Christian Bonatti

Yash Lodha

Michele Triestino

Hyperbolicity as an obstruction to smoothability for one-dimensional actions

Introduction

A few examples are known of groups that admit no sufficiently smooth action on a one-dimensional manifold. Following the direction of Zimmer program, typical examples come from lattices in higher rank Lie groups [START_REF] Burger | Bounded cohomology of lattices in higher rank Lie groups[END_REF][START_REF] Ghys | Actions de réseaux sur le cercle[END_REF][START_REF] Witte | Arithmetic groups of higher Q-rank cannot act on 1-manifolds[END_REF], or more generally from groups with Kazhdan's property (T ) [START_REF] Navas | Actions de groupes de Kazhdan sur le cercle[END_REF][START_REF]Quelques nouveaux phńomènes de rang 1 pour les groupes de difféomorphismes du cercle[END_REF].

Other interesting examples appear in [START_REF] Baik | Unsmoothable group actions on compact one-manifolds[END_REF][START_REF] Calegari | Dynamical forcing of circular groups[END_REF][START_REF]Nonsmoothable, locally indicable group actions on the interval[END_REF][START_REF]A finitely generated, locally indicable group with no faithful action by C 1 diffeomorphisms of the interval[END_REF][START_REF] Parwani | C 1 actions on the mapping class groups on the circle[END_REF].

In this work we address the problem of the existence of smooth actions of groups of piecewise projective homeomorphisms of the real line. Our principal interest comes from the existence of groups of this kind which are negative solutions to the so-called Day-von Neumann problem, as shown by Monod and Lodha-Moore [START_REF] Lodha | A nonamenable finitely presented group of piecewise projective homeomorphisms[END_REF][START_REF] Monod | Groups of piecewise projective homeomorphisms[END_REF]. On the other hand, partially motivated by his work on Kazhdan groups acting on the circle, Navas raised the problem to find obstructions for a group of piecewise linear homeomorphisms of the interval to admit smooth actions (cf. [START_REF] Bonatti | Rigidity for C 1 actions on the interval arising from hyperbolicity I: solvable groups[END_REF][START_REF]A finitely generated, locally indicable group with no faithful action by C 1 diffeomorphisms of the interval[END_REF]). With this work, we illustrate relatively elementary tools which apply to a large variety of examples of such groups. Our techniques rely on some classical facts on one-dimensional dynamics and the recent work by Bonatti, Navas, Rivas and Monteverde on actions of abelian-by-cyclic groups [START_REF] Bonatti | Rigidity for C 1 actions on the interval arising from hyperbolicity I: solvable groups[END_REF].

A classical obstruction to have C 1 actions on the interval is Thurston's Stability Theorem [START_REF] Thurston | A generalization of the Reeb stability theorem[END_REF]: a group of C 1 diffeomorphisms of the interval is locally indicable, namely every finitely generated subgroup has a nontrivial morphism to Z. This obstruction does not apply in our setting: the group of piecewise projective homeomorphisms of the real line is locally indicable. Therefore our results exhibit new examples of locally indicable groups that have no C 1 action on the interval.

As an appetizer, even before introducing the notions and definitions which are necessary for presenting our main results, we start with two results whose statement are very easy to understand, and which illustrate the spirit of the paper. Fix λ > 1 and consider:

• the linear map f λ : R → R defined as x → λx,

• the map h λ : R → R defined as h λ (x) =

x if x ≤ 0, λx if x > 0,

• the translation g : x → x + 1.

Let G λ be the subgroup f λ , g, h λ ⊂ Homeo + (R).

Theorem 1.1. For any λ > 1 which is rational (in formula: λ ∈ Q ∩ (1, +∞)) and any morphism ρ : G λ → Diff 1 + ([0, 1]) one has :

the commutator [g, h λ gh -1 λ ] belongs to the kernel of ρ.

In particular ρ cannot be injective. The same holds for any morphism ϕ : G λ → Diff 1 + (S 1 ), where S 1 is the circle.

As a corollary, looking at the relations in G λ , one obtains the stronger conclusion that for any representation ρ : G λ → Diff 1 + ([0, 1]), the image ρ(G λ ) is a metabelian group (that is, a solvable group with abelian derived subgroup).

We do not know if the same occurs for λ / ∈ Q. Nevertheless, consider the natural realization ρ 0 : G λ → Homeo + (S 1 ) defined as follows:

• one considers S 1 as being the projective space RP 1 ,

• ρ 0 (f λ ) acts on S 1 as the projective action of the matrix λ 0 0 1 ,

• ρ 0 (g) acts on S 1 as the projective action of the matrix 1 1 0 1 ,

• ρ 0 (h λ ) coincides with ρ 0 (f λ ) on the half circle [0, +∞] and with the identity map on the half circle [-∞, 0].

Theorem 1.2. With the notation as above, it does not exist any homeomorphism φ : S 1 → S 1 so that φρ 0 (f λ )φ -1 , φρ 0 (h λ )φ -1 and φρ 0 (g)φ -1 belong to Diff 1 + (S 1 ).

In other words, the natural action of G λ on S 1 is not smoothable, and furthermore, if λ ∈ Q, then every C 1 action of G λ on the circle or the interval are (non-faithful) metabelian actions.

For more precise statements, see Theorems 6.7 and 6.9.

The paper is organized as follows. In Section 2 we introduce the basic objects and fix some notation. In Section 3 we roughly explain the different strategies that we develop in this work, showing which are the main applications. In Section 4 we illustrate the main motivation of our work, which is the recent construction by Monod of nonamenable groups without free subgroups. In Section 5 we study the C 1 actions of Monod's groups and the finitely presentable group defined by Lodha-Moore. Section 6 contains the main part of this work, namely the study of C 1 actions of the groups G λ introduced above. Finally, in Section 7 we use different techniques that work in C 2 regularity.

Some definitions and notation

Definition 2.1. Let M be a manifold and Homeo(M ) the group of homeomorphisms of M . A group

G ⊂ Homeo(M ) is C r -smoothable (r ≥ 1) if it is conjugate in Homeo(M ) to a group in Diff r (M ), the group of C r diffeomorphisms of M . Remark 2.2. Even if a certain group G ⊂ Homeo(M ) is not C r -
smoothable, it is still possible that the group G, as abstract group, admits C r actions on the manifold M .

Throughout this work we shall only be concerned by one-dimensional manifolds. We restrict our discussion to orientation-preserving homeomorphisms, which form a subgroup Homeo + (M ) of index two in Homeo(M ). We will not make much distinction between the groups Homeo + (R) and Homeo + ([0, 1]). Notice however that the groups Diff r + (R) and Diff r + ([0, 1]) are different and for this reason we sometimes identify the interval [0, 1] to the compactified real line [-∞, +∞]. Choosing the affine chart t → [t : 1], we consider R as the affine line in the projective space RP 1 ∼ = R ∪ {∞}, which is topologically the circle S 1 . The group Homeo + (R) can be identified to a subgroup of Homeo + (S 1 ), for instance as the stabiliser of the point ∞ of S 1 ∼ = RP 1 .

The projective special linear group PSL(2, R) = SL(2, R)/{±id} naturally acts on the projective real line RP 1 by Möbius transformations: from now on, we shall always suppose that PSL(2, R) acts on the circle in this way.

Definition 2.3. A circle homeomorphism

h ∈ Homeo + (RP 1
) is piecewise projective if there exists a finite partition RP 1 = I 1 ∪ . . . ∪ I of the circle into intervals, such that every restriction h| I k , k = 1, . . . , , coincides with the restriction of a Möbius transformation.

A breakpoint of h is a point b ∈ RP 1 such that the restriction of h to any neighbourhood of b does not coincide with the restriction of a Möbius transformation.

The group of all orientation-preserving piecewise projective homeomorphisms of the circle is denoted by PP + (RP 1 ). Similarly, we define the group of piecewise projective homeomorphisms of the real line PP + (R), identifying it to the stabiliser of ∞ inside PP + (RP 1 ).

We recall that a fixed point p ∈ R for a diffeomorphism f ∈ Diff 1 + (R) is a hyperbolic fixed point if f has derivative at p which is not 1. We shall say that a group G ⊂ Diff 1 + (R) has hyperbolic fixed points if there exists an element f ∈ G with hyperbolic fixed points. This notion is related to the notion of hyperbolic elements in PSL(2, R). A nontrivial projective transformation in PSL(2, R) has at most two fixed points. If it has exactly two fixed points, it is called hyperbolic, and if it has only one fixed point it is called parabolic.

A matrix M in SL(2, R) is hyperbolic if | Tr(M )| > 2, parabolic if | Tr(M )| = 2 and elliptic if | Tr(M )| < 2.
Then the corresponding projective transformation is respectively hyperbolic, parabolic and elliptic.

Given a subgroup Γ ⊂ PSL(2, R), we say that a real r ∈ R is a hyperbolic fixed point for Γ if there is a γ ∈ Γ such that γ is hyperbolic and γ(r) = r. Similarly, we define the notion of a parabolic fixed point for Γ. We consider the sets H Γ and P Γ of hyperbolic fixed points and parabolic fixed points of elements of Γ, respectively. When Γ = PSL(2, A) = SL(2, A)/{±Id}, for some subring A ⊂ R, we simply write H A and P A . Here SL(2, A) is the group of invertible (2 × 2)-matrices with determinant 1 and coefficients in A.

The mechanisms

The aim of this work is to present three different techniques which provide a variety of examples of non-smoothable groups in PP + (R). The three techniques rely on the rigid hyperbolicity of the actions: there are groups G ⊂ Diff 1 + (R) such that, no matter how one (topologically) conjugates them inside Diff 1 + (R), will always have hyperbolic fixed points. More precisely, suppose that in G ⊂ Diff 1 + (R) there is an element f having a hyperbolic fixed point p ∈ R. Consider another group G ⊂ Diff 1 + (R) to which G is topologically conjugate by some homeomorphism φ: φGφ -1 = G. The point φ(p) is a fixed point for φf φ -1 , but since φ is just a homeomorphism, we cannot ensure that it is a hyperbolic fixed point. However, there are some topological mechanisms that guarantee hyperbolicity.

The first one is when there are linked pairs of fixed points in G. We now define this notion. Denote by Fix(g) the set of fixed points of a homeomorphism g. A pair of successive fixed points of G is a pair a, b ∈ R, a < b such that there is an element g ∈ G and (a, b) is a connected component of R \ Fix(g). A linked pair of fixed points consists of pairs a, b and c, d such that:

1) There are elements f, g ∈ G such that a, b is a pair of successive fixed points of f and c, d is a pair of successive fixed points of g.

2) Either {a, b} ∩ (c, d) or (a, b) ∩ {c, d} is a point.

In this case hyperbolicity is obtained by a probabilistic argument. Some element h in the semigroup generated by f and g will have a hyperbolic fixed point somewhere. This is the so-called Sacksteder's Theorem, in its version for C 1 -pseudogroups [START_REF] Deroin | Sur la dynamique unidimensionnelle en régularité intermédiaire[END_REF][START_REF]Groups of circle diffeomorphisms[END_REF]. This method applies to large groups of piecewise projective homeomorphisms, as the Monod's groups (see Definition 5.1):

Theorem 3.1. The following holds for Monod's groups H(A) and G(A).

(1) For any subring A ⊂ R, Monod's groups H(A) and G(A) are not C 1 -smoothable.

(2) If A is such that the group PSL(2, A) contains the homothety

f λ : x → λx,
for some rational λ > 1, then there exists no injective morphism ρ :

H(A) → Diff 1 + ([0, 1]).
The second one is when there is an exponential growth of orbits. In this case we can ensure that a specific point is always a hyperbolic fixed point. This applies for example to the dyadic affine group t → t + 1, t → 2t , which is isomorphic to the Baumslag-Solitar group BS(1, 2), as described in [START_REF] Bonatti | Rigidity for C 1 actions on the interval arising from hyperbolicity I: solvable groups[END_REF]. From this, it is easy to build examples of finitely generated groups in PP + (R) which are not C 1 -smoothable. This method applies to the finitely presentable Lodha-Moore group (see § 5.2), for which we do not only prove that its action is not C 1 -smoothable, but also that it has no nontrivial C 1 action on the interval: Theorem 3.2. Every morphism from the Lodha-Moore group G 0 to Diff 1 + ([0, 1]) has abelian image.

The third one relies on the nature of stabilisers, and here we require that the regularity of the group G is C 2 . If there exists a point x ∈ R such that the (right, for instance) germs of elements g ∈ G fixing x define a group which is dense in R, then we can use the Szekeres vector field to have a well-defined local differentiable structure, by means of which we ensure that the hyperbolic nature of a fixed point cannot change after topological conjugacy to another C 2 action. This method applies to examples of groups in PP + (R) that are naturally in Diff

Historical motivation 4.1 Thompson's groups F and T

In the 50s, Richard J. Thompson introduced three groups F , T and V , which have many nice properties (cf. [START_REF] Cannon | Introductory notes on Richard Thompson's groups[END_REF]). These groups are finitely presented and [F, F ], T , V are simple. They have been among the first known examples sharing these properties. Since only F and T act by homeomorphisms on the circle, we restrict our attention to them. Definition 4.1. Thompson's group T is the group of all piecewise linear homeomorphisms of the circle S 1 ∼ = R/Z such that all derivatives are powers of 2 and the breakpoints are dyadic rationals, i.e. points of the form p/2 q , p, q ∈ N. Thompson's group F is the stabiliser of the point 0 in T .

It has been proved by Ghys and Sergiescu in [START_REF] Ghys | Sur un groupe remarquable de difféomorphismes du cercle[END_REF] that the piecewise linear action of T (and hence of F ) on S 1 is C ∞ -smoothable. On the other side, it is not difficult to find C ∞ faithful actions (a priori not topologically conjugate to the standard one) of Thompson's group.

We recall Thurston's interpretation of T as a group of piecewise projective homeomorphisms of RP 1 (cf. [START_REF] Cannon | Introductory notes on Richard Thompson's groups[END_REF]). Definition 4.2. T is the group of piecewise PSL(2, Z) homeomorphisms of RP 1 with breakpoints in P Z (which is the set of rational numbers together with the point at infinity). T is generated by PSL(2, Z) and an additional element c defined as

c(t) =              t if t ∈ [∞, 0], t 1-t if 0 ≤ t ≤ 1 2 , 3 -1 t if 1 2 ≤ t ≤ 1, t + 1 if t ∈ [1, ∞].
It is particularly striking that the element c has continuous first derivative. As the action of PSL(2, Z) is even real-analytic, Thurston's interpretation gives a natural C 1 -smoothing of T . 2 In this model, F is the group of piecewise PSL(2, Z) homeomorphisms of RP 1 with breakpoints in P Z , that also fix infinity. So F is the stabiliser of ∞ in T . F is generated by t → t + 1 together with c from above. Recall that the group PSL(2, Z) is isomorphic to the free product Z 2 * Z 3 , freely generated by the order two element a : t → -1 t and the order three element b : t → 1 1-t . Now we sketch a proof that F admits a C ∞ action inspired by [START_REF] Kim | Chain groups of homeomorphisms of the interval and the circle[END_REF] (see also [START_REF] Bleak | Groups of fast homeomorphisms of the interval and the ping-pong argument[END_REF]). Note that this is weaker than proving it is C ∞ -smoothable, which is a consequence of the theorem of Ghys-Sergiescu.

Given any homeomorphism h :

[0, 1] → [0, 2], if we define the element c(t) =        t if t ∈ [∞, 0], h(t) if t ∈ [0, 1], t + 1 if t ∈ [1, ∞]
then the group generated by t → t + 1 and c is isomorphic to T . If we choose h to be C ∞ , infinitely tangent to the identity at 0 and to t → t + 1 at 1, then the modified element c is C ∞ . The algebraic properties of F guarantee that the group generated by t → t + 1 and c is isomorphic to F . 3 However, it is not guaranteed that one can choose h and hence c such that the action of the group t → t + 1, c is actually conjugate to the standard action of F .

A very important remark is that this strategy is morally possible because 0 and 1 are not hyperbolic fixed points (they are parabolic). This allows to slow-down the dynamics near these points and make c infinitely tangent to the identity. This feature already appeared in the work of Ghys and Sergiescu. Hyperbolicity is a typical obstruction for such modifications in differentiable dynamics.

One open problem: The Day-von Neumann problem for

Diff 2 + (R)
One of the main motivations for our work is understanding amenable groups of diffeomorphisms of the circle. There are several equivalent definitions of amenability and an extensive literature on the topic (see [START_REF] Ceccherini-Silberstein | Cellular automata and groups[END_REF] for an elementary introduction). We provide one definition: Definition 4.3. A discrete group G is amenable if it admits a finitely additive, left translation invariant probability measure.

Here is an equivalent definition, à la Krylov-Bogolyubov, which is more natural from the viewpoint of dynamical systems: Definition 4.4. A discrete group G is amenable if every continuous action on a compact space has an invariant probability measure.

The class of amenable groups includes finite, abelian and solvable groups. Amenability is closed under extensions, products, direct unions and quotients. Subgroups of amenable groups are amenable. On the other hand, groups containing nonabelian free subgroups are nonamenable. The so-called Day-von Neumann problem (popularized by Day in the 50s) is about the converse statement: does every nonamenable group contain a nonabelian free subgroup? If one restricts the question to linear groups, then the well-known Tits alternative gives a positive answer: any non virtually solvable linear group contains nonabelian free subgroups.

The problem has been solved with negative answers and currently various negative solutions are known. These include Tarski monsters, Burnside groups, and Golod-Shafarevich groups. In this article we are interested in a particular class of such groups, discovered by Monod [START_REF] Monod | Groups of piecewise projective homeomorphisms[END_REF] and Lodha-Moore [START_REF] Lodha | A nonamenable finitely presented group of piecewise projective homeomorphisms[END_REF], which are subgroups of PP + (R). Among them, there are examples that are in Diff 1 + (R). For instance, the group generated by t → t + 1 2 together with the element c from Definition 4.2 above provides such an example.

Interestingly, no negative solution to the Day-von Neumann problem is known among groups of Diff 2 + (R). Motivated by this question, in this work we prove (Theorems 5.8 and 5.9) that the natural actions of these groups are not C 2 -smoothable. However, we have to stress that a priori there could be smooth actions of such nonamenable groups that are not topologically conjugate to the standard actions (cf. Remark 2.2).

The moral consequence of our results is that the Day-von Neumann problem in Diff 2 + (R) is strictly harder than in Diff 1 + (R). This is not so surprising, since there are important differences between C 2 and C 1 diffeomorphisms in one-dimensional dynamics. We end this section by recalling a couple of tantalising longstanding open questions in this direction. Question 4.5. Is F amenable? Question 4.6. Does the Tits alternative hold for the group of real-analytic diffeomorphisms of the real line?

A second open problem: Higher rank behaviour

Definition 4.7. Let 1 < n 1 < • • • < n k be natural numbers such that the group Λ = n i ⊂ R * + is an abelian group of rank k. Denote by A the ring Z[ 1 m ],
where m is the least common multiple of the n i 's.

Thompson-Stein's group T (n 1 , . . . , n k ) is the group of all piecewise linear homeomoprhisms of the circle S 1 ∼ = R/Z such that all derivatives are in Λ and the breakpoints are in A. Thompson-Stein's group F (n 1 , . . . , n k ) is the stabiliser of the point 0 in T (n 1 , . . . , n k ).

With the above definition, the group T (2) is the classical Thompson's group T . It has been proved by Stein [29] that these groups share many group-theoretical properties with the classical Thompson's groups, such as being finitely presentable (cf. [START_REF] Bieri | On groups of PL-homeomorphisms of the real line[END_REF]).

However there are important differences from the dynamical viewpoint. In [START_REF] Minakawa | Exotic circles of PL+(S 1 )[END_REF][START_REF]Classification of exotic circles of PL+(S 1 )[END_REF], Minakawa discovers that PL + (S 1 ) contains "exotic circles", namely topological conjugates of SO(2) that are not one-parameter groups inside PL + (S 1 ), in the sense that they are not PL conjugates of SO(2). In particular, Liousse shows in [START_REF] Liousse | Rotation numbers in Thompson-Stein groups and applications[END_REF] that T (n 1 , . . . , n k ) contains an abelian group of rank k -1 that is contained in a topological conjugate of SO(2), but not in a PL conjugate of SO(2). Whence Navas suggested the following: Question 4.8. Does T (n 1 , . . . , n k ), k ≥ 2, satisfy Kazhdan's property (T )? 4On the other hand Navas proved in [START_REF] Navas | Actions de groupes de Kazhdan sur le cercle[END_REF] that the only groups of C r diffeomorphisms, r > 3/2, that have property (T ) are finite. Focusing our attention on one particular example, in [START_REF] Liousse | Rotation numbers in Thompson-Stein groups and applications[END_REF], Liousse proves, among other things, that every action of T (2, 3) on S 1 by C 9 diffeomorphisms is trivial and with Corollary 3.4 we improve this result to C 2 regularity. It would be very interesting to prove that T (2, 3) has no C 1 action on the circle, as this would confirm that this group is a good candidate for finding an infinite Kazhdan group of circle homeomorphisms.

Naturally, there could be also good candidates among groups of piecewise projective homeomorphisms.

Nonamenable groups of piecewise projective homeomorphisms

Monod's groups

Generalizing a well-known result by Brin and Squier [START_REF] Brin | Groups of piecewise linear homeomorphisms of the real line[END_REF], Monod showed in [START_REF] Monod | Groups of piecewise projective homeomorphisms[END_REF] that PP + (R) does not contain nonabelian free subgroups. One key feature is that given any r ∈ R, the group of germs of elements in PP + (R) fixing the point r is isomorphic to the affine group.

Definition 5.1 (Monod's groups). Let A be a subring of R. G(A) is defined as the group of all piecewise PSL(2, A) homeomorphisms of the circle with breakpoints in H A . The group H(A) is the stabiliser of ∞ inside G(A).

Observe that the groups G(R) and H(R) coincide with PP + (RP 1 ) and PP + (R) respectively. Relying on the fact that for any A = Z, the group PSL(2, A) contains dense free subgroups, Monod proved in [START_REF] Monod | Groups of piecewise projective homeomorphisms[END_REF] that for any A = Z, the group H(A) is nonamenable. Therefore these groups give negative answer to the Day-von Neumann problem.

Remark 5.2. The previous definition can be generalized, considering any subgroup Γ ⊂ PSL(2, R). Elements in G(Γ) are piecewise Γ and the breakpoints are in H Γ . For any non-discrete Γ ⊂ PSL(2, R), the group H(Γ) does not contain free subgroups and is nonamenable. Theorem 3.1 can be extended to these groups as well.

We shall now demonstrate Theorem 3.1, namely that Monod's examples are not C 1 -smoothable. For part [START_REF] Baik | Unsmoothable group actions on compact one-manifolds[END_REF], it is enough to prove the following:

Theorem 5.3. Monod's group H(Z) is not C 1 -smoothable.
On the other hand, part (2) relies on Theorem 1.1.

Proof of Theorem 3.1. Let us first prove [START_REF] Baik | Unsmoothable group actions on compact one-manifolds[END_REF]. Any subring A ⊂ R contains Z, therefore PSL(2, Z) is a subgroup of any PSL(2, A). Therefore we have inclusions H(Z) ⊂ H(A) ⊂ G(A). As H(Z) is not C 1 -smoothable (Theorem 5.3), neither are H(A) and G(A).

Next, we demonstrate part [START_REF] Bekka | Kazhdan's property (T)[END_REF]. Let λ > 1 be a rational number such that f λ : x → λx belongs to PSL(2, A). As Z ⊂ A, the translation g : x → x+1 belongs to PSL(2, A) as well. Moreover, f λ being a hyperbolic element in PSL(2, A), we have that H A contains its fixed point 0. Therefore Monod's group H(A) contains the piecewise defined element

h λ : x → x if x ≤ 0, λx if x > 0. We have just shown that G λ = f λ , g, h λ is a subgroup of H(A). Let ρ : H(A) → Diff 1 + ([0, 1]) be a representation. Theorem 1.1 implies that [g, h λ gh -1 λ ]
is in the kernel of ρ. Therefore ρ cannot be injective, as desired.

The dynamical ingredient we need for Theorem 5.3 is the following Sacksteder-like result, originally due to Deroin, Kleptsyn, Navas [START_REF] Deroin | Sur la dynamique unidimensionnelle en régularité intermédiaire[END_REF] (see [START_REF]Groups of circle diffeomorphisms[END_REF]Prop. 3.2.10] and also [5, § 4.5] for a simplified proof). Proof of Theorem 5.3. Let us assume by way of contradiction that there is a homeomorphism φ : R → R such that G := φH(Z)φ -1 is a group of C 1 diffeomorphisms of R. First we observe that there are elements f, g ∈ H(Z) that have linked pairs of fixed points. For example, consider the hyperbolic element γ defined as the projective transformation

γ = 2 -1 -1 1 , whose fixed points a, b satisfy that a < -3 2 < 1 2 < b. Now define f (t) = t if t / ∈ [a, b], γ(t) if t ∈ [a, b], g(t) = f (t -1) + 1.
Note that the pairs a, b and a + 1, b + 1 are linked. 5Now the elements

f 1 = φf φ -1 , g 1 = φgφ -1
in G have fixed points φ(a), φ(b) and φ(c), φ(d) respectively. This forms a linked pair. By Proposition 5.4, there is an element g ∈ G with a fixed point such that the derivative of g at x is not equal to 1. Now let g 1 = φ -1 gφ be the corresponding element in H(Z). Note that g 1 fixes y = φ -1 (x).

We claim that y is a fixed point of a hyperbolic matrix in PSL(2, Z). If y is a breakpoint of g 1 , then this is true because the set of breakpoints of elements in H(Z) is exatly H Z . We consider the case when y is not a breakpoint of g 1 , so there exists an element γ 1 ∈ PSL(2, Z) whose restriction to a neighbourhood U of y coincides with the restriction g 1 | U .

Observe that since x is a hyperbolic fixed point for g, the corresponding point y must be a topological attractor or repellor for γ 1 ∈ PSL(2, Z) that acts locally like g 1 around y, and hence g 1 must be hyperbolic and y is hence a hyperbolic fixed point for PSL(2, Z). Now consider an element g 2 ∈ H(Z) which is the identity on (-∞, y) and agrees with g 1 on [y, ∞). Then g 3 = φg 2 φ -1 ∈ G has right derivative λ = 1 at x and a left derivative that equals 1 at x. This contradicts the assumption that g 3 is C 1 . Hence our original assumption that H(Z) is C 1 -smoothable must be false.

The Lodha-Moore example

Lodha and Moore constructed a finitely presented subgroup G 0 of Monod's group. This example provides the first torsion free finitely presentable example solving the Day-von Neumann problem. The group G 0 is generated by t → t + 1 together with the following two homeomorphisms of R:

c(t) =              t if t ≤ 0, t 1-t if 0 ≤ t ≤ 1 2 , 3 -1 t if 1 2 ≤ t ≤ 1, t + 1 if 1 ≤ t, d(t) = 2t 1+t if 0 ≤ t ≤ 1, t if t / ∈ [0, 1].
The following was proved in [START_REF] Lodha | A nonamenable finitely presented group of piecewise projective homeomorphisms[END_REF]:

Theorem 5.5. The group G 0 is nonamenable and does not contain nonabelian free subgroups. Moreover, it is finitely presentable with 3 generators and 9 relations.

In [START_REF] Lodha | A nonamenable finitely presented group of piecewise projective homeomorphisms[END_REF] a combinatorial model for G 0 is constructed by means of a faithful action of G 0 by homeomorphisms of the Cantor set {0, 1} N . This model was used to prove that G 0 is finitely presentable. Here {0, 1} N is the Cantor set of infinite binary sequences, viewed as the boundary of the infinite rooted binary tree. We denote by {0, 1} <N as the set of all finite binary sequences, which are addresses of nodes in the infinite rooted binary tree.

Consider the map Φ : {0, 1} N → R ∪ {∞} given by:

11 a 0 0 a 1 1 a 2 ... → a 0 + 1 a 1 + 1 a 2 +... , 00 a 0 1 a 1 0 a 2 ... → -a 0 + 1 a 1 + 1 a 2 +...
. This function is one-to-one except on sequences ξ which are eventually constant. On sequences which are eventually constant, the map is two-to-one: Φ(s01 ∞ ) = Φ(s10 ∞ ) and Φ(0

∞ ) = Φ(1 ∞ ) = ∞.
It was shown in [START_REF] Lodha | A nonamenable finitely presented group of piecewise projective homeomorphisms[END_REF] that upon conjugating a, b, c by Φ one obtains the following combinatorial model. We start with the following map x : {0, 1} N → {0, 1} N as:

x(00ξ) = 0ξ, x(01ξ) = 10ξ, x(1ξ) = 11ξ
and also, recursively, the pair of mutually inverse maps y, y -1 : {0, 1} N → {0, 1} N as:

y(00ξ) = 0y(ξ), y -1 (0ξ) = 00y -1 (ξ), y(01ξ) = 10y -1 (ξ), y -1 (10ξ) = 01y(ξ), y(1ξ) = 11y(ξ), y -1 (11ξ) = 1y -1 (ξ).
From these functions, we define the functions x s , y s : {0, 1} N → {0, 1} N for s ∈ {0, 1} <N which act as x and y localised to binary sequences which extend s:

x s (ξ) = sx(η) if ξ = sη, ξ otherwise, y s (ξ) = sy(η) if ξ = sη, ξ otherwise.
If s is the empty-string, it will be omitted as a subscript. The group G 0 is generated by functions in the set

S = x t , y s | s, t ∈ {0, 1} <N , s = 0 k , s = 1 k , s = ∅
In fact, G 0 is generated by x, x 1 , y 10 which correspond respectively to conjugates of the functions a, b, c defined above by Φ. (See [START_REF] Lodha | A nonamenable finitely presented group of piecewise projective homeomorphisms[END_REF] for details.) It is important to note that G 0 acts on the boundary of the infinite rooted binary tree, but not on the tree itself.

Recall from the introduction that we are denoting by G 2 the group generated by f 2 , g, h 2 , where f 2 is the scalar multiplication by 2, g is the translation by 1, and h 2 is the element which agrees with f 2 to the right of zero and is the identity elsewhere. We obtain the following obstruction to smoothability of G 0 .

Lemma 5.6. The three elements y -1 100 y 101 , y 101 and x 10 generate an isomorphic copy of G 2 in the Lodha-Moore group G 0 .

Proof. It was demonstrated in [START_REF] Lodha | A nonamenable finitely presented group of piecewise projective homeomorphisms[END_REF] that the elements x and y -1 0 y 1 are conjugate respectively to t → t + 1 and t → 2t by Φ. Hence they generate an isomorphic copy of BS(1, 2). In particular, y -1 0 y 1 , y 1 , x 10 generate an isomorphic copy of G 2 .

It is easy to see that the groups y -1 100 y 101 , y 101 , x 10 and y -1 0 y 1 , y 1 , x are isomorphic, since their respective actions on boundaries of the binary trees, T 1 rooted at the empty sequence and T 2 rooted at the sequence 10, are the same.

More explicitly, one can verify that the elements y -1 100 y 101 , y 101 , x 10 correspond via Φ to the following piecewise projective transformations: Proof of Theorem 3.2. As a consequence of Lemma 5.6, the group G 0 contains a subgroup H isomorphic to G 2 . Let ρ : G 0 → Diff 1 + ([0, 1]) be a morphism. By a direct application of Theorem 1.1, we obtain that the kernel of ρ contains some nontrivial element of H. Thus ρ is not injective. Now, it has been proven in [START_REF] Burillo | Commutators in groups of piecewise projective homeomorphisms[END_REF] that every proper quotient of G 0 is abelian, whence we get our result: as we have just shown that the kernel is not trivial, then the image must be abelian, as we wanted to prove.

x 10 ∼                                              1 0 -1 1 on 0, 1 3 , 4 -1 5 -1 on 1 3 , 1 2 , 0 1 -1 2 on 1 2 , 1 , id on R \ [0, 1], y 101 ∼              3 -1 2 0 on 1 2 , 1 , id on R \ 1 2 , 1 , y -1 100 y 101 ∼                              1 0 -2 2 on 0, 1 2 , 3 -1 2 0 on 1 2 , 1 , id on R \ [0, 1].

Further examples

An interesting family of nonamenable groups is obtained adding translations in top of F (defined as in Definition 4.2). Mimicking Monod's argument, it is not difficult to prove the following: Proposition 5.7. For any α ∈ (0, 1), the group of piecewise projective homeomoprhisms generated by F and the translation t → t + α is nonamenable.

Observe that the groups F, t → t + α appearing in the above statement are naturally of C 1 diffeomorphisms. Theorem 5.8. For any irrational α ∈ (0, 1), the action of the group of piecewise projective homeomorphisms F, t → t + α on the compactified real line [-∞, +∞] is not C 2 -smoothable.

Proof. We denote by T α the translation by α. If α is irrational, then T 1 and T α generate an abelian free group of rank 2 of C 2 (even real analytic) diffeomorphisms of R. The maps f = T -1 and g = T -|α| are contractions on R. Consider any element h ∈ F, T α with a C 2 discontinuity point on R. Then Theorem 7.3 implies directly that the action of F, T α on [-∞, +∞] is not C 2 -smoothable.

For rational translations T α , we can extend the previous argument and prove that even the action on the non-compactified real line (-∞, +∞) is not C 2 -smoothable. Theorem 5.9. For any rational α ∈ (0, 1), the action of the group of piecewise projective homeomorphisms F, t → t + α on R is not C 2 -smoothable.

Proof. We consider the conjugate of c by T α :

T α cT -1 α (t) =              t if t ≤ α, t-α 1-(t-α) + α if α ≤ t ≤ 1 2 + α, 3 -1 t-α + α if 1 2 + α ≤ t ≤ 1 + α, t + 1 if 1 + α ≤ t.
In restriction to the interval α, 1 2 + α , the element T α cT -1 α coincides with the projective transformation

1 -α α 2 -1 1 + α , which is a parabolic element in PSL(2, Z[α]) fixing α. It is not in PSL(2, Z).
Inside Thompson's F we can find an element f such that:

• f fixes α,
• the restriction of f to the interval α,

1 2 + α is C 2 ,
• f is a contraction of the interval α, 1 2 + α , namely f (t) < t for any t in the right neighbourhood α, 1 2 + α of α. Indeed, since α is rational, there exists a parabolic element in PSL(2, Z) with α as fixed point, and we can take for f any element of F which coincides with this element (or its inverse) in restriction to α, 1 2 + α . Finally, consider an element h ∈ F which has a C 2 discontinuity point p on α, 1 2 + α , with h(p) ∈ α, 1 2 + α . It is straightforward to verify that f , g = T α cT -1 α -1 and h satisfy the requirements of Theorem 7.3 (when considering the interval α, 1 2 + α as the interval [0, a] in the statement). Thus the theorem is proved.

C 1 actions of affine and piecewise affine groups 6.1 C 1 actions of the Baumslag-Solitar groups

Let n > 1 be an integer. The classical Baumslag-Solitar groups BS(1, n) are defined by the presentations

BS(1, n) = a, b | aba -1 = b n .
They are naturally realized as subgroups of the affine group Aff + (R) ⊂ PSL(2, R), generated by the homothety a(x) = nx, and the translation b(x) = x + 1.

Similarly, for any rational λ = p/q there is a morphism from the Bausmlag-Solitar group

BS(q, p) = a, b | ab q a -1 = b p
to the subgroup of Aff + (R) generated by a(x) = λx and b(x) = x + 1. However, when p/q is not an integer, this morphism is not an isomorphism. We refer to this morphism as to the standard affine action of BS(q, p).

For general λ > 1, we define A λ to be the subgroup of Aff + (R) generated by a(x) = λx and b(x) = x + 1. For irrational λ's, one easily sees that the group A λ is abstractly isomorphic to the wreath product Z Z.

In [START_REF] Bonatti | Rigidity for C 1 actions on the interval arising from hyperbolicity I: solvable groups[END_REF], the authors study the C 1 actions on the interval of the groups A λ . The following result appears in [6, § 4.3]: Proposition 6.1. Fix λ > 1 and let φ : R → R be a homeomorphism such that φA λ φ -1 is in Diff 1 + ([0, 1]). Then φaφ -1 has derivative equal to λ at its interior fixed point φ(0).

For rational values of λ's, the authors prove a much stronger statement (see [6, Thm. 1.7]): Theorem 6.2. Let λ = p/q > 1 be rational and consider a C 1 action of the Baumslag-Solitar group BS(q, p) on the closed interval, without global fixed points in its interior. Then either the action is topologically conjugate to the standard affine action of BS(q, p), or the action is abelian.

The groups G λ

Inspired by the definition of Monod's groups, we consider an analogous construction starting from these affine groups. Here we repeat the definition already given in the introduction: Definition 6.3. For any λ > 1, we define G λ to be the subgroup of PP + (R) generated by the elements

a(x) = λx, a + (x) = x if x ≤ 0 λx if x > 0 , b(x) = x + 1.
We also set a -= aa -1 + , which agrees with a to the left of 0 and is the identity elsewhere.

Remark 6.4. In the introduction, we were denoting a, b, a + by f λ , g, h λ respectively.

Remark 6.5. The algebraic structure of G λ is highly complicated. For instance, in the case λ = 2, inside the group G 2 , the elements b and [a + , b] are the generators of Thompson's F , in its natural piecewise linear action on R.

Example 6.6. There are two canonical standard affine actions of the group G λ on the real line that factor through the affine group A λ . First, as every element in G λ fixes ±∞, we can consider the germs of elements of G λ at this two points. This gives two surjective homomorphisms

ρ ± : G λ → A λ .
It is clear from the definition of G λ that we have

ρ ± (a ∓ ) = id, ρ ± (a ± ) = ρ(a)
for these two morphisms. More generally, every element of G λ that is the identity outside a compact interval belongs to the kernels of both morphisms ρ ± . This is the case for the commutator [b, a + ba -1 + ] that appears in the statement of Theorem 1.1.

On the other hand, as Z 2 is the abelianization of G λ (generated by the images of a ± ), there are plenty of abelian actions of G λ on the real line. In this case the translation b is always in the kernel, so the commutator [b, a + ba -1 + ] always acts trivially.

C 1 actions of G λ

The following result is essentially the one contained in Theorem 1.2: Theorem 6.7. For any λ > 1, the natural action of G λ on the compactified real line

[-∞, +∞] is not C 1 -smoothable.
Proof. We argue by contradiction. After Proposition 6.1, if there existed a homeomorphism φ : R → R such that φA λ φ -1 was in Diff 1 + ([-∞, +∞]), then φaφ -1 would have derivative equal to λ at p = φ(0) and φa + φ -1 would not be C 1 at p. Hence the group is not C 1 -smoothable. Remark 6.8. In the previous statement, it is fundamental to consider the action of G λ on the compactified line. Indeed, the statement is no longer true if one simply considers the action on R (see [START_REF] Bonatti | Rigidity for C 1 actions on the interval arising from hyperbolicity I: solvable groups[END_REF]Remark 4.14]).

Our second result, more precise than the statement in Theorem 1.1, says that every C 1 action of G λ on the interval, for rational λ > 1, is always described by combining the examples above. Theorem 6.9. Let λ = p/q > 1 be rational. Let ρ :

G λ → Diff 1 + ([0, 1]
) be a nontrivial homorphism. Then there exists finitely many pairwise disjoint subinterval I 1 , . . . , I n ⊂ [0, 1] such that [START_REF] Baik | Unsmoothable group actions on compact one-manifolds[END_REF] for any i = 1, . . . , n, the image ρ(G λ ) preserves the interval I i , [START_REF] Bekka | Kazhdan's property (T)[END_REF] for any i = 1, . . . , n, the restriction of ρ(G λ ) to I is topologically conjugate to one of the two canonical actions on R given by ρ ± : G λ → A λ ,

(3) the restriction of ρ(G λ ) to the complement [0, 1] \ n i=1 I i is abelian. In particular the group G λ admits no faithful C 1 action on the closed interval. Remark 6.10. Relying on the work by Guelman and Liousse [START_REF] Guelman | C 1 -actions of Baumslag-Solitar groups on S 1[END_REF] (cf. also [START_REF] Bonatti | Rigidity for C 1 actions on the interval arising from hyperbolicity I: solvable groups[END_REF]), we could provide a similar statement for C 1 actions of G λ on the circle S 1 . Indeed, every nonabelian action of A λ has a global fixed point, so every nonabelian action of G λ reduces to an action on the interval. Remark 6.11. The proof of Theorem 6.9 would be much simpler for representations ρ :

G λ → Diff 1+α
+ ([0, 1]) of the group G λ into the group of C 1 diffeomorphisms with α-Hölder continuous derivative. Indeed, it is classical that any C 1 element commuting with a C 1+α hyperbolic contraction of an interval lies in a one parameter flow (cf. Theorem 7.1: when the fixed point of the contraction is hyperbolic, Szekeres theorem requires only C 1+α regularity).

Let us sketch the proof under the assumption of C 1+α regularity. Assume that the image ρ(G λ ) is nonabelian. Then the image ρ(A λ ) is also nonabelian (cf. Lemma 6.21). From Theorem 6.2 and Proposition 6.1 we deduce that the element ρ(a) behaves as the corresponding scalar multiplication in restriction to some interval I ⊂ [0, 1] and has a hyperbolic fixed point s ∈ I. As the elements ρ(a ± ) commute with ρ(a), we deduce from Szekeres theorem that in restriction to the interval I, also these elements behave like scalar multiplications (as the one parameter flow containing a scalar multiplication is exactly the one parameter flow of all scalar multiplications). This implies that the group ρ(G λ ) acts like an affine group in restriction to the interval I.

The proof of Theorem 6.9 will occupy the rest of the section.

Elementary ingredients

When working with C 1 actions on the interval, hyperbolic fixed points do not often give rigidity (one usually needs C 1+α regularity, cf. Remark 6.11). Indeed there are only a few dynamical tools that work in C 1 regularity. For this reason our proof relies mainly on very elementary arguments. A first tool is the following: Lemma 6.12. Let α ∈ Diff 1 + ([0, 1]) be a diffeomorphism. For any δ > 1, there are only finitely many points s ∈ [0, 1] that are fixed by α and such that α (s) > δ.

Proof. Suppose α has infinitely many fixed points {s n | n ∈ N} in [0, 1], such that α (s n ) > δ for any n ∈ N. Let s * ∈ [0, 1] be an accumulation point of the sequence {s n }. By continuity of the derivative, we must have α (s * ) ≥ δ. On the other hand, let {s n k } be a subsequence converging to s * ; by the very definition of the derivative we must have α (s * ) = 1. This is a contradiction.

Then we state and prove a second crucial elementary fact. However, since α and β commute, indeed λ n = α and hence λ n (s) = α (s). It follows that α (s) = α (β n (s)) for each n ∈ N.

Since the set S is infinite, the claim is in contradiction with Lemma 6.12.

A particular case: no global fixed points for A λ

Before dealing with a general statement as in Theorem 6.9, we study actions on the interval without global fixed points. For the statement, recall that we denote by A λ ⊂ G λ the subgroup generated by a and b. Proposition 6.14. Let λ = p/q > 1 be rational. ρ : G λ → Diff 1 + ([0, 1]) be a morphism satisfying the following:

(1) the image ρ(A λ ) is nonabelian, [START_REF] Bekka | Kazhdan's property (T)[END_REF] the action of ρ(A λ ) has no global fixed point in (0, 1).

Then ρ(G λ ) is topological conjugate to one of the two canonical representations ρ

± : G λ → A λ .
The end of the proof is inspired by [START_REF] Bonatti | Centralizers of C 1 -contractions of the half line[END_REF]: in a centraliser of a hyperbolic element, like h, there cannot be elements with hyperbolic fixed points, and therefore by Proposition 5.4, there cannot be linked pairs of successive fixed points. Lemma 6.19. Suppose that k is not the identity and let s, t ∈ [s 0 , 1] be a pair of successive fixed points of k. There exists a dyadic rational r > 0 such that the pair g r (s), g r (t) together with s, t defines a linked pair of fixed points for g r kg -1 r and k.

Proof. For any dyadic rational r, the points g r (s), g r (t) define a pair of successive fixed points for the conjugate g r kg -1 r . An element g r , r > 0, moves every point in (s 0 , 1) to the right, and using the fact that g r is topologically conjugate to the translation by r, we can choose r > 0 sufficiently small such that s < g r (s) < t ≤ g r (t)

(with equality t = g r (t) if and only if t = 1).

Suppose that k is not the identity. Then, from Lemma 6.19 and Proposition 5.4, we realize that the subgroup g r kg -1 r , k contains an element γ with a hyperbolic fixed point p in (s 0 , 1). Since g r kg -1 r , k commute with h, it follows that γ commutes with h. So by Lemma 6.13 h must fix the point p. This contradicts the conclusion of Lemma 6.16.

Therefore we must have that k = ρ(a + ) is the identity, and so ρ(a -) = ρ(a). Thus the representation ρ :

G 2 → Diff 1 + ([0, 1]
) is topologically conjugate to the canonical representation ρ -: G 2 → A 2 . This finishes the proof of Proposition 6.14.

Equivalent properties

Now we consider almost the same statement as in Proposition 6.14, but we only make assumptions on the global dynamics of G λ , rather than on the one of A λ . Proposition 6.20. Let λ = p/q > 1 be rational and ρ : G λ → Diff 1 + ([0, 1]) be a morphism satisfying the following:

(1) the image ρ(G λ ) is nonabelian, (2) the action of ρ(G λ ) has no global fixed point in (0, 1).

Then ρ(G λ ) is topological conjugate to one of the two canonical representations ρ

± : G λ → A λ .
The proof follows directly from the following two lemmas. Proof. Clearly (2) implies (1). On the other hand, if the image ρ(A λ ) is abelian, the translation b belongs to the kernel of ρ and hence ρ(G λ ) itself is abelian. Lemma 6.22. Let λ = p/q > 1 be rational and ρ : G λ → Diff 1 + ([0, 1]) be a morphism with nonabelian image. Then the following properties are equivalent:

(1) the action of ρ(G λ ) has no global fixed point in (0, 1),

(2) the action of ρ(A λ ) has no global fixed point in (0, 1).

Proof. Again, (2) easily implies [START_REF] Baik | Unsmoothable group actions on compact one-manifolds[END_REF]. Assume [START_REF] Baik | Unsmoothable group actions on compact one-manifolds[END_REF]. Since the image ρ(G λ ) is nonabelian, by Lemma 6.21 also the image ρ(A λ ) is nonabelian. Using Theorem 6.2, we find at least one interval I = [x, y] ⊂ [0, 1] such that (1) I is preserved by ρ(A λ ),

(2) ρ(A λ ) has no global fixed point in the interior of I,

(3) the restriction ρ(A λ )| I is nonabelian and therefore it is topologically conjugate to the standard affine action on R.

Moreover, by Proposition 6.1, there exists a unique point s 0 ∈ (x, y) in the interior of I which is a hyperbolic fixed point for ρ(a), with derivative ρ(a) (s 0 ) = λ. Proceeding as in Lemma 6.15, the elements ρ(a ± ) must fix the point s 0 and we can suppose that s 0 is a hyperbolic fixed point for ρ(a -), with derivative ρ(a -) (s 0 ) > 1. Let s -be the first fixed point of ρ(a -) which lies to the left of s 0 .

If s -∈ (x, s 0 ), then {ρ(a) -n (s 0 ) | n ∈ N} is a sequence of fixed points for ρ(a -) that converges to s 0 as n → ∞. But this is not possible because the derivative of ρ(a -) at s 0 is not 1 (cf. Lemma 6.13).

Similarly, if s -∈ [0, x), then {ρ(a -) -n (x) | n ∈ N} is a sequence of fixed points for ρ(a) that converges to s 0 as n → ∞. Again, this is not possible.

Thus s -= x and so x is a global fixed point for ρ(G λ ). As we are assuming (1), this implies x = 0. Similarly, denoting by s + the first fixed point of ρ(a -) which lies to the right of s 0 , we obtain that s + = y and so y = 1. This is what we wanted to prove.

Proof of Proposition 6.20. The statement follows directly from Lemmas 6.21 and 6.22.

General case

We proceed now to the proof of Theorem 6.9.

Proof of Theorem 6.9. Let ρ : G λ → Diff 1 + ([0, 1]) be a homomorphism. If the image ρ(G λ ) is abelian, there is nothing to prove. Hence we can assume that ρ(G λ ) is nonabelian and by Lemma 6.21 this is equivalent to saying that ρ(A λ ) is nonabelian, where A λ is denoting the subgroup generated by a and b.

If ρ(A λ ) is nonabelian, then after Theorem 6.2, there exists at least an interval I ⊂ [0, 1] which is preserved by ρ(A λ ) and such that ρ(A λ ) is topologically conjugate to the standard affine action.

Claim. There are only finitely many pairwise disjoint intervals I 1 , . . . , I n that are preserved by ρ(A λ ) and such that for any i = 1, . . . , n the restrictions ρ(A λ )| I i is nonabelian. Proof of Claim. Let I be an interval preserved by ρ(A λ ) and such that ρ(A λ )| I is nonabelian. By Theorem 6.2, the action is topologically conjugate to the standard action of A λ and by Proposition 6.1 there exists a point s ∈ I which is fixed by ρ(a) and such that ρ(a) (s n ) = λ > 1. Then Lemma 6.12 implies that there can only be finitely many such intervals, whence the first statement.

Claim. The restriction of ϕ to (0, a] is C 1 and takes the Szekeres vector field X of f , defined on [0, a], to Y:

ϕ * X = Y.

Proof of Claim. For any x ∈ [0, a] and α ∈ A, α > 0, we have φ α Y (ϕ(x)) = ϕ (φ α X (x)) . Now, φ α X (x) = x for any α > 0, x ∈ (0, a]. Thus for any x ∈ (0, a] and α ∈ A, α > 0, we have

φ α Y (ϕ(x)) -ϕ(x) α = ϕ (φ α X (x)) -ϕ(x) α = ϕ (φ α X (x)) -ϕ(x) φ α X (x) -x • φ α X (x) -x α . Hence ϕ (φ α X (x)) -ϕ(x) φ α X (x) -x = φ α Y (ϕ(x)) -ϕ(x) α • φ α X (x) -x α -1
.

Taking the limit on both sides as α ∈ A, α > 0, goes to 0 (recall that A is dense in R), we get on the left hand side the derivative ϕ (x) and on the right hand side the ratio Y (ϕ(x)) /X (x) (here we identify X , Y to C 1 functions). Observe that the ratio Y (ϕ(x)) /X (x) is well-defined because X has no singularities on (0, a). This gives the two desired statements. Indeed, as Y (ϕ(x)) /X (x) is C 0 on (0, a], so is ϕ . Hence ϕ is C 1 on (0, a]. Moreover, taking X (x) to the left hand side, we get ϕ (x) • X (x) = Y (ϕ(x)) for any x ∈ [0, a], that is ϕ * X = Y, as wanted. Now we can conclude the proof. From the previous Claim, we write ϕ (x) = Y (ϕ(x)) X (x) for every x ∈ (0, a].

Moreover the previous Claim gives that the right hand side in the last expression is at least C 1 on (0, a] and therefore the same holds for ϕ . This implies that ϕ is C 2 on (0, a], as desired.

Thompson-Stein groups

We finally apply the previous result to prove that the Thompson-Stein groups are not C 2smoothable.

Proof of Theorem 3.3. In the group F (n 1 , . . . , n k ), k ≥ 2, it is possible to find two elements f, g fixing 0 such that f (0) = 1/n 1 , g (0) = 1/n 2 . Let a ∈ (0, 1) be such that f, g are linear contractions in restriction to [0, a]. Consider any element h which is the identity in restriction to [0, a/2] but not in restriction to [0, a]. Then there exists t ∈ [a/2, a) which is a C 1 discontinuity point of h with h(t) ∈ [a/2, a) (actually we may take for t the leftmost point in the support of h). Thus we apply Theorem 7.3 and conclude that the natural action on [0, 1] of the group generated by f, g and h is not C 2 -smoothable. In particular the natural action of F (n 1 , . . . , n k ) on [0, 1] is not C 2 -smoothable.

Proof of Corollary 3.4. From [19,Theorem 3.A], every faithful C 2 action of T (2, n 2 , . . . , n k ) on S 1 is topologically conjugate to its standard piecewise linear action. However T (2, n 2 , . . . , n k ) contains F (2, n 2 , . . . , n k ) as a subgroup, whose standard action on S 1 cannot be conjugate to a C 2 action, after our Theorem 3.3. Therefore a C 2 action of T (2, n 2 , . . . , n k ) cannot be faithful.

As in [19, Theorem 3.B'], if we assume furthermore n 2 = 3, the simplicity of T (2, 3, n 3 , . . . , n k ) allows to conclude that every C 2 action of such a group is trivial.

Proposition 5 . 4 .

 54 Let G = f, g be a group acting by C 1 diffeomorphisms on a compact onedimensional manifold. If {a, b} and {c, d} are linked pairs of successive fixed points for f, g, then G contains an element with a hyperbolic fixed point in (a, b) ∩ (c, d).

Figure 1 :

 1 Figure 1: The generators y -1 100 y 101 (yellow), y -1 101 (blue), x 10 (green) in a neighbourhood of [0, 1].

Lemma 6 . 13 .

 613 Let α, β ∈ Diff 1 + ([0, 1]) be two commuting C 1 diffeomorphisms. Let s ∈ [0, 1] be a hyperbolic fixed point of α. Then β fixes s.Proof. Let us assume by way of contradiction that β does not fix s. For each n ∈ Z we haveα(β n (s)) = β n (α(s)) = β n (s)This means that α fixes each point in the set S = {β n (s) | n ∈ Z}.Claim. Each t ∈ S is a hyperbolic fixed point of α and α (t) = α (s) for all t ∈ S.Proof of Claim. Let λ n be the formal word β n αβ -n . Using the chain rule, we find λ n (s) = α (β n (s)).

Lemma 6 . 21 .

 621 Let λ > 1 and ρ : G λ → Diff 1 + ([0, 1]) be a morphism. Then the following properties are equivalent: (1) the image ρ(G λ ) is nonabelian, (2) the image ρ(A λ ) is nonabelian.

Theorem 3.3. The

  Thompson-Stein groups F (n 1 , . . . , n k ), k ≥ 2, are not C 2 -smoothable. Assume n 1 = 2. The Thompson-Stein groups T (2, n 2 , . . . , n k ), k ≥ 2, have no faithful C 2 action on S 1 .

	1 + (R), e.g. the group generated by Thompson's group F (which is C 1 in PP + (R)) together with t → t + 1 2 , for which we establish that their actions are not C 2 -smoothable. It also applies to the Thompson-Stein groups F (n 1 , . . . , n k )
	and T (n 1 , . . . , n k ) (see Definition 4.7), extending a previous work by Liousse [19]:
	Corollary 3.4.
	(1)

(2) Assume n 1 = 2, n 2 = 3. Every C 2 action of T (2, 3, n 3 , . . . , n k ) on S 1 is trivial. This holds in particular for T (2, 3).

Another way of seeing this is that C 1 continuity follows from the choice of P Z for the set of breakpoints.

To see this, first check that the relations of F are satisfied and conclude using the property F satisfies that every proper quotient is abelian.

We do not define property (T ) here, we refer the reader to[START_REF] Bekka | Kazhdan's property (T)[END_REF].

We remark that in general a linked pair may not look like it does in this situation, for instance such maps may have components of support lying outside (a, b) and (a + 1, b + 1) respectively.

Acknowledgements

The authors thank Isabelle Liousse for her valuable comments on the first version of the paper (Corollary 3.4 is due to her). This work has been done during visits of the authors to the Institut de Mathématiques de Bourgogne and the École Polytechnique Fédérale de Lausanne. The authors thank these institutions for the welcoming atmosphere. We are also grateful to the members of the IMB participating to the working seminar "Dynamique des actions de groupes" during Autumn 2017, as well as to the organizers of the conference Dynamics Beyond Uniform Hyperbolicity held in BYU, Provo, June 2017. M.T. was partially supported by PEPS -Jeunes Chercheur-e-s -2017 (CNRS).

In the following, we restrict to the case λ = 2, the general case being similar. We let s 0 denote the hyperbolic fixed point of ρ(a) in (0, 1), ensured by Theorem 6.2 and Proposition 6.1 above. For simplicity of notation, we also write ρ(a) = f, ρ(b) = g, ρ(a -) = h, ρ(a + ) = k. Lemma 6.15. With the notation as above, the elements h and k fix the point s 0 and we have

Proof. By Lemma 6.13, the two elements h, k fix the point s 0 , as they commute with f . Remarking that hk = f , applying the chain rule we deduce that

The previous lemma implies that s 0 is a hyperbolic fixed point for at least one among h and k. Without loss of generality, we assume h (s 0 ) > 1. The following lemma says that h behaves like a hyperbolic element on the whole interval [0, 1]: Lemma 6.16. With the notation as above, suppose h (s 0 ) > 1. Then s 0 is the only point of (0, 1) which is fixed by h.

Proof. If h had a fixed point s different from s 0 , since h and f commute, the images f -n (s) would form a sequence of fixed points for h that converge to s 0 . This would imply that the derivative of h at s 0 should be equal to 1. Contradiction.

Given a dyadic rational r = p/2 q , we denote by g r the element ρ(a -q b p a q ), namely the image of the element which is the translation by p/2 q when thinking of A 2 as the dyadic affine group. We have g -r = g -1 r . By Theorem 6.2 these elements are actually topologically conjugate to the corresponding translations. Lemma 6.17. Let r = p/2 q be a dyadic rational, r > 0. The conjugate g r kg -1 r commutes with h.

Proof. This is actually a statement about relations of the group G 2 : we prove the relation looking at the standard action of G 2 on the real line. The support of a + is [0, +∞), therefore the support of the conjugate of a + by the translation by r is [r, +∞), which is disjoint from (-∞, 0], which is the support of a -. Lemma 6.18. With the notation as above, the restriction of k to [0, s 0 ] is the identity.

Proof. The element k r = g r kg -1 r commutes with h by Lemma 6.17. We claim that k r fixes s 0 for each r > 0. Since

this means that h fixes k r (s 0 ). Now k r (s 0 ) ∈ (0, 1), and since the only fixed point of h in this set is s 0 , we conclude that k r (s 0 ) = s 0 . In particular, it follows that k fixes g -r (s 0 ) for every dyadic rational r > 0. By density of positive dyadic rationals in (0, +∞), we obtain the statement.

Claim. Let I 1 , . . . , I n be the intervals provided by the previous claim. Then ρ(G λ ) preserves I i for any i = 1, . . . , n.

Proof of Claim.

Let I be an interval as above. Let J ⊂ I be a interval which is preserved by ρ(G λ ) and such that ρ(G λ ) has no global fixed point in its interior. By Proposition 6.22, we must have the equality I = J.

After Proposition 6.20, we deduce that the restriction of the action of G λ to any of the intervals I 1 , . . . , I n is topologically conjugate to one of the two canonical affine actions ρ ± : G λ → A λ . This is what we wanted to prove.

C 2 actions with locally non-discrete stabilisers 7.1 Szekeres vector field

The method that we present in this section is inspired by [START_REF] Bonatti | Rigidity for C 1 actions on the interval arising from hyperbolicity I: solvable groups[END_REF]Prop. 4.17] and relies on the following important result in one-dimensional dynamics, due to Szekeres [START_REF] Szekeres | Regular iteration of real and complex functions[END_REF]. Here we state it as in [27, § 4.1.3]: Theorem 7.1 (Szekeres). Let f be a C 2 diffeomorphism of the half-open interval [0, 1) with no fixed point in (0, 1). Then there exists a unique C 1 vector field X on [0, 1) with no singularities on (0, 1) such that 1) f is the time-one map of the flow {φ s X } generated by X ,

2) the flow {φ s X } coincides with the C 1 centraliser of f in Diff 1 + ([0, 1)).

An obstruction to C 2 smoothability

The criterion we provide holds in a framework that is far more general that the one of piecewiseprojective dynamics. First we need a statement of differentiable rigidity for the conjugacy of some particular actions. Proposition 7.2. Take a ∈ (0, 1) and assume that two homeomorphisms f, g ∈ Homeo + ([0, 1]) satisfy the following properties:

1) the restrictions of f and g to [0, a] are C 2 contractions, namely the restrictions are C 2 diffeomorphisms such that f (x) < x, g(x) < x for every x ∈ (0, a],

2) f and g commute in restriction to [0, a], that is

3) the C 2 germs of f and g at 0 generate an abelian free group of rank 2.

Then for every homeomorphism ϕ ∈ Homeo + ([0, 1]) such that ϕf ϕ -1 , ϕgϕ -1 are C 2 in restriction to [0, ϕ(a)], one has that the restriction of ϕ to (0, a] is C 2 .

Before giving the proof of the proposition, which encloses the main arguments, we present the main result of the section: Theorem 7.3. Assume a ∈ (0, 1), f, g ∈ Homeo + ([0, 1]) satisfy the properties of Proposition 7.2 above. Moreover, assume that there exists h ∈ Homeo + ([0, 1]) such that there exists t ∈ (0, a) which is a C 2 discontinuity point of h and h(t) ∈ (0, a).

Then the natural action of f, g, h

Proof of Theorem 7.3. We argue by contradiction. If there was a homeomorphism ϕ ∈ Homeo + ([0, 1]) such that ϕ f, g, h ϕ -1 is in Diff 2 + ([0, 1]), then ϕ would satisfy the requirements of Proposition 7.2, whence ϕ would be C 2 in restriction to (0, a].

However h has a C 2 discontinuity point t ∈ (0, a) and hence the conjugate ϕhϕ -1 would have ϕ(t) as C 2 discontinuity point, against the assumption that ϕhϕ -1 is C 2 .

Proof of Proposition 7.2. As f is a C 2 contraction when restricted to [0, a], Szekeres' Theorem 7.1 applies: we denote by X the Szekeres vector field of f , which is C 1 , defined on [0, a) and with no singularities on (0, a). We have the assumption 2) that f and g commute in restriction to [0, a], so by Szekeres' theorem g belongs to the Szekeres flow {φ s X }. Let λ > 0 be such that g = φ λ X . Then by assumption 3), the subgroup A := 1, λ ⊂ R is a dense abelian group of rank 2.

As f and g are contractions, we have that for any positive power n ∈ N, the restrictions of the iterates f n and g n to the interval [0, a] coincide with the times φ n X and φ nα X respectively (however such a statement is in general not true for negative powers of f and g). More generally, we have the following:

Claim. Denote by A the rank 2 abelian subgroup of R generated by 1 and λ. For every α ∈ A, α > 0 there exists an element h α ∈ f, g such that

Proof of Claim. Let , m ∈ Z be such that α = + mλ. There exists y > 0 such that the element f g m is equal to φ α X on the right neighbourhood [0, y]. If y ≥ a, then we set h α = f g m and we are done.

Otherwise, we have y < a. As f is a contraction on [0, a], there exists a positive integer

]. We conclude that for any x ∈ [0, a] we have

The claim implies that the group generated by f and g contains a one-parameter flow in its local C 0 -closure. Suppose that ϕ is a homeomorphism such that ϕ f, g, h ϕ -1 is in Diff 2 + ([0, 1]). The element ϕf ϕ -1 is a C 2 contraction on a right neighbourhood [0, ϕ(a)] of 0, thus Szekeres' theorem applies again. Let Y denote the Szekeres vector field of ϕf ϕ -1 and let {φ s Y } be the associated one-parameter flow defined on [0, ϕ(a)]. The elements ϕh α ϕ -1 's commute with ϕf ϕ -1 on [0, ϕ(x)], hence by Szekeres' theorem we must have that their restrictions to [0, ϕ(a)] are contained in the flow {φ s Y } s≥0 . Moreover they are densely contained because A is dense in R.


Tom Hohweiller
email: tom.hohweiller@creatis.insa-lyon.fr

Nicolas Ducros

Françoise Peyrin

Bruno Sixou

A Kullback-Leibler approach for 3D reconstruction of spectral CT data corrupted by Poisson noise

While standard computed tomography (CT) data do not depend on energy, spectral computed tomography (SPCT) acquire energy-resolved data, which allows material decomposition of the object of interest. Decompositions in the projection domain allow creating projection mass density (PMD) per materials. From decomposed projections, a tomographic reconstruction creates 3D material density volume. The decomposition is made possible by minimizing a cost function. The variational approach is preferred since this is an ill-posed non-linear inverse problem. Moreover, noise plays a critical role when decomposing data. That is why in this paper, a new data fidelity term is used to take into account of the photonic noise. In this work two data fidelity terms were investigated: a weighted least squares (WLS) term, adapted to Gaussian noise, and the Kullback-Leibler distance (KL), adapted to Poisson noise. A regularized Gauss-Newton algorithm minimizes the cost function iteratively. Both methods decompose materials from a numerical phantom of a mouse. Soft tissues and bones are decomposed in the projection domain; then a tomographic reconstruction creates a 3D material density volume for each material. Comparing relative errors, KL is shown to outperform WLS for low photon counts, in 2D and 3D. This new method could be of particular interest when low-dose acquisitions are performed.

INTRODUCTION

Spectral computed tomography (SPCT) is a new modality that is expected to be the future of computed tomography (CT). While conventional CT data are obtained integrating the whole energy spectrum received, new detectors, called photo-counting detectors, allow counting how many photons have hit the detectors in several energy bins. Since material attenuation depends on the incident energy of the photon the concentration of the object in a basis of different materials can be estimated from energy-resolved data. Thus SPCT produces more information from a single imaging exam than standard CT. This new modality is expected to have many applications in the medical field, such as imaging high Z contrast agents. [START_REF] Feuerlein | Multienergy photon-counting k-edge imaging: potential for improved luminal depiction in vascular imaging 1[END_REF][START_REF] Cormode | Atherosclerotic plaque composition: analysis with multicolor ct and targeted gold nanoparticles 1[END_REF] From 2D energy-resolved projections, three differents approaches are available to estimate the 3D volumes of material density for a chosen basis of materials. A first approach consists in performing tomographic reconstruction for the different energy bins, thus creating 3D energy-resolved volumes. From those, materials decomposition can be achieved [START_REF] Maaß | Image-based dual energy ct using optimized precorrection functions: A practical new approach of material decomposition in image domain[END_REF] in the object domain. A second approach consists in solving the joint problem of decomposition and tomographic reconstruction, and can be referred as the one-step approach. [START_REF] Long | Multi-material decomposition using statistical image reconstruction for spectral ct[END_REF] The last possibility is to decompose materials in the projection domain, [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting k-edge imaging in pre-clinical computed tomography[END_REF][START_REF] Ducros | Regularization of nonlinear decomposition of spectral x-ray projection images[END_REF][START_REF] Roessl | Cramér-rao lower bound of basis image noise in multiple-energy x-ray imaging[END_REF] creating material density map (called projected mass density, PMD) and performorm the tomographic reconstruction from the decomposed data to created the 3D volumes of material density. This decomposition approach is highly parallelizable since each can be processed independently of the others. Decomposition in projection domain also embeds all the non-linearities of the forward model. While tomographic reconstruction is linear. For all these reasons, we have selected this approach, thus performing material decomposition in the projection domain followed by the standard filtered-back-projection to create 3D volumes of material density.

Material decomposition in SPCT is a non-linear ill-posed inverse problem. It can be solved by minimizing a cost function, composed of a data fidelity term (representing the distance between the data and the forward model) and a regularization term (which make the solution stable regarding the noise). Material decomposition was achieved by minimizing such cost function with different type of algorithms. Both zero-th and first-order algorithms succeeded to that task. [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting k-edge imaging in pre-clinical computed tomography[END_REF][START_REF] Roessl | Cramér-rao lower bound of basis image noise in multiple-energy x-ray imaging[END_REF][START_REF] Schirra | Spectral ct: a technology primer for contrast agent development[END_REF][START_REF] Xu | Sparsity-regularized image reconstruction of decomposed k-edge data in spectral ct[END_REF] However, recent work proposed a second-order iterative algorithm (Gauss-Newton scheme) and proved to be more efficient than lower order algorithms. [START_REF] Ducros | Regularization of nonlinear decomposition of spectral x-ray projection images[END_REF] Ducros et al. introduced a cost function with a weighted least squares (WLS) data fidelity term, that was minimized with a Gauss-Newton algorithm. However, the noise in SPCT is Poisson distributed, and WLS is adapted to Gaussian noise. In this work, we consider a data fidelity term based on the Kullback-Leibler distance to take into account the Poisson nature of the noise. [START_REF] Resmerita | Joint additive kullback-leibler residual minimization and regularization for linear inverse problems[END_REF] From this change, one can expect to improve the quality of the decomposition. This paper is organized as follow. In Section 2, the forward model, the inverse problem, and the minimization algorithm are presented. Section 3 presents the KL distance as new data fidelity. Numerical experiments are described in Section 4. Section 5 displays results of experiments and discusses them. Finally, Section 6 concludes this paper.

MATERIAL DECOMPOSITION IN SPECTRAL CT

Object model

Alvarez and Macovski showed that 11 the object of interest can be decomposed on a basis of M materials. This allows to write the linear attenuation coefficient of the object as the following linear combination:

µ(E, x) = M m=1 ρ m (x)τ m (E) (1)
where µ(E, x) is the linear attenuation coefficient (in cm -1 ) at point x and energy E, ρ m (x) is the mass density (in g.cm -3 ) of the m-th material at point x, and τ m (E) is the mass attenuation (in cm 2 .g -1 ) of the m-th material.

Forward model

The standard SPCT forward model 6, 12 is written as

s i (u) = R d i (E)n 0 (E) exp - M m=1 a m (u)τ m (E) dE (2)
where s i (u) is the number of photons hitting the detector in the i-th energy bin at pixel u, d i (E) is the response function of the i-th energy bin of the detector, 5 n 0 (E) is the emitted X-ray spectrum and a m (u) is the projection of all the mass of the m-th material along the x-ray path L(u):

a m (u) = L(u) ρ m (x)dx (3)

Discretization and noise

The detector is assumed to have P pixels, with u p the location of the p-th pixel, and I energy bins. We define the measurement vector

s ∈ R IP as s = [s 1,1 , . . . s i,p , . . . s I,P ] (4)
where s i,p is the photon count measured in the i-th energy bin at pixel u p . Similarly, the projected mass density (PMD) vector a ∈ R M P is defined by

a = [a 1,1 , . . . a m,p , . . . a M,P ] (5)
where a m,p is PMD of the m-th material at pixel u p .

Data is assumed to be corrupted by Poisson noise. Let s * be the noiseless data vector, s the noisy data, and P(γ) the Poisson distribution with mean γ. We have:

s ∼ P(γ = s * ). ( 6
)

Inverse problem

Material decomposition is an ill-posed problem that consists in recovering the PMD a from the data s. Let s = F(a) be the forward model defined in (2). The PMD can be recovered by minimizing the following cost function

C(s, F(a)) = D(s, F(a)) + αR(a) (7)
where D(s, F(a)) is the fidelity term, α the regularization parameter and R(a) the regularization term used to stabilize the solution. The regularization terms used was chosen as 6

R(a) = ||∆a soft || 2 2 + ||∇a bone || 2 2 ( 8
)
where ∆ and ∇ are first-and second-order differential operator. The usual WLS fidelity term is defined by

D WLS (s, F(a)) = 1 2 ||W(s -F(a))|| 2 (9)
where ||.|| 2 is the 2 -norm and W is the weighting matrix defined by

W = diag( 1 √ s+1
) where diag(x) represents a diagonal matrix with the elements x.

Minimization of the cost function

In this work, the cost function ( 7) is minimized by using a Gauss-Newton's algorithm. This second-order algorithm provides faster convergence than first order methods. Starting with an initial guess a 0 , the solution is computed iteratively with the following update formula

a k+1 = a k + λ k δa k (10)
where a k is the solution at the iteration k, δa k is the Newton's step and λ k a scalar optimizing the minimization in the direction δa k . The Newton's step is computed by solving

(J Z h J + αH reg )δa k = -(J Z g (F(a k ) -s) + αg reg ) ( 11
)
where J is the Jacobian of the forward problem, H reg and g reg is the Hessian and the gradient of the regularization term, respectively. Z h and Z g are diagonals matrices which depend on the fidelity term. The step length λ k is obtained by solving

λ k = argmin λ C(s, F(a k + λδa k )) (12)
Z g and Z h are, in the WLS data fidelity term case, as follow [START_REF] Favati | Performance analysis of maximum likelihood methods for regularization problems with nonnegativity constraints[END_REF] WLS:

Z g = W T W Z h = W T W 3. A NEW APPROACH: THE KULLBACK-LEIBLER DISTANCE 3.

Noise in spectral CT

The standard data fidelity term WLS is adapted to Gaussian noise. However, in SPCT the noise is Poisson distributed since it is a comes from the detection process of the photons. The WLS is thus not well adapted to our problem for low signal-to-noise ratios. That is why in this work the cost function uses the Kullback-Leibler distance as data fidelity term. The KL term allows taking into account the statistics of the noise more faithfully. [START_REF] Resmerita | Joint additive kullback-leibler residual minimization and regularization for linear inverse problems[END_REF]

A noise adapted fidelity term

The KL distance between two Poisson-distributed data s and F(a) is written 14

D KL (s, F(a)) = i (s i + ζ) log s i + ζ F(a) i + ζ + F(a) i -s i ( 13
)
where ζ is a small integer that avoids dividing by zero and having the log of zero. The Gauss-Newton algorithm requires the computation of the Jacobian and the approximation of the Hessian, as seen in Equation (11). For KL, we have:

KL: Z g = Y -1 Z h = Y -2 S
where Y = diag(F(a) + ζ) and S = diag(s + ζ).

NUMERICAL SIMULATIONS

Both methods are tested on the numerical phantom DigiMouse. [START_REF] Dogdas | Digimouse: a 3d whole body mouse atlas from CT and cryosection data[END_REF] We consider a X-ray spectrum generated by a 90KeV source using the spekCalc software. [START_REF] Poludniowski | Spekcalc: a program to calculate photon spectra from tungsten anode x-ray tubes[END_REF] Three bins of energy (I = 3) are used by the photon-counting detectors to detect photons. We define N 0 as the number of photons send at each pixel detector:

N 0 = R n 0 (E)dE (14)
N 0 controls the signal-to-noise ratio, thus a large range of N 0 values is used:

N 0 ∈ [10 2 , 10 4 ].
The WLS decomposition is performed using the SPRAY Matlab toolbox. [START_REF]Spray matlab toolbox[END_REF] The initial guess of the iterative algorithm are uniform PMD maps with a 0 soft = 2 g.cm -2 and a 0 bones = 1 g.cm -2 . The algorithm stops when the relative decrease of the cost function is lower than 0.1%, if λ k is lower than 10 -3 or if the maximum number of iteration is reached. This maximum number of iterations is set at 50. A large range of regularization parameter is used in order to find the best decomposition, α ∈ [10 -2 , 10 4 ].

We define the relative decomposition error ξ θ of the θ-th projection as:

ξ θ = M m=1 ||a dec m (θ) -a true m (θ)|| ||a true m (θ)|| ( 15
)
where a dec m (θ) is the decomposed PMD of the m th material and a true m (θ) is the ground truth for the material m. Similarly we define δ Z as the relative reconstruction error in the z-th slice of the object as:

δ Z = M m=1 ||ρ rec m (z) -ρ true m (z)|| ||ρ true m (z)|| ( 16
)
where ρ true m (z) is the m-th material density ground-truth and ρ rec m (z) is the reconstructed m-th material density. For each projection angle θ, αθ represents the regularization parameter that minimizes the relative error, i.e., αθ = argmin

α ξ θ (α) (17)
The 3D volumes (154 × 154 × 496 voxels) are reconstructed from the Filtered Back Projection algorithm implemented by the iradon function of Matlab. We used the Hann filter and a v5cubic interpolation method (cubic interpolation from Matlab 5). 1: Optimal α value, time per iteration and number of iterations required to converge for α at N 0 = 10 2.2 for two different projection angle θ

RESULTS AND DISCUSSION

The PMD are first obtained on the DigiMouse phantom. The computation of the relative errors is very relevant to evaluate the quality of the decompositions.

Figure 1 shows the evolution of the relative error ξ on the projections for the optimized αθ and for the projection angles θ = 0 • (left) and θ = 90 • (right) with an increasing numbers of photons N 0 . At a low number of photons, the WLS is not a good approximation of the KL distance, which takes account of the photonic noise. For both projection angles, KL outperforms WLS at a small number of photons. At a higher signal-to-noise ratio, the quadratic approximation is good, and both WLS and KL lead to the same relative error. Similar curves can be found by considering each material (soft tissues and bones).

Table 1 provides quantitative results from the decompositions at N 0 = 10 2.2 photons, for θ = 0 • and θ = 90 • . This table presents the optimal value of α, the time of computation of one iteration and the number of iteration required to converge. KL and WLS are taking almost the same CPU time (3.5 GHz Xeon E5-1650 v3 and 64 GiB of RAM) to make an iteration and also the same number of iterations to converge. This comes from the calculation of the Jacobian and Hessian where they are almost the same.

Previous results show that KL outperforms WLS at a low number of photons. To highlight this trend, and to validate it, decomposition of soft tissues and bones were done for θ ∈ [0, 360], at N 0 = 10 2.2 photons. Evolution of relative error ξ, for αθ , for different θ is depicted on Figure 2 (top). It shows that KL outperforms WLS for any projection angle. On the last two rows decomposition of soft tissues (middle row) and bones (bottom row) are shown for θ = 180 • . The first column corresponds to the ground truth, the second one to WLS and the last to KL. From those images we can see that KL retrieves more faithfully the PMD from the data. A video is available to see the evolution of the PMD along the angle projection.

A tomographic reconstruction was performed from each material. Figure 3 shows the relative error δ for each slice z. This figure presents the slice 274 since it highlights the best performance of KL with respect to WLS. On soft tissues, KL allows having more details inside the volume than WLS. On bones, KL retrieves more faithfully the value of interest ρ bones . Yet, bones and soft tissus regions are mixed in some slices. A video is also available to see the evolution of the error with the slice number.

CONCLUSION

In this work, a new algorithm of material decomposition in the projection domain for SPCT is presented. The cost function uses the KL distance instead of the standard WLS data fidelity term. This allows taking into account the Poisson noise statistics. Since WLS is adapted to Gaussian noise and KL to Poisson noise, the latest divergence should produce better decompositions. Numerical experiments on a mouse phantom, with a large range of signal-to-noise ratios (which controls the approximation of the noise as Gaussian), allow comparing the WLS and KL methods. We obtain PMD maps with the two decomposition methods on the projection domain.

Results from experiments show that KL outperforms WLS, considering the relative decomposition error, for any angle of projection θ, at a low number of photons. This efficiency comes from the fact that at a low signal-to-noise ratio the KL distance is more adapted to the noise structure. A tomographic reconstruction is carried out to create a 3D volume of materials. Those volumes are reconstructed from low signal-to-noise projection, where KL is a more adapted data term. In the object domain, KL still has a lower relative error for each slice of the object. In the future, we plan to assess the method with other phantoms, a larger number of materials and to develop methods for automatic selection of the regularization parameter.

Figure 1 :

 1 Figure 1: Evolution of the relative error with the number of photons at 0 • (left) and 90 • (right) θ( • ) αθ Time (sec) Iteration number WLS 0 10 2 3.37 5 90 10 1.53.49 4 KL 0 10 1.4 2.95 4 90 10 3.56 4 Table 1: Optimal α value, time per iteration and number of iterations required to converge for α at N 0 = 10 2.2 for two different projection angle θ

Figure 2 :

 2 Figure 2: Video1: Ground truth of the material density map of soft tissues and bones in the projection domain (left), decomposition with WLS (middle) and with KL (right) for N 0 = 10 2.2 photons for α at θ = 180 • . http://dx.doi.org/doi.number.goes.here

Figure 3 :

 3 Figure 3: Video2: Ground truth of the material volume of soft tissues and bones in the object domain (left), decomposition with WLS (middle) and with KL (right) for N 0 = 10 2.2 photons for α at the slice number 274. http://dx.doi.org/doi.number.goes.here

  It is a volume of 104 × 190 × 496 voxels composed by M = 2 materials: soft tissues and bones. As mentioned earlier, PMD are first decomposed, then, a tomographic reconstruction is performed to retrieve a 3D volume per material. The projection data are obtained with the Radon function of Matlab (219 × 496 pixels). 360 projections angles are used over 360 • .

Acknowledgement

This work was supported by the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). It was also supported by the ANR grant ANR-11-INBS-0006.


Lianming Xia

Xiangyou Wang

Duayang Geng

Qingfeng Zhang
email: zhangqingfeng1844@163.com

Sliding liding liding liding Monitoring Monitoring Monitoring Monitoring System System System System for for for for Ground Ground Ground Ground Wheel Wheel Wheel Wheel Based Based Based Based on on on on ATMEGA16 ATMEGA16 ATMEGA16 ATMEGA16 for for for for No-tillage No-tillage No-tillage No-tillage Planter Planter Planter Planter----CT246 CT246 CT246 CT246

Keywords: no-tillage planter, sliding, photoelectric sensor, rotary encode, single-chip microcomputer

The seed-metering device of the no-tillage planter was driven by the ground wheel. Sliding was unavoidable when the ground was covered by the straw. High sliding brought seeding absence and affected the quality of sowing. In order to monitor the sliding of the ground wheel, a monitoring system composed of hardware and software was designed based on ATMEGA16 Microcomputer. In the system, the practical and the theoretical driving distance of the ground wheel were measured by the photoelectric sensor and rotary encoder respectively, then the sliding can be calculated and displayed on the LED. The system could achieve the data of sliding when the planter worked, and this could insure that the driver may recognize the working conditions at any moment. So it had very important significance which can improve the precision of sow.

Introduction Introduction Introduction Introduction

Conservation tillage has been defined as tillage and planting system that retains at least 30% of the cover residues on the soil surface after planting operation is completed. Erosion is reduced by at least 50% in these soils as compared to bare soils [START_REF] Karayel | Performance of a Modified Precision Vacuum Seeder for No-till Sowing of Maize and Soybean[END_REF][2] [START_REF] Barut | Determination of the Optimum Working Parameters of a precision Vacuum Seeder[END_REF]. Retain crop residues on the soil surface provide a source of plant nutrients, improves organic matter level in the soil, and increase soil water content by reducing evaporation and increasing infiltration rate [START_REF] Wang | Controlled traffic farming with no tillage for improved fallow water storage and crop yield on the Chinese Loess Plateau[END_REF]. In the last three decades, no-till sowing practices that promote soil and water conservation have slowly become an accepted alternative to conventional tillage systems. The precision and no-till seeder has been a key factor in the successful shift to no-till sowing [START_REF] Baker | No-tillage Seeding and Practice[END_REF]. In the precision and no-till seeder, the seed-metering device which is driven by the ground wheel places seeds at the required spacing. The sliding which can bring seed absence is unavoidable when the field is covered with crop residue and this can affect the quality of sowing.

Moreover, in response to the recent trend toward improving the quality of sowing, many researchers have focused their research on the development of precision seedmetering device and little work has been done to evaluate the sliding for the no-till precision planter. Although it was realized that sliding of ground wheel could affect the quality of sowing, what could do was only to let the tractor drive for a certain distance and then calculate the sliding by counting the number of turns. Then the quantity of sowing can be adjusted based on it. This could only improve the average sliding to a certain extent. However, the quality of sowing could not be improved by only adjusting the quantity of sowing with the varying sliding. In this research, a sliding monitoring system based on the technologies of single-chip microcomputer and sensor was designed to help the driver realize the sliding at real-time.

A A A Analysis nalysis nalysis nalysis of of of of the the the the S S S Sliding liding liding liding M M M Mon on on oni i i itor tor tor toring ing ing ing T T T Theory heory heory heory

The sliding of the ground wheel is caused by the different driving distances between the tractor and the ground wheel. So, the sliding can be calculated by the formula below % 100

1 2 1 × - = s s s η (1)
Where s1 is the practical distance the tractor drives, s2 the theoretical distance the ground wheel drives [START_REF] Gao | Ground wheel Slide of no Tillage Seeding-machine[END_REF]. A five-wheel gauge was usually used to measure the practical distance of the ground wheel drove. However, the result was often affected by the factors such as the bounce of the fifth wheel brought by the ground flatness, the difference between the arc and lineal length of the ground surface, the sliding which was caused by the no-pure rolling between the fifth wheel and the ground. In the no-tillage sowing, the ground is covered by a lot of crop residue. So, the error would be very obvious when it was measured by the five-wheel gauge [7]. In this paper, the practical driving distance is measured by the photoelectric sensor and this is showed in Fig1. From Fig2 and Fig3, it can be seen that the photoelectric sensor and the rotary encode are connected to PD2 and PD3 which are the external interruption pins of ATMEGA16. The output level sent out from PD2 is high when the photoelectric sensor is near the bamboo pile and low when it is away from it. The PD3 pin sends out high LEVEL when the rotary encode revolves and low or high level when it does not revolve [START_REF] Cui | A Design Instance of Measurement System for Rotary Speed Based on Photoelectric Encoder[END_REF]. All interruptions are triggered at the falling edge. Although the photoelectric sensor and the rotary encode can send out changing level, it can not absorb and send out too high current, or it can not make the current of the interruption pin be low. So a PNP9013 is used as switch in Fig3.

D D D Design esign esign esign of of of of the the the the LCD LCD LCD LCD C C C Circuit ircuit ircuit ircuit

In the monitoring system, LCD12864 with FLASH is used to show the sliding of the ground wheel. Its wiring diagram is showed in Fig4. In the main program, four global variables and one global array are set. In the array, only three elements are used. The interrupt 0 and 1 are used to count the numbers of Pulse signals from the photoelectric sensor and the rotary encode. In the interruption system, flag bit num0 and num1, flag0 and flag1 are needed to be updated. The screen will blink when interruption0 counts one pulse signals and interruption1 counts ten pulse signals. The value of the sliding will refresh when the interruptions count twice and at the same time all flag bits are cleared. In this system, rate0 and rate1 are used to indicate the distances when one impulse signal is interrupted. The flow chart of the main program is showed in Fig6 [START_REF] Taner | Virtual instrumentation:a solution to the problem of design complexity in intelligent[END_REF]. In the experiments, the factors which can affect the sliding are the soil conditions, the parameters of the ground wheel and the vertical load on the wheel [START_REF] Walsh | Towards a Standard for Controlled Traffic Machinery, proceeding of the Second National controlled Traffic Conference of Australia[END_REF]. In order to test the performance of the monitoring system, some ground wheels with different parameters and vertical loads were chosen to work in different soils. The parameters of the ground wheel and the vertical load were showed in Table3.

Table3. Table3. Table3. Table3. The parameters of the ground wheel According to the parameters of Table3, five ground wheels with different parameters were chosen to test the performance of the system, the parameters and results were showed in Table4.

Table4 Table4 Table4 Table4. The results of the experiments with different ground wheels From the results above, it could be seen that the monitoring system correctly tested the sliding of the ground wheels with different parameters.

C C C Conc onc onc onclu lu lu lusions sions sions sions

In this paper, an automatic instrument based on the technologies of sensor and singlechip microcomputer was used for precision planter. It could measure the sliding of the ground wheel when the tractor drove on the soil which was covered by the straws at real-time. The driver could recognize the sliding of the ground wheel and this would help him control the quality of sowing timely.

Fig

  Fig Fig Fig Fig. . . .1 1 1 1. . . . The theory for measurement of the practical driving distance

Fig Fig Fig Fig. 3 . . 3 . . 3 . . 3 .

 3333 Fig Fig Fig Fig.3. .3. .3. .3. The circuits of the sensors

FFF

  Fig ig ig ig. . . .4. 4. 4. 4. Wiring diagram of LCD12864 As showed in Fig4, port LCDDATA is used as the I/O port of LCD, LED and the four-digit digital diode. It is also the extension of PA. Its ports are composed of ten groups of two-pin jumpers which are linked to the pin MAMBD. The pins of the LCDDATA from left to right are showed in Table1. Table1 Table1 Table1 Table1. The pins of LCDDATA LCDControl is the extended port of PB which is the control port of LCD12864 or LCD1602. The ports of LCDControl are composed of five group two-pin jumpers which are linked to pin PB0 to PB4 of M2MD1.0. The pins of port LCDControl are showed in Table2. Table2 Table2 Table2 Table2. The pins of LCDControlThe wiring diagram of the monitoring system is showed in Fig5.

F F F Fig. 5 .

 5 ig.5. ig.5. ig.5. Wiring diagram of the monitoring system 4. 4. 4. 4. D D D Design esign esign esign of of of of S S S Software oftware oftware oftware System System System System

F F F Fig. 6 .

 6 ig.6. ig.6. ig.6. Flow chart of the main program 5. 5. 5. 5. Experiments Experiments Experiments Experiments and and and and Discussion Discussion Discussion Discussion