ pushes I" away from all basins and does not destroy what was gained in the first step. So, cumulating the isotopies related to each positive inflection that is initially contained in a basin makes all basins B(m), m ∈ β, free from positive contacts. The order in which the concerned inflections are numbered is irrelevant. □ Now we are ready for the decisive part of the proof of Theorem (A'). It will consists of decreasing the lower complexity κ -(f, ℓ) until it vanishes. By the symmetry z → 1 -z, the same holds true about the upper complexity. Proposition 5.7. If the set of negative minima contacts of the pair (f, ℓ) is non-empty there exists an isotopy supported in {0 < f < 1/2} that carries ℓ to a function ℓ ′ such that

(5.1) κ -(f, ℓ ′ ) < κ -(f, ℓ).
Proof. There are several cases depending on how the complexity is topologically made. 1) Assume µ > 0 and ν = 0. We continue with the upper negative birth saddle-min inflection I 0 and the arc β of index 0 that it generates. By Lemma 5.6, every basin B(m), m ∈ β, is free of positive contact. In this case, for every minimum m ∈ β the cone C L (m) is standard. 22 If the negative λ-saddle arc α emanating from I 0 and the minimum arc β close up in a common cancellation inflection point then the union α ∪ β forms a simple loop in the sense of Definition 3.2. By Proposition 3.3 there is an isotopy supported in a neighbourhood of ∪ m∈β C L (m) that eliminates this simple loop and decreases µ by 1.

Let us show that there are no other configurations with ν = 0. Let J 1 be the inflection ending α; let I 1 , distinct from J 1 , be the inflection ending β and let I 2 be the birth inflection of the λ-arc, named α ′ , descending from I 1 . We have f (I 0 ) > f (I 2 ) by definition of I 0 (Figure 15.) Moreover, f (J 1 ) > f (I 1 ); if not, the unique separatrix converging to J 1 starts from a minimum m 1 ∈ β, and hence ν > 0.

When ν = 0 and f (J 1 ) > f (I 1 ), every m ∈ β ends one separatrix descending from some saddle s(m) ∈ α and one separatrix descending from some saddle s ′ (m) ∈ α ′ . The latter two claims prove similarly: the existence of s(m) is clear near the inflection I 0 and extends to β since no accident happens along this arc; the existence of s ′ (m) is clear near I 1 and also extends to β for the same reason.

When m is close to I 0 one has f (s ′ (m)) > f (s(m)) and when s ′ (m) is close to I 1 one has f (s ′ (m)) < f (s(m)). Then there is m 0 ∈ β such that 2) We assume ν > 0 in the rest of the proof. Consider the first accidental minimum m 0 in β when coming from I 0 and assume that the curve ∂C L (m 0 ) contains two saddles s 0 and s ′ 0 , necessarily negative by the "cleaning" Lemma 5.6. Then we have the saddle-center-saddle configuration; item (3) from Definition 3.4 is fulfilled by the choice of I 0 and the cleaning of the I 0 -basins. Let α and α ′ denote the contact arcs containing s 0 and s ′ 0 respectively; let s t , s ′ t and m t denote local parametrizations of α, α ′ and β respectively near t 0 such that these three contact points are in the same leaf for every t close to 0. If α comes from I 0 and knowing that m 0 is the first accidental point on β, one has f (m t ) < f (s t ) < f (s ′ t ) for every t < 0. In other words, the f -values of s t and s ′ t are ordered as in Figure 2. Thus every saddle in the sub-arc (I 0 , s 0 ) ⊂ α is cancellable with the corresponding minimum in β; for every small t > 0, the pair (s ′ t , m t ) can be cancelled. Proposition 3.6 applies and its effect is shown in Figure 4. In particular, this bypass creates a simple loop which contains I 0 and a new birth inflection I 1 with the following properties:

-f (I 1 ) > f (I 0 ).

-The minimum arc β ′ emanating from I 1 has less accidental minima β since it coincides with β in {f > f (I 1 ) + ε} for ε small enough (Figure 16.) Once the simple loop has been cancelled the number of birth inflections remains equal to the initial µ but the entry ν of the lower complexity decreases by 1: the arc β ′ has one less accidental minimum than β. 3) With the same notation, we assume ν > 0 and the first accidental minimum m 1 ∈ β has a cone with an inflection point J 1 in its boundary. This inflection cannot be a birth since it lies at a higher level than I 0 . So, this is a cancellation inflection, which is a negative contact due to the cleaning Lemma 5.6. Let β denote the index-0 arc starting from I 0 and let I 1 be its upper end. We have I 1 ̸ = J 1 since an arc that is transverse to L cannot meet the same leaf twice. Here there are two cases.

3-1) Assume that the saddle arc α starting from I 0 ends at J 1 . Here, we are in the configuration min-saddle-min (Definition 3.7). Let β ′ be the arc of index 0 descending from J 1 . The isotopy from Proposition 3.8 modifies the contact arcs as, including their names, it is shown in Figure 6. The outcome is a simple loop containing I 0 and an arc of minima. After the simple loop has been cancelled, the number of negative birth saddle-min inflections has decreased by one and hence the complexity decreases (Figure 17.) 3-2) Assume that α does not end at J 1 . Let α ′ and β ′ respectively denote the saddle arc and the minimum arc ending at J 1 . The arcs (β, α ′ , β ′ ) and the leaf L that contains m 0 ∈ β slightly below m 1 also presents a configuration min-saddle-min. The effect of the isotopy from Proposition 3.8 does not produce a simple loop that could be cancelled. But it keeps I 0 as the upper negative birth inflection and the index-0 arc β emanating from I 0 contains no accidental minimum. Therefore, the complexity decreases from (µ, ν) to (µ, 0) (Figure 18.) □

When the two complexities vanish, no negative contacts are remaining and the Moser trick (Lemma 1.3) completes the proof of Theorem (A').

□

2. 2 .

 2 Excellence. The pair of functions (f, ℓ) is said to be excellent if the following conditions are fufilled. (E1) Let a and b be two distinct quadratic contact points at the same level and in the same leaf; let (a(t), b(t)) is a pair of local parametrizations of the respective contact arcs such that a(0) = a, b(0) = b and ℓ(a(t)) = ℓ(b(t)) for every t close to 0. Then we have d dt [f (a(t)) -f (b(t))] ̸ = 0 at t = 0.

Figure 1 .

 1 Figure 1. A simple loop max-saddle.

Figure 4 .

 4 Figure 4. Bypass in configuration saddle-center-saddle.

Proposition 3 . 8 .

 38 In the configuration min-saddle-min, there is an isotopy supported in a neighbourhood N of the ball B L (m L ) that, at time 1, carries ℓ to a function whose contact curves in N have the behaviour presented in the right hand side of Figure6.

Figure 5 .

 5 Figure 5. Another bypassing of the cusp singularity.

Figure 6 .

 6 Figure 6. Min-saddle-min configuration.

( 2 )Ĉ0Figure 7 .

 27 Figure 7. Ĉ0 denotes the connected component of s in the level curve F 0 ∩ L 0 . The dashed (resp. dotted) curves lie in L 0 and in level sets above (resp. below) z 0 .

Figure 8 .

 8 Figure 8. The unique example of a chain in S 2 . Uniqueness up to isotopy does not hold in R 2 .

Figure 10 .

 10 Figure 10. The very thick lines are lower boundaries of descending saturated sets of saddles s(0), s ′ (0) and α ′′ ∩ {f = z 0 -ε}.

Definition 4 . 1 .

 41 (Dehn twist and Dehn modification.) 1) A left Dehn twist on the standard annulus A := S 1 × [1, 2] is a diffeomorphism τ that, in polar coordinates, reads (4.1) (θ, r) τ -→(θ + φ(r -1), r)

Proposition 4 . 4 . 2 ) 3 ) 4 )

 44234 With this notation, the following holds true. 1) Every D ∈ D A defines a unique diffeomorphism ψ D : M τ → M , up to isotopy. For every pair (D 0 , D 1 ) of elements in D A , then ψ D 1 is isotopic to ψ D 0 . There are two discs D 0 and D 1 , elements of D A , such that (ψ D 0 ) * df = df τ , and (ψ D 1 ) * dℓ = dℓ τ . The functions f and ℓ are isotopic if and only if the functions f τ and ℓ τ are isotopic.

Figure 12 .

 12 Figure 12. Example of meridional twist. Here, w A (D) = -1.

(4. 6 )Figure 13 .

 613 Figure13. T ′′ is marked by S z 1 ( α′′ ) where α′′ is a saddle interval of type λ in L1 and f (α ′′ ) > f (α ′ 1 ).

Definition 5 . 2 .Figure 14 .

 5214 Figure 14. I 0 is a birth positive saddle-min inflection; α and β are respectively the saddle and the min locus emanating from I 0 ; γ and γ ′ form the boundary of the unstable manifold of I 0 . The saddle s is negative.

(5. 2 ) 2 αFigure 15 .

 2215 Figure 15. The horizontal lines stand for level sets.

Figure 16 .

 16 Figure 16. creating a simple loop.

1 Figure 17 .

 117 Figure 17. Bypass in configuration min-saddle-min.

Figure 18 .

 18 Figure 18. Variant of Figure 17.

These facts hold true on S n × [0, 1] for every dimension.

In the case of non-vanishing closed one-forms, a very simple proof is given in [5, Appendice I].

A bi-one-jet is just a pair of two one-jets.

By the Stokes formula this point is unique when it exists.

See the definition in subsection 1.5.

These two points are close to I 0 and I 1 respectively, depending on the distance of ∂N to ∪ m L ∈β B L (m L ).

This proof is inspired by my proof of a theorem of Morse about the cancellation of a pair of critical points of one real function[START_REF] Laudenbach | A proof of Morse's theorem about the cancellation of critical points[END_REF].

See subsection 3.1 for notation B L (m L ).

See more details in [5, chap. III §4].

There are two possibilities for the pair (C 0 , C ′ 0 ): either it is the limit of the dotted curve from Figure7or it is the limit of the dashed curve.

Here, (F 0 , L 0 ) = (F, L).

For the index of an inflection see Subsection 2.1.

Inequalities of intervals mean that their are disjoint and ordered as it is written.

The uniqueness of its threshold holds since there are no X-saddles (compare Figure7.)

This pair is meant to measure the topological complexity of L in the domain {0 < f < 1/2}.

Indeed, for z = z I ′ , the threshold has a separatrix coming from m ′ for some choice of the Riemannian metric. Hence, the same holds in nearby basins.

Possibly, I" belongs to Σ ′ . Hence, Φ 1 (I ′′ ) is already away from any basin and hence this inflection can be skipped.

See Subsection 3.1 for notation.

Similarly, some upper complexity, κ + (f, ℓ), is defined that measures the topological complexity of L in the domain {1/2 < f < 1}. Its vanishing means that there are no negative contacts in {f > 1/2}. By symmetry, any result about negative λ-saddles/minima applies to negative Y -saddles/maxima-and conversely. Therefore we are focusing on the first case. Lemma 5.5. By the choice of I 0 , denoting by β the arc of minima that I 0 generates, the following holds.

(1) Let B * be a co-basin joint to a basin B(m) with m ∈ β. Every (piece of ) leaf in the interior of B * contains no negative minimum. (2) The positive contact arcs descending from contact points in the interior of ∂ L B * descends to an inflection in the interior of B(m). In particular, m is the absolute minimum of the basin that it defines.

Proof. 1) Let L be a leaf passing through B * and m ′ be a negative minimum contact in L ∩ B * . The contact arc β ′ passing through m ′ cannot cross ∂ L B * which only contains positive contacts. So, β ′ descends to an inflection in B * . But such an inflection should be higher than m, the negative minimum of B(m), that itself is higher than I 0 . This contradicts the choice of I 0 .

2) Suppose such an arc γ crosses ∂ L B(m) twice. The integral of dℓ on γ, oriented in the direction of decreasing z, is positive. This prevents γ from having two distinct points on the same leaf. Hence the lower point of γ is an inflection in the interior of B(m). □

The topic of the next lemma is to clean up the basins we are going to deal with in the proof of Theorem (A'), namely to make sure that no co-basins are joint to them. Again, I 0 is the upper birth negative saddle-min inflection and β is the contact arc of minima that I 0 generates. Lemma 5.6. In this setting, let I ′ be a positive birth saddle-min inflection located in ∂ L B(m) for some m ∈ β. Then there exists a contact conjugating 19 isotopy Φ t , t ∈ [0, 1], fulfilling the following:

(i) the isotopy is supported in {0 < f < 1/2} and keeps the negative contacts fixed;

(ii) Φ 1 (I ′ ) is away from every so-called I 0 -basin, that is defined by some minimum in β. After having iterated such isotopies, every I 0 -basin is free of positive contact.

Proof. For the first part we recall that when the negative threshold of a basin goes down on its own contact arc, meanwhile the positive threshold of the joint co-basin generated by I ′ goes up on its own contact arc. Hence, there is a level z I ′ which is the common level of both the threshold of some basin B(m ′ ), m ′ ∈ β, and the threshold of the joint co-basin B * I ′ generated by I ′ . So, it is natural to consider the ascending saturated set Σ ′ := S z I ′ +ε (I ′ ) for some small enough ε. Here, we recall that every ascending separatrix of a λ-saddle reaches the level {z = 1/2} except if it is bounded from above by a cancellation inflection or another λ-saddle-which will be incorporated to Σ ′ . Claim. Σ ′ does not approach the negative contacts.

Indeed, by Lemma 5.5, there is no negative contact locus in the interior of Σ ′ -what is equivalent to the interior of a co-basin. What about α, the locus of saddles emanating from I 0 when 19 A diffeomorphism Φ of S 2 × [0, 1] is said to be contact conjugating if Φ maps every contact point of the pair (F, L) to a contact point of the pair (F, Φ(L)) and conversely.


Alain Ayong Le Kama

Aude Pommeret
email: aude.pommeret@cityu.edu.hk

Mitigation and adaptation are not enough: turning to emissions reduction abroad

Keywords: climate change, mitigation, adaptation, CDM, pollution threshold Q5, Q52, Q56, Q58

In this paper we focus on a long-term dynamic analysis of the optimal adaptation/mitigation mix in the presence of a pollution threshold above which adaptation is no longer e¢ cient.

We account for accumulation in abatement capital, greenhouse gases, and adaptation capital in order to better capture the arbitrage between abatement and adaptation investments.

Pollution damages arise from the emissions due to the country consumption but also from the emissions of the rest of the world (ROW). A pollution threshold is then introduced, above which adaptation is no longer e¢ cient. We obtain that if this threshold is lower than the steady-state level of pollution, there is no way for the modeled economy to avoid it. In particular, such a situation will appear if the ROW's emissions are high. Next step is then to introduce another type of investment allowing for lower ROW pollution ie. emissions reduction abroad through CDM for instance. We obtain that CDM may be a means to avoid a pollution threshold above which adaptation becomes of no use.

Introduction

The issue of climate change could be tackled at two levels: one can seek either to avoid it through mitigation or to avoid its consequences through adaptation. Mitigation refers to "an anthropogenic intervention to reduce the sources or enhance the sinks of greenhouse gases" (IPCC, 2007) while adaptation concerns measures that reduce the vulnerability of natural and human system. A usual example of adaptive measures is the construction of levees to prevent ‡ooding of plain areas as a result of rising seawater but could also include the development of crop varieties better suited to the new climate characteristics. However, if climate change could become too large for adaptation to be e¤ective, current mitigation and adaptation policies should be a¤ected by this ine¤ectiveness threat. In this paper we focus on a long-term dynamic analysis of the optimal adaptation/mitigation mix in the presence of a pollution threshold above which adaptation is no longer e¢ cient.

Both mitigation and adaptation are costly but an interesting di¤erence between the two is that adaptation may be e¢ cient even if implemented in a decentralized uncoordinated way. However, adaptation has long been left aside. Motivations were clear: it may be strategically dangerous to focus the attention on an alternative to mitigation while mitigation is already so di¢ cult to organize because it deals with a global and intertemporal problem. Moreover, mitigation is viewed as a proactive policy while adaptation policies are rather reactive and could therefore always be considered in case mitigation had failed. This is probably why the economic and environmental science literature has largely analyzed the cost and e¤ectiveness of mitigation but adaptation to an already changed climate has only recently entered the picture. Having crossed the red line, it becomes more and more usual to consider them together as complements or substitutes. However, adaptation e¤ectiveness is limited: it may be too costly or even impossible to adapt to a too large climate change. Studies available in the literature provide estimates (World Bank, 2009) that reach 100 billion USD per year for the adaptation cost between 2010 and 2050 to an only 2 C warmer world by 2050. Moreover, some catastrophic consequences often described (Keller et alii. Climatic change, 2008) seems di¢ cult to cope with.

Only a few studies explicitly consider both adaptation and mitigation as policy responses to climate change in long term dynamic analysis. 1 Brechet et alii (2013) propose a theoretical framework with accumulation in physical capital, greenhouse gases, and adaptation capital, to show that the issue of substitutability between the two instruments depends on the stage of development. In particular, the relationship between adaptation and economic e¢ ciency is inverted U-shaped implying that adaptation should be low or nil for poor or developed countries but of a signi…cant size for medium-developed ones. Tsur and Whitagen (2012) extend the framework to account for catastrophic (abrupt) uncertain climate change in a model with exogenous growth.

The timing of this change depends on atmospheric GHG concentration that is assumed to be exogenous to a single country and there is therefore no pollution accumulation in the model. They obtain that for a stationary economy adaptation actions may be suboptimal depending on the discount rate, the aversion to intergenerational inequality, the catastrophic risk and the severity of the catastrophic damage. However a growing economy can more easily a¤ord the adaptation investment cost and is more vulnerable to the climate change. It is therefore always optimal to invest in adaptation capital (up to the point where it is no longer e¤ective) in such economies.

Finally, Zemel (2014) consider optimal emission when adaptation is possible in the presence of an uncertain pollution damage. Less pollution reduces the risk of damage occurrence while adaptation capital reduces the size of this potential damage. The nature of the pollution and adaptation processes is studies. Cases are exhibited in which …rst only one of these processes evolves for some period of time and second pollution or adaptation is not monotonic.

In line with this literature this paper provides a long term analysis of the optimal mitiga-tion/adaptation mix by considering the social planner problem of a polluting country. Following the …rst part of Tsur and Whitagen (2012), we restrict our attention to economies with neither growth nor technological progress 2 but we account for accumulation in abatement 3 capital, in greenhouse gases, and in adaptation capital in order to better capture the arbitrage between abatement and adaptation investments. Pollution damages arise from the emissions due to the country consumption but also from the emissions of the rest of the world (ROW). We obtain that an increase in ROW's emissions unambiguously reduces mitigation and increases adaptation. Moreover, a larger revenue leads to a larger share devoted to mitigation. Finally, a more e¢ cient adaptation may lead to more adaptation, in contrast to what is obtained in [START_REF] Benchekroun | Adaptation E¤ectiveness and Free-Riding Incentives in International Environmental Agreements[END_REF]. We introduce a critical threshold of pollution (in the same vein as [START_REF] Prieur | The Environmental Kuznets curve in a world of irreversibility[END_REF][START_REF] Naevdal | Optimal Regulation of Natural Resources in the Presence of Irreversible Threshold E¤ects[END_REF], above which adaptation is longer e¢ cient. We obtain that if this threshold is lower than the steady-state level of pollution, there is no way for the modeled economy to avoid it. In particular, such a situation will appear if the ROW's emissions are high. Next step is then to introduce another type of investment allowing for lower ROW pollution ie. emissions reduction abroad through, for instance, clean development mechanisms (CDM). The model is therefore extended to capture the arbitrage between adaptation investment (necessarily domestic), domestic mitigation investment and investment in CDM that aims at reducing ROW's emissions. We obtain that CDM may be a means to avoid a pollution threshold above which adaptation becomes of no use.

The rest of the paper is as follows. The general model is presented in section 2. It is solved for the optimal adaptation and abatement investments in section 3. CDM are considered in section 4. Section 5 concludes.

2 Adding growth would prevent from analytically solving the model. Moreover, such an assumption together with the existence of a pollution threshold would lead to a degenerate solution.

3 There exists a di¤erence between mitigation and abatement: the former refers to a reduction in net emissions of greenhouse gases while the latter refers to a reduction in gross emissions. In this paper, we integrate only emission abatement opportunities as it is largely the case in the literature. In the rest of the paper we will indi¤erently refer to mitigation or abatement.

The model

We consider a country-level economy where consumption, c, is the source of polluting emissions.

Households derive utility from consumption but are damaged by climate change due to GHG stock. However, damages occurring because of climate change of a given size depend on the adaptation e¤orts made by the country. To capture these features, the instantaneous utility function is de…ned over consumption and a damage function D whose arguments are the stock of pollution (that gives a measure of climate change) and the level of adaptation:

U (c; S; A) = u(c) D(S; A) (1)
where c is consumption, and S and A are respectively the stock of GHG and the adaptation capacity. Moreover, u0 > 0; u 00 < 0;

D S > 0; D SS 0; D A < 0; D AA > 0; D AS < 0. Note also that, D(0; A) = 0; D(S; 0) = D(S) > 0; D(1; A) = 1; D(S; 1) = 0.
The stock of GHG evolves according to:

S t = c t M (K t ) + E S t (2)
E 0 is exogenous and corresponds to the net emissions of the rest of the world. It can be seen as the di¤erence between total emissions of the rest of the world and mitigation e¤orts of the rest of the world. is the rate of natural pollution decay. Note that we model a reduction in net emissions of greenhouse gases that could more rigorously be referred to as abatement rather than mitigation. 4 In case the rest of the world would make very signi…cant e¤orts, it is possible

to have E = 0. Finally M (K) is the level of mitigation that depends on the stock of mitigation capital. An example of such a capital is the amount of renovations in buildings aimed at reducing energy consumption.

The economy is endowed with a …xed revenue Y and with a resource that is available in in…nite quantity. Final consumption only comes from the free resource use that generates GHG emissions and contributes therefore to climate change. Revenue Y is used for expenses that have to be shared between mitigation and adaptation:

Y = I Kt + I At
where I Kt and I At denote investment in mitigation and adaptation respectively. The share of revenue …nancing mitigation, t , is a control variable. The remaining share (1 t ) is used to adapt the economy to climate change:

Y = t Y + (1 t )Y with t Y = I Kt and (1 t )Y = I At (3)
Adaptation infrastructures (such as dykes) are accumulated according to:

A t = [(1 t )Y ] A A A t (4)
with A being the depreciation rate of adaptation capital. Recall that the adaptation capital stock reduces the damage of a given pollution stock: D is function of A.

Investment in mitigation infrastructures (such as energy e¢ cient investments in buildings) accumulates according to:

K t = ( t Y ) M M K t (5)
with M being the depreciation rate of mitigation capital. Recall that the mitigation capital stock reduces the stock of pollution: S t is a function of K.

In order to provide analytical results, we make the following simplifying assumptions:

u(c) = c ! , 0 < ! < 1 D(S; A) = A S M (K) = K m m 1 A = M = < 1; A = M =
3 Optimal adaptation and mitigation e¤orts

In this section, we solve for the optimal adaptation and mitigation e¤orts. We then perform some comparative statics to study the e¤ect of ROW emissions, revenue and adaptation e¢ ciency on the steady-state levels of adaptation and mitigation. We …nally appraise whether a pollution threshold above which adaptation becomes ine¢ cient can be avoided.

Solving for the optimal adaptation and mitigation e¤orts

The policy maker of the economy chooses the optimal sequences of consumption fc t g 1 t=0 and of the share of revenue devoted to mitigation f t g 1 t=0 in order to solve:

max fc; g Z +1 0 exp t U (c; S; A)dt
subject to(2), ( 4) and (5) S 0 ; A 0 ; K 0 given with 2 (0; 1), the discount rate. The corresponding Hamiltonian is:

H = c ! SA + c S K m + E + A [[(1 )Y ] A] + M [( Y ) K]
and associated FOCs lead to

= = (c) c=c = ( + ) A c 1 ! =! (6) = 1 1 1 M ( =(1 )) 1 SA 1 !c ! 1 mK m 1 =( ) (7)
where (c) = u 00 (c)c=u 0 (c). A steady-state (SS) is such that A=A = 0 K=K = 0; S=S = 0; : c=c = 0; and = = 0. Equations ( 4), ( 5) and ( 6) gives:

K = (Y ) = = K ( ) (8) A = (1 ) Y = = A ( ) (9) c = (( + )!A ( ) = ) 1 1 ! = c( ) (10) Note that 1 is equivalent to K K, with K = Y = . Moreover, (2) implies S( ) = 1 2 4 ( + )!K ! 1 1 ! (1 ) 1 ! K m m + E 3 5 (11)
Finally ( 7) leads to

S( ) = mK m ( + ) (1 ) m 1 | {z } G( ) with G 0 ( ) < 0; lim !0 G(0) = +1; G(1) = 0 and S 0 ( ) < 0 with S(0) = ( + )!K 1 1 ! + E and S(1) = K m + E that

Comparative statics

Let us determine the e¤ect of more ROW emissions on domestic behavior. An increase in E a¤ects the SS through S( ) but has no e¤ect on G( ). Since @( S( ))=@E = , it shifts the S( ) curve upwards. An increase in the ROW emissions E generates a decrease in the steady-state level of the share of revenue devoted to mitigation thus increasing adaptation.This result is obtained analytically but …gures 2 provides a graphic illustration. Following an increase in revenue, G( ) shifts upwards and S( ) rotates clockwise, and …gure 3 shows that for a large initial one cannot rule out the case in which there is a larger share devoted to mitigation. meaning that a more e¢ cient adaptation could lead to less adaptation. Note that this latter possible result is consistent with the result of [START_REF] Benchekroun | Adaptation E¤ectiveness and Free-Riding Incentives in International Environmental Agreements[END_REF]. Intuition runs as follows.

If adaptation becomes more e¢ cient, it is then possible to invest less in adaptation, to reach a higher or equivalent level of A; therefore releasing resources for more mitigation. Finally, note that b is independent from E, while a change in the ROW emissions shifts the S curve upwards.

There exists a high enough value for these emissions such that the steady state is smaller than b , implying a non-ambiguous positive e¤ect of adaptation e¢ ciency on adaptation. One can …nd the condition on E that ensures < b :

G( b ) < S( b ) E > (1 K 1= ) m 1 K m 1 + m ( + ) ( + ) K 1= ( + )! 1 1 !
The set of parameters ensuring both the steady state existence and < b is obviously non-empty.

Therefore, under this latter condition, a more e¢ cient adaptation leads to more adaptation: in our framework, there exist levels of ROW emissions such that the conclusion obtained by Benchekroun is not valid. Note that it is especially likely to be the case for small emitting countries. However, a more e¢ cient adaption leads to more adaptation if and only if:

G( b ) < S( b ) with b = 1 K 1= 3.

Introducing a deterministic threshold

In this subsection we take account of a threshold of pollution above which adaption is no longer e¢ cient. In this paper, the stock of pollution is a proxy for climate change and what we want to address here is the fact that for a too large climate change the e¤ect of damages on welfare can no longer be reduced using adaption investment. This is consistent with the …ndings of de Bruin et al. ( 2009) that showed that the higher the current value of climate damage, the less relevant adaptation is as a policy option. Therefore, for S(t) > S; the adaptation alternative disappears.

Note that this irreversibility only plays a role if the threshold is lower than the steady-state level of pollution derived in the previous section, ie. S < S : However there exist two types of solution.

Even if S < S , it may be optimal, knowing the threshold, to change decisions in order to have pollution remaining below S. Such a solution is said to be reversible. By contrast, a solution is irreversible if the pollution stock enters the irreversible region in …nite time. We …rst show that the only reversible solution consists in reaching the threshold. Such a result suggests that it could be worth considering other climate policies such as devoting money to reduce emissions abroad (this will be considered in the next section).

Once the threshold has been reached, the main assumption a¤ected in our model speci…cations concerns the damage function, that becomes such that D 0 A = 0, and consistent with this new assumption we now adopt D(S; A) = S: Since adaptation generates no bene…ts but is costly (it requires a fraction (1 ) of revenue Y ), it is of no interest to engage into any adaptation activities and variable A is simply removed from the model.

If considering the reversible solution, the program is: 4) and (5) S 0 ; A 0 ; K 0 given and S t < S for all t

max fc; g Z +1 0 exp t U (c; S; A)dt subject to(2), (
The corresponding Hamiltonian is then:

H = c ! SA + c S K m + E + A [[(1 )Y ] A] + M [( Y ) K] + (S S)
The associated FOCs are the same as in the case without threshold except for the one with respect to S, and the additional condition related to the threshold constraint:

= = + + A = +
((S S) = 0; 0 If = 0, the solution reduces to the one without threshold. Since we have assumed that S < S , the threshold cannot be avoided.

If > 0, S t = S and

c t K m t + E S = 0
consumption growth rate is then:

c=c = 1 (c) ( + ) A c 1 ! =! +
The Lagrange multiplier a¤ects the arbitrage through the marginal value of the stock. Indeed is larger and we expect 0 to be smaller since we have reached the threshold (see also Amigues and Moreaux, 2012). The threshold cannot be avoided. In particular, a steady-state with adaptation is no longer possible. Proposition 3 summarizes the e¤ect of a pollution threshold:

Proposition 3 In the current model with emissions generated by the ROW, accumulation in pollution, in mitigation capital and in adaptation capital, there is no way to avoid a threshold of pollution above which adaption is no longer e¢ cient if the steady state level of pollution (in a model that ignores this threshold) is higher than the threshold.

The fact that the threshold cannot be avoided comes from ROW emissions (and pollution natural decay) that cannot be controlled for by the country. Devoting money to CDM would be a means to control for the ROW emissions. This is why it seems to be an especially relevant policy to consider. We turn to it in the next section.

Introducing ROW emissions reduction

We now consider that part of the resources of the country can be devoted to CDM leading to less ROW emissions. In a …rst step, we simply focus on the e¤ect of devoting resources to CDM on the trade-o¤ between adaptation and mitigation, still using the previous framework. The fraction of Y used for CDM is then taken as exogenous and we do not consider any e¤ect of these CDM on ROW emissions. Rationale for such an assumption is that little is known about the e¢ ciency of CDM on global emissions. We obtain, in particular, that one cannot rule out the case for which less resources reduces so much more adaptation (and therefore consumption and emissions) than mitigation that the steady-state level of pollution is lower even if we ignore any e¤ect on the ROW emission level. Analytical computation leads to @S @Y > 0 or < 0 depending on the parameters Considering now some qualitatively negative e¤ect on ROW emissions, the function G( ) moves up but function F ( ) is not a¤ected, that leads to a larger . Since @A =@E = @A =@ :@ =@E with @A =@ < 0 adaptation is reduced. One can deduce, using equation ( 11), that following a decrease in the ROW emissions, the total stock of pollution is smaller, due to more mitigation, less adaptation and the direct e¤ect of ROW emissions.

For parameters values such that @S =@Y > 0 investing in CDM is unambiguously a way to lower the steady-state level of pollution and threfore a means to avoid the threshold above which adaptation becomes ine¤ective. For parameters values such that @S =@Y < 0 two opposite e¤ects exist and a lower S may only be reachable if CDM are in reality su¢ ciently e¢ cient in reducing E.

In a second step, investment in CDM becomes an optimal choice. The corresponding model is presented in the next subsection that is followed by the corresponding comparative statics.

The model

Considering CDM as an optimal choice implies to formally introduce the bene…ts of CDM ie. to endogeneize ROW emissions E t that become dependent on the amount devoted to CDM:

E t = E [(1 t )Y ] s
where is the share of revenue devoted to domestic management of pollution (i.e. either adaptation or mitigation). s is the CDM e¢ ciency in reducing ROW emissions. We assume s > 1 to ensure the concavity of the pollution accumulation process. We impose (1 t )Y > 1 such that a larger s, by rising mitigation e¢ ciency abroad, e¤ectively reduces ROW emissions. Moreover, we assume that Y s < E, meaning that even if all the revenue is devoted to CDM, emissions cannot be negative. The share is a new control variable. The adaptation and mitigation processes become then:

A t = [ t (1 t )Y ] A t (12) K t = ( t t Y ) K t (13)
and the corresponding Hamiltonian writes:

H = c ! SA + c S K m + E [(1 )Y ] s + A [[ (1 )Y ] A] + M [( Y ) K]
FOCs with respect to consumption, , pollution adaptation and mitigation stocks remain unaffected by the fact that ROW emissions are endogeneized. However, there is a new FOC, with respect to :

eY s (1 ) s 1 1 = M 1 Y
The FOCs lead to

: 1 + (s 1) 1 = (1 )
:

(1 !)

: c c ( + ) + !m M c ! 1 K m 1 (14)
A steady-state is such that A=A = 0 K=K = 0; S=S = 0;

: c=c = 0; : = = 0 and = = 0.
Equations ( 12), ( 13) and (6) gives:

K ( ; ) = ( ) K (15)
A ( ; ) = (1 ) K (16) c ( ; ) = ( + )! (1 ) K 1 1 ! (17)
Using the pollution accumulation process and the fact that at the steady-state : S=S = 0, we have:

S ( ) = ( + )! (1 ) K 1 1 ! ( ) m K m + E (1 ) s Y s
Moreover, using ( 14) and the fact that at the steady-state : = = 0 (and knowing that at this steady-state we also have

: = = : c=c = 0): ( + ) m eY s (1 ) s 1 K 1 m = ( ) m 1 (18) ) d d > 0 i¤ > e with e = 1 m s m < 1
For a large (resp. small) ,there exists some substitutability (resp. complementarity) between home emissions reduction and ROW emissions reduction : a higher (resp. lower) , that means less (resp. more) ROW emissions reduction, goes with a higher (resp. lower) , that means a larger (resp. smaller) share of home mitigation with respect to adaptation). Moreover

lim !0 ! +1 and lim !1 ! +1
The share devoted to home investments can never be equal to zero or unity, meaning that it is never optimal to use all revenue only for CDM or only for home climate policies. We have ( ) > 0 8 , and the condition ensuring that there exist ( ) 1 is

( e ) 1 , Y s m ( + ) K m 1 e m 1 (1 e ) 1 s
The revenue must be su¢ ciently high so that it is worth devoting some money to home and foreign climate policies. From now on we restrict the de…nition domain for to be [ ; ] with ; and implicitly determined by ( ) = 1.

Finally, using the fact that : = = 0 we obtain:

S( ( ); ) m | {z } F ( ) = mK m ( + ) (1 ( )) ( ) m 1 | {z } G( )
with G( ) that is the same as previously (without CDM). Since G 0 ( ) = @G( ) @ @ @ , G( ) is hump-shaped with a max in e . Moreover

E > max[ m K m + (1 ) s Y s ; m K m + (1 ) s Y s ]
and G( e ) > F ( e ) with e = 1 m s m :

Comparative statics

Having characterized the SS, we can now turn to some comparative statics. An increase in BAU (Business As Usual) shifts F ( ) upwards. As a result, a steady-state located in the decreasing (resp. increasing) part of G( ) will be reduced (resp. increased). The e¤ect of BAU ROW emissions i.e. ROW emissions that would prevail in the absence of CDM can be analyzed as follows:

@ @E = S 0 @ " G= (" S= m) | {z } F E 1 A
where " G= is the elasticity of G with respect of , and S= is the elasticity of S with respect of . Note that @ @E > 0 i¤ " G= > 0 since F E is only a feedback e¤ect potentially reducing or reinforcing a direct positive e¤ect (see …gure 5). This means that larger pre-CDM ROW's emissions leads to less revenue devoted to CDM (ie. a larger ) if and only if a large fraction of Y was initially used for CDM. This tranlates then into a smaller share of mitigation in home policy irrespective of the initial level of foreign mitigation. Note that the introduction of CDM reverses the e¤ect of BAU ROW on home mitigation. Figure 5 provides a numerical illustration.

The e¤ect of a more e¢ cient adaptation is even more complicated to study than in the absence of CDM since the relationship between and is not straightforward. Moreover, a larger shifts G( ) downwards but @F ( )=@ > 0 i¤ (1 ( )) > K 1= . In particular, @F ( )=@ j = = 0 and @F ( )=@ j = = 0. But interestingly in the case described in …gure 7, starting from a low , a more e¢ cient adaptation generates more home policies and more adaptation (since d =d < 0). Starting from a high , more e¢ cient adaptation generates less home policies and more adaptation (since d =d > 0). Therefore the result obtained by [START_REF] Benchekroun | Adaptation E¤ectiveness and Free-Riding Incentives in International Environmental Agreements[END_REF] is not valid for parameters used to design …gure 7. We also study the e¤ect of an increase in CDM e¢ ciency. We obtain that more e¢ cient CDM unambiguously shifts F ( ) downwards and G( ) upwards. Focusing then on a small (such that < e ), an increase in CDM e¢ ciency unambiguously reduces the SS share of revenue devoted to home policies and increases the share of mitigation in home policy. For an initially large , less.revenue is used for policy abroad if the e¢ ciency of this policy rises, and this comes with more domestic mitigation as a share home policy (and in level as well). The …nal question consists in appraising whether CDM are a means to avoid the "adaptation threshold". To address it, we compare steady-state pollution with foreign mitigation (denoted by S ) and without such mitigation (denoted by S ):

S S = ( + )!K (1 ) ! 1 1 ! ( 1) | {z } <0 + m K m (1 m ) | {z } >0 (1 ) s Y s | {z } <0 A su¢ cient condition for S < S is : m K m (1 m ) < (1 ) s Y s (19)
since @ @s < 0 (see equation ( 18)), a su¢ ciently high value for s ensures that condition (18) is satis…ed. Therefore, we can conclude that CDM are an e¤ective way to lower the probability of reaching a pollution threshold provided foreign mitigation is su¤ciently e¢ cient. Comparative statics results are summarized in proposition 5:

Proposition 5 In the current model, conducting mitigation abroad is a means to avoid a pollution threshold provided foreign mitigation is su¢ ciently e¢ cient. Moreover, comparative statics are the following: an increase in BAU ROW reduces the share of mitigation in home policy; it increases foreign mitigation if and only if foreign mitigation was initially low; a more e¢ cient adaptation does not necessarily result in less adaptation; a more e¢ cient foreign mitigation increases the share of mitigation in home policy; it reduces foreign mitigation if and only if it was initially low therefore leading to a further lower level of foreign mitigation.

This paper provides a long term analysis of the optimal mitigation/adaptation/CDM mix by considering the social planner problem of a polluting country. We account for accumulation in mitigation capital, in greenhouse gases, and in adaptation capital. Notably, we obtain that a more e¢ cient adaptation does not necessary lead to a substitution of adaptation for mitigation that contradicts the results in [START_REF] Benchekroun | Adaptation E¤ectiveness and Free-Riding Incentives in International Environmental Agreements[END_REF]. In our framework, if a pollution threshold exists, above which adaptation is no longer e¢ cient we show that there is no way for the modeled economy to avoid it. We therefore extend our framework to allow a fraction of revenue to be devoted to CDMs and we show that they may be a means to avoid the pollution threshold. Moreover, our results concerning the e¤ect of adaptation e¢ ciency are robust to the introduction of CDM. Finally we show that more e¢ cient CDMs generate an expansion in foreign mitigation only if this policy was already largely developped. Some extensions of the paper could be considered. In particular, the pollution threshold above which adaptation is of no use is not perfectly known. Studying the consequence for mitigation and adaptation of an uncertain threshold would be very interesting since we know that uncertain thresholds lead to dynamics that drastically di¤er from those that prevail with deterministic thresholds (see Ayong Le Kama et alii; 2013). Further research could also extend the model to allow for a share of revenue used to develop a non polluting resource such that consumption would be fully free. This would clearly a¤ect the mitigation/adaptation arbitrage as well as the risk to reach the pollution threshold.

  Figure 1

Figure 2 :

 2 Figure 2: increase in ROW emissions (in red)

Figure 4 :

 4 Figure 4: more e¢ cient adaptation (in red)

Proposition 4

 4 s ]: Therefore there exist at least two steady-states 6 if and only if G( e ) > F ( e ) that will require E not to be too large (see comparative statics with respect to E below). The existence and uniqueness results are established in proposition 4 and illustrated in …gure 5.7 In the current model with accumulation in pollution, in adaptation capital and in mitigation in the home country and abroad, su¢ cient conditions for the existence of a steady-state denoted (c ; ; ; A ; K ; S ) are:

Figure 5

 5 Figure 5

Figure 6 :

 6 Figure 6: e¤ect of E Figure 7: e¤ect of a more e¢ cient adaptation

Figure 9 :

 9 Figure 9: More e¢ cient CDM (in red)

  and Fankhauser S., (2008), Economic Aspects of Adaptation to Climate Change: Costs, Bene…ts and Policy Instruments, OECD, Paris. Amigues J-P. and Moreaux M., (2012), "Potential Irreversible Catastrophic Shifts of the Assimilative Capacity of the Environment » , IDEI Working Paper, n 697, 8 février 2012.

There exists however descriptive studies (e.g.Kane and Yohe,2000;[START_REF] Smit | An anatomy of adaptation to climate change and variability[END_REF]

2000; Agrawal and Fankhauser, 2008; EEA,2007; UNFCCC, 2007), game-theoretic papers[START_REF] Shalizi | To mitigate or to adapt; is that the question? Observations on an appropriate response to the climate change challenge to development strategies[END_REF] Ikefuji, Mangnus and Sakamoto, 2014;[START_REF] Shogren | Linking adaptation and mitigation in climate change policy[END_REF][START_REF] Buob | To mitigate or to adapt: how to combat with global climate change[END_REF] or static modeling[START_REF] Ingham | Can Adaptation and Mitigation be complements?[END_REF].

Mitigation rather refers to a reduction in gross emissions, ie. mitigation e¤orts would lower in equation (2).

It means that if doing no adaptation, home mitigation and international agreements are not enough to induce a pollution stock that is decreasing in time.

In the special case m = 1, which means that income Y has a linear impact on pollution accumulation, G 0 ( ) < 0 and F 0 ( ) < 0 that leads to a unique steady state and is constant. Moreover, it provides hints for the selection of the steady-state that should be considered for combinations of and m such that m is not too far from unity.

For clarity of the …gure, we have chosen parameters such that F 0 ( ) > 0 but the same reasoning applies for a non-monotonous F ( ) that could lead to more than 2 steady-states.