J Ér

Ôme Buzzi

DIMENSIONAL ENTROPIES AND SEMI-UNIFORM HYPERBOLICITY

We describe dimensional entropies introduced in [6], list some of their properties, giving some proofs. These entropies allowed the definition in [7, 10] of entropy-expanding maps. We introduce a new notion of entropyhyperbolicity for diffeomorphisms. We indicate some simple sufficient conditions (some of them new) for these properties. We conclude by some work in progress and more questions.

Introduction

We are interested in using robust entropy conditions to study chaotic dynamical systems. These entropy conditions imply some "semi-uniform" hyperbolicity. This is a type of hyperbolicity which is definitely weaker than classical uniform hyperbolicity but which is stronger than Pesin hyperbolicity, that is, non vanishing of the Lyapunov exponents of some relevant measure. This type of conditions allows the generalization of some properties of interval maps and surface diffeomorphisms to arbitrary dimensions.

In this paper, we first explain what is known in low dimension just assuming the non-vanishing of the topological entropy h top (f ). Then we give a detailed description of the dimensional entropies. These are d + 1 numbers, if d is the dimension of the manifold,

0 = h 0 top (f ) ≤ h 1 top (f ) ≤ • • • ≤ h d top (f ) = h top (f ).
h k top (f ) "counts" the number of orbits starting from an arbitrary compact and smooth k-dimensional submanifold. We both recall known properties and establish new ones. We then recall the definition of entropy-expanding maps which generalize the complexity of interval dynamics with non-zero topological entropy. We also introduce a similar notion for diffeomorphisms: Definition 1. A diffeomorphism of a d-dimensional manifold is entropy-hyperbolic if there are integers d u , d s such that:

• h du top (f ) = h top (f ) and this fails for every dimension k < d u ; • h ds top (f -1 ) = h top (f ) and this fails for every dimension k < d s ;

• d u + d s = d.
We give simple sufficient conditions for entropy-expansion and entropy-hyperbolicity. Finally we announce some work in progress and state a number of questions.

Date: Talk at ICMP -Rio de Janeiro, Brazil, August 2006.

1

We now recall some classical notions which may be found in [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]. A basic measure of orbit complexity of a map f : M → M is the entropy. The topological entropy h top (f ) "counts" all the orbits and the measure-theoretic entropy (also known as Kolmogorov-Sinai entropy or ergodic entropy) h(f, µ) "counts" the orbits "relevant" to some given invariant probability measure µ. They are related by the following rather general variational principle. If, e.g., f is continuous and M is compact, then

h top (f ) = sup µ h(f, µ)
where µ ranges over all invariant probability measures. One can also restrict µ to ergodic invariant probability measures. This brings to the fore measures which realize the above supremum, when they exist, and more generally measures which have entropy close to this supremum.

As µ → h(f, µ) is affine, µ has maximum entropy if and only if almost every ergodic component of it has maximum entropy. Hence, with respect to entropy, it is enough to study ergodic measures. Definition 2. A maximum measure is an ergodic and invariant probability measure µ such that h(f, µ) = sup ν h(f, ν).

A large entropy measure is an ergodic and invariant probability measure µ such that h(f, µ) is close to sup ν h(f, ν).

The Lyapunov exponents for some ergodic and invariant probability measure µ are the possible values µ-a.e. of the limit lim n→∞ 1 n log T x f n .v where • is some Riemannian structure and T x f is the differential of f and v ranges over the non-zero vectors of the tangent space T x M .

A basic result connecting entropy and hyperbolicity is the following theorem (proved by Margulis for volume preserving flows):

Theorem 1 (Ruelle's inequality). Let f : M → M be a C 1 map on a compact manifold. Let µ be an f -invariant ergodic probability measure. Let λ 1 (µ) ≥ . . . be its Lyapunov exponents repeated according to multiplicity. Then,

h(f, µ) ≤ d i=1 λ i (µ) +
In good cases (with enough hyperbolicity), the entropy is also reflected in the existence of many periodic orbits: Definition 3. The periodic points of some map f : M → M satisfy a multiplicative lower bound, if, for some integer p ≥ 1:

lim inf n→∞,p|n e -nhtop(f ) #{x ∈ [0, 1] : f n x = x} > 0.
Recall that many diffeomorphisms have infinitely many more periodic orbits (see [START_REF] Kaloshin | Generic diffeomorphisms with superexponential growth of number of periodic orbits[END_REF][START_REF] Kaloshin | Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms. I[END_REF]).

The following type of isomorphism will be relevant to describe all "large entropy measures". Definition 4. For a given measurable dynamical system f : M → M , a subset S ⊂ M is entropy-negligible if there exists h < sup µ h(f, µ) such that for all ergodic and invariant probability measures µ with h(f, µ) > h, µ(S) = 0.

An entropy-conjugacy between two measurable dynamical systems f : M → M and g : N → N is a bi-measurable invertible mapping ψ : M \ M 0 → N \ N 0 such that: ψ is a conjugacy (i.e., g • ψ = ψ • f ) and M 0 and N 0 are entropy-negligible.

Low Dimension

Low dimension dynamical systems here means interval maps and surface diffeomorphisms -those systems for which non-zero entropy is enough to ensure hyperbolicity of the large entropy measures.

2.1. Interval Maps. Indeed, an immediate consequence of Ruelle's inequality on the interval is that a lower bound on the measure-theoretic entropy gives a lower bound on the (unique) Lyapunov exponent. Thus, invariant measures with nonzero topological entropy are hyperbolic in the sense of Pesin. One can obtain much more from the topological entropy:

Theorem 2. Let f : [0, 1] → [0, 1] be C ∞ . If h top (f ) > 0 then f
has finitely many maximum measures. Also the periodic points satisfy a multiplicative lower bound.

This was first proved by F. Hofbauer [START_REF] Hofbauer | On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II[END_REF][START_REF] Hofbauer | Periodic points for piecewise monotonic transformations[END_REF] for piecewise monotone maps (admitting finitely many points

a 0 = 0 < a 1 < • • • < a N such that f |]a i , a i+1
[ is continuous and monotone). It was then extended to arbitrary C ∞ maps in [START_REF] Buzzi | Intrinsic ergodicity of smooth interval maps[END_REF]. In both settings, one builds an entropy-conjugacy to a combinatorial model called a Markov shift (which is a subshift of finite type over an infinite alphabet). One can then apply some results of D. Vere-Jones [START_REF] Vere-Jones | Ergodic properties of non-negative matrices[END_REF] and B. Gurevič [START_REF] Gurevich | Shift entropy and Markov measures in the path space of a denumerable graph (Russian)[END_REF].

We can even classify these dynamics. Recall that the natural extension of f :

M → M is f : M → M defined as M := {(x n ) n∈Z ∈ M Z : ∀n ∈ Z x n+1 = f (x n )} and f ((x n ) n∈Z ) = (f (x n )) n∈Z .
Recall that π : (x n ) n∈Z → x 0 induces a homeomorphism between the spaces of invariant probability measures which respects entropy and ergodicity.

Theorem 3. The natural extensions of C ∞ interval maps with non-zero topological entropy are classified up to entropy-conjugacy by their topological entropy and finitely many integers (which are "periods" of the maximum measures).

The classification is deduced from the proof of the previous theorem by using a classification result [START_REF] Boyle | Almost isomorphism for countable state Markov shifts[END_REF] for the invertible Markov shifts involved.

The C ∞ is necessary: for each finite r, there are C r interval maps with non-zero topological entropy having infinitely many maximum measures and others with none.

Remark 5. These examples show in particular that Pesin hyperbolicity of maximum measures or even of large entropy measures (which are both consequences of Ruelle's inequality here) are not enough to ensure the finite number of maximum measures.

Surface Transformations.

As observed by Katok [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF], Ruelle's inequality applied to a surface diffeomorphism and its inverse (which has opposite Lyapunov exponents) shows that a lower-bound on measure-theoretic entropy bounds away from zero the Lyapunov exponents of the measure. Thus, for surface diffeomorphisms also, nonzero entropy implies Pesin hyperbolicity.

It is believed that surface diffeomorphisms should behave as interval maps, leading to the following folklore conjecture:

Conjecture 1. Let f : M → M be a C ∞ surface diffeomorphism. If h top (f ) > 0 then f has finitely many maximum measures.
I would think that, again like for interval maps, finite smoothness is not enough for the above result. However counter-examples to this (or to existence) are known only in dimension ≥ 4 [START_REF] Misiurewicz | Diffeomorphism without any measure with maximal entropy[END_REF].

The best result for surface diffeomorphisms at this point is the following "approximation in entropy" [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF]: Theorem 4 (A. Katok). Let f : M → M be a C 1+ǫ surface diffeomorphism. For any ǫ > 0, there exists a horseshoe 1

Λ ⊂ M such that h top (f |Λ) > h top (f ) -ǫ.
In particular, the periodic points of f satisfy a logarithmic lower bound:

lim sup n→∞ 1 n log #{x ∈ M : f n (x) = x} ≥ h top (f ).
Katok in fact proved a more general fact, valid for any C 1+ǫ -diffeomorphism of a compact manifold of any dimension. Namely, if µ is an ergodic invariant probability measure without zero Lyapunov exponent :

lim sup n→∞ 1 n log #{x ∈ M : f n (x) = x} ≥ h(f, µ).
On surfaces, Ruelle's inequality and the variational principle imply the theorem as explained above.

I have proved the conjecture for a model class, which replaces distortion with (simple) singularities [START_REF] Buzzi | Measures of Maximum Entropy for Piecewise Affine Surface Homeomorphisms[END_REF]: Theorem 5. Let f : M → M be a piecewise affine homeomorphism. If h top (f ) > 0 then f has finitely many maximum measures.

Dimensional Entropies

We are going to define the dimensional entropies for a smooth self-map or diffeomorphism f : M → M of a d-dimensional compact manifold. We will then investigate these quantities by considering other growth rates obtained from the volume and size of the derivatives. Finally we shall establish the topological variational principle stated in the introduction by a variant of Pliss Lemma.

3.1. Singular disks. The basic object is:

Definition 6. A (singular) k-disk is a map φ : Q k → M with Q k := [-1, 1] k . It is C r if it can be extended to a C r map on a neighborhood of Q k .
We need to define the C r size φ C r of a singular disk φ for 1 ≤ r ≤ ∞ as well as the corresponding topologies on the space of such disks. This involves some technicalities as vectors in different tangent spaces are not comparable a priori. We refer to Appendix A for the precise definitions, which are rather obvious for finite r. For r = ∞, we need an approximation property (which fails for some otherwise very reasonable definitions of C r size), Fact 40, which is used to prove Lemma 10 below.

From now on, we fix some C r size arbitrarily on the manifold M . We will later check that the entropies we are interested in are in fact independent of this choice.

1 A horseshoe is an invariant compact subset on which some iterate of f is conjugate with a full shift on finitely many symbols.

Notations. It will be convenient to sometimes write φ instead of φ(Q k ), e.g.,

h top (f, φ) instead of h top (f, φ(Q k )).
3.2. Entropy of collections of subsets. Given a collection D of subsets of M , we associate the following entropies. Recall that the (ǫ, n)-covering number of some subset S ⊂ M is:

r f (ǫ, n, S) := min{#C : x∈S B f (ǫ, n, x) ⊃ S} where B f (ǫ, n, x) := {y ∈ M : ∀0 ≤ k < n d(f k y, f k x) < ǫ} is the (ǫ, n)-dynamic ball. The classical Bowen-Dinaburg formula for the topological entropy of S ⊂ M is h top (f, S) = lim ǫ→0 lim sup n→∞ 1 n log r f (ǫ, n, S) and h top (f ) = h top (f, M ). Definition 7.
The topological entropy of D is:

h top (f, D) := sup D∈D h top (f, D) = sup D∈D lim ǫ→0 lim sup n→∞ 1 n log r f (ǫ, n, D)
The uniform topological entropy of D is:

H top (f, D) := lim ǫ→0 lim sup n→∞ 1 n log sup D∈D r f (ǫ, n, D). Clearly h top (f, D) ≤ H top (f, D).
The inequality can be strict as shown in the following examples (the first one involving non-compactness, the second one involving non-smoothness).

Example 1. Let T : T 2 → T 2 be a linear endomorphism with two eigenvalues Λ 1 , Λ 2 with 1 < |Λ 1 | < |Λ 2 |. Let L be the set of finite line segments. We have 0 < h top (T, L) = log |Λ 2 | < H top (T, L) = log |Λ 1 | + log |Λ 2 |. Example 2. There exist a C ∞ self-map F of [0, 1] 2 and a collection C of C r curves with bounded C r norm such that 0 < h top (F, C) < H top (F, C
). This can be deduced from the example with h 1 top (f × g) > max(h top (f ), h top (g)) in [START_REF] Buzzi | Ergodicité intrinsèque de produits fibrés d'applications chaotiques unidimensionelles[END_REF] by considering curves with finitely many bumps converging C r-1 to the example curve there, which has infinitely many bumps.

Definitions of the dimensional entropies.

We can now properly define the dimensional entropies. Recall that we have endowed M with a C r size. Definition 8. For each 1 ≤ r ≤ ∞, the standard family of C r singular k-disks is the collection of all C r singular k-disks. For finite r, the standard uniform family of C r singular k-disks is the collection of all C r singular k-disks with C r size bounded by 1.

Definition 9. The C r , k-dimensional entropy of a self-map f of a compact manifold is: h k,C r top (f ) := h top (f, D k r ) where D k r is a standard family of C r k-disks of M . We write h k top (f ) for h k,C ∞ (f ) top and call it the k-dimensional entropy. The C r , k-dimensional uniform entropy H k,C r top (f ) is obtained by replacing h top (f, D k r ) with H top (f, D k r )
in the above definition where D k r is the standard uniform family. We write

H k top (f ) for H k,C ∞ top (f ) and call it the k-dimensional uniform entropy.
Observe that h k,C r top (f ) and H k,C r top (f ) are non-decreasing functions of k and nonincreasing functions of r. Indeed, (1)

D k r ⊃ D k s and D k r ⊃ D k s if r ≤ s; (2) for any 0 ≤ ℓ ≤ k ≤ d, restricting a k-disk to [0, 1] ℓ × {0} k-ℓ does not increase its C r size. Observe also that h 0,C r top (f ) = 0 and h d top (f ) = h top (f ).
Lemma 10. Let f : M → M be a C ∞ self-map of a compact manifold. We have:

H k top (f ) = lim r→∞ H k,C r top (f )
and the limit is non-increasing.

We shall see later in Proposition 26 that the same holds for h k top (f ).

Proof. We use one of the (simpler) ideas of Yomdin's theory. For each n ≥ 1, we divide Q k into small cubes with diameter at most (ǫ/4)

1/r φ -1/r C r Lip(f ) -n/r . We need (ǫ/4) -k/r √ k k φ k/r C r Lip(f ) k r n such cubes.
Let q be one of them. By Fact 40, there exists a

C ∞ k-disk φ q such that φ q C ∞ ≤ 2 φ C r and ∀t ∈ q d(φ q (t), φ(t)) ≤ φ C r t -t q r ≤ φ C r × ǫ 2 φ -1 C r Lip(f ) -n ≤ ǫ 2 Lip(f ) -n
It follows that r f (ǫ, n, φ ∩ q) ≤ r f (ǫ/2, n, φ q ). Thus,

r f (ǫ, n, D k r ) ≤ √ k k (ǫ/4) -k/r φ k/r C r Lip(f ) k r n r f (ǫ/2, n, D k ∞ ) Hence, writing lip(f ) := max(log Lip(f ), 0), H k,C r top (f ) ≤ k r lip(f ) + H k top (f ).
The inequality

H k,C r top (f ) ≥ H k top (f ) is obvious, concluding the proof.
Lemma 11. The numbers H k,C r top (f ) do not depend on the underlying choice of a C r size.

Proof. Using Lemma 10, it is enough to treat the case with finite smoothness. Let D 1 , D 2 be two standard families of k-disks, defined by two C r sizes

• 1 C r , • 2 C r . By Fact 39, there exists C < ∞ such that • 1 C r ≤ C • 2 C r . Hence setting K := ([C] + 1) k , for any k-disk φ 1 ∈ D 1 can be linearly subdivided 2 into K k-disks φ 1 2 , . . . , φ K 2 ∈ D 2 . Thus ∀n ≥ 0 r f (ǫ, n, φ 1 ) ≤ K max j r f (ǫ, n, φ j 2 ). It follows immediately that H(f, D 1 ) ≤ H(f, D 2 ).
γ(f, φ) := lim sup n→∞ 1 n log vol(f n • φ) with vol(ψ) := Q k Λ k ψ(x) dx
where Λ k ψ is the Jacobian of ψ : Q k → M wrt the obvious Riemannian structures. The volume growth exponent of f in dimension k is:

γ k (f ) := sup φ∈D k r γ(f, φ), γ(f ) := max 0≤k≤d γ k (f ) is simply called the volume growth of f .
Observe that the value of the growth rates defined above are independent of the choice of the Riemannian structure, by compactness of the manifold.

The volume growth dominates the entropy quite generally:

Theorem 6 (Newhouse [START_REF] Newhouse | Entropy and volume, Ergodic Theory Dynam. Systems[END_REF]). Let f : M → M be a C 1+α , α > 0, smooth self-map of a compact manifold. Then:

h top (f ) ≤ γ(f ).
Remark 13. More precisely, his proof gave

h top (f ) ≤ γ d cu (f )
where d cu is such that the variational principle h top (f ) = sup µ h(f, µ) still holds when µ is restricted to measures with exactly d cu nonpositive Lyapunov exponents. For C 1+1 -diffeomorphisms, deeper ergodic techniques due to Ledrappier and Young are available and Cogswell [START_REF] Cogswell | Entropy and volume growth[END_REF] has shown that, for any ergodic invariant probability measure µ, there exists a disk ∆ such that h top (f, µ) ≤ h top (f, ∆) ≤ γ(f, ∆). More precisely, the dimension of this disk is the number of positive Lyapunov exponents. For C ∞ diffeomorphisms (more generally if there is a maximum measure) there exists a disk ∆ max such that

h top (f ) = h top (f, ∆ max ) = γ(f, ∆ max ).

I do not know if Newhouse's inequality fails for C 1 maps.

The proof of Newhouse inequality involves ergodic theory and especially Pesin theory. Indeed, this type of inequality does not hold uniformly: Example 3. There exist a C ∞ self-map F of a surface and a C ∞ curve φ such that, for some sequence n i → ∞,

lim i→∞ 1 n i log r F (ǫ, n i , φ) > lim i→∞ 1 n i log vol(F ni • φ).
Proof. Let α > 0 be some small number. Let

I := [0, 1]. Let f : I → I be a C ∞ map such that: (i) f (0) = f (1) = 0; (ii) f (1/2) = 1; (iii) f |[0, 1/2] is increasing and f |[1/2, 1] is decreasing; (iv) f ′ |[0, 1/2 -α] = 2(1 + α) and f ′ |[1/2 + α] = -2(1 + α). As α is small, 1/2 has a preimage in [0, 1/2 -α]. Let x -n be the leftmost preimage in f -n (1): x 0 = 1, x -1 = 1/2, and for all n ≥ 2, x -n = 2 -n (1 + α) -n+1 . Let g : I → I be another C ∞ map such that: (i) g(0) = 0; (ii) 0 < g ′ < 1; (iii) g(x -n ) = x -n-1 for all n ≥ 0.
Consider the following composition of length 3n for some n ≥ 1:

(1)

I -→ f n 2 n × I -→ g n 2 n × [0, x -n ] -→ f n 2 n × I.
Observe that after time 2n, the length of the curve

g n • f n is 2 n • x -n = (1 + α) -n+1
whereas the number of (ǫ, n)-separated orbits is less than ǫ -1 n2 n . After time 3n, the curve f n • g n • f n has image I with multiplicity 2 n . It is therefore easy to analyze the dynamics of compositions of such sequences. We build our example by considering a skew-product for which the curve will be a fiber over a point which will drive the application of sequences as above.

Let h : S 1 → S 1 be the circle map defined by h(θ) = 4θ mod 2π. Let F : 1 6 ] and

S 1 × I → S 1 × I be a C ∞ map such that: F (θ, x) = (h(θ), f (x)) if θ ∈ [0,
F (θ, y) = (h(θ), g(x)) if θ ∈ [ 1 2 , 2 3 ] Recall that the expansion in basis 4 of θ ∈ S 1 is the sequence a 1 a 2 • • • ∈ {0, 1, 2, 3} N such that θ = 2π k≥1 a k 4 -k . We write θ = 0.a 1 a 2 a 3 . . . 4 .
Observe that, whenever θ has only 0s and 2s in its expansion,

h n (θ) ∈ [0, 1 6 ] = [0, 0.02222 . . . 4 ]
whenever its nth digit is 0 and

h n (θ) ∈ [ 1 2 , 2 3 ] = [0.2000 . . . 4 , 0.222 . . . 4 ]
whenever its nth digit is 2. Thus we can specify the desired compositions of f and g just by picking θ ∈ S 1 with the right expansion. We pick:

θ 1 = 0.0 n1 2 n1 0 n1+n2 2 n2 0 n2+n3 2 n3 0 n3+n4 .... 4
so that we shall have a sequence of compositions of the type (1). We write

N i := 3n 1 + • • • + 3n i . We set n i := i! so that n i+1 /N i → ∞. Let φ 1 : Q 1 → S 1 × I be defined by φ 1 (s) = (θ 1 , (s + 1)/2).
The previous analysis shows that

F Ni •φ 1 has image I with multiplicity 2 n1+•••+ni = 2 1 3 Ni . F Ni+ni+1 • φ 1 has image I with multiplicity 2 1 3 Ni × 2 ni+1 . F Ni+ni+1 • φ 1 has image [0, x -ni+1 ] with multiplicity 2 2 3 Ni × 2 ni+1 . It follows that, setting t i := N i + 2n i+1 ≈ 2n i+1 , log r F (ǫ, t i , φ 1 ) ≈ ( 1 3 N i + n i+1 ) log 2 whereas vol(F ti • φ 1 ) = x -ni+1 × 2 1 3 Ni+ni+1 = (1 + α) -ni+1 2 1 3 Ni . Hence, 1 t i log r F (ǫ, t i , φ 1 ) ≈ 1 2 log 2 whereas 1 t i log vol(F ti • φ 1 ) - 1 2 α,
as claimed.

Remark 14. The inequality in the previous example is obtained as the length is contracted after a large expansion. For curves, this is in fact general and it is easily shown that, for any C 1 1-disk φ with unit length, for any 0 < ǫ < 1:

(2)

∀n ≥ 0 ǫ • r f (ǫ, n, φ) ≤ max 0≤k<n vol(f k • φ) + 1.
(2) implies that, for curves,

(3) h top (f, φ) ≤ γ(f, φ),
both quantities being defined by lim sup (this would fail using lim inf). However, one can find similarly as above, a C ∞ self-map of a 3-dimensional compact manifold and a C ∞ smooth 2-disk such that (2) fails though (3) seems to hold.

We ask the following:

Question. Let f : M → M be a C ∞ self-map of a compact d-dimensional manifold.
Is it true that, for any singular k-disk ψ (0

≤ k ≤ d) h top (f, ψ) ≤ max φ⊂ψ γ(f, φ) ?
(both rates being defined using lim sup and φ ranging over singular ℓ-disks,

0 ≤ ℓ < k, with φ(Q ℓ ) ⊂ ψ(Q k ))? Is it at least true that h k top (f ) ≤ max 0≤ℓ≤k γ ℓ (f ) ?
These might even hold for finite smoothness for all I know.

Conversely, entropy also provides some bounds on volume growth Theorem 7 (Yomdin [START_REF] Yomdin | Volume growth and entropy[END_REF]). Let f : M → M be a C r , r ≥ 1, smooth self-map of a compact manifold. Let α > 0. Then there exist C(r, α) < ∞ and ǫ 0 (r) > 0 with the following property. Let φ :

Q k → M be any C r singular k-disk with unit C r size for some 0 ≤ k ≤ d. Then, for any n ≥ 0, vol(f n • φ) ≤ C(r, α)Lip(f ) ( k r +α)n • r f (ǫ 0 , n, φ).
In particular,

γ k (f ) ≤ h k top (f ) + k r lip(f ).
Remark 15. The above extra term is indeed necessary as shown already by examples attributed by Yomdin [START_REF] Yomdin | Volume growth and entropy[END_REF] to Margulis: there is f : [0, 1] → [0, 1], C r with h top (f ) = 0 and γ(f ) = lip(f )/r.

Remark 16. Yomdin's estimate is uniform holding for each disk and each iterate. Its proof involves very little dynamics and no ergodic theory, in contrast to

Newhouse's inequality quoted above.

Corollary 17. Let f : M → M be a self-map of a compact manifold. If f is C ∞ , then h top (f ) = γ(f ).
Let f * : H * (M, R) → H * (M, R) be the total homological action of f . Let ρ(f * ) be its spectral radius. As the ℓ 1 -norm in homology gives a lower bound on the volume, we have γ(f ) ≥ log ρ(f * ). Hence, the following special case of the Shub Entropy Conjecture is proved:

Corollary 18 (Yomdin [27]). Let f : M → M be a self-map of a compact manifold. If f is C ∞ , then log ρ(f * ) ≤ h top (f ).

Resolution entropies.

The previous results of Yomdin and more can be obtained by computing a growth rate taking into account the full structure of singular disks. A variant of this idea is explained in Gromov's Bourbaki Seminar [START_REF] Gromov | Entropy, homology and semialgebraic geometry[END_REF] on Yomdin's results. We build on [START_REF] Buzzi | Représentation markovienne des applications régulières de l'intervalle[END_REF].

Definition 19. Let r ≥ 1. Let φ : Q k → M be a C r singular k-disk. A C r - resolution R of order n of φ is a collection of C r maps ψ ω : Q k → Q k , for ω ∈ Ω
with Ω a finite collection of words of length at most n with the following properties. For each ω ∈ Ω, let

Ψ ω := ψ σ |ω|-1 ω • • • • • ψ ω (Q k ). We require: (1) |ω|=n Ψ ω (Q k ) = Q k ; (2) ψ ω C r ≤ 1 for all ω ∈ Ω; (3) f |ω| • Ψ ω C r ≤ 1 for all ω ∈ Ω,
The size |R| of the resolution is the number of words in Ω with length n.

Condition (2) added in [START_REF] Buzzi | Représentation markovienne des applications régulières de l'intervalle[END_REF] much simplifies the link between resolutions and entropy. It no longer relies on Newhouse application of Pesin theory and becomes straightforward:

Fact 20. Let R := {ψ ω : Q k → Q k : ω ∈ Ω} be a C r -resolution of order n of φ : Q k → M . Let ǫ > 0 and Q k ǫ be ǫ-dense in Q k , i.e., Q k ⊂ t∈Q k ǫ B(x, ǫ). Then {Ψ ω (t) : t ∈ Q k ǫ and ω ∈ Ω with |ω| = n} is a (ǫ, n)-cover of φ(Q k ).
On the other hand, the notion of resolution induces entropy-like quantities:

Definition 21. Let 1 ≤ r < ∞ and let f : M → M be a C r self-map of a compact manifold. Let R f (C r , n, φ
) be the minimal size of a C r -resolution of order n of a C r singular disk φ. The resolution entropy of φ is:

h R (f, φ) := lim sup n→∞ 1 n log R f (C r , n, φ).
If D is a collection of C r singular disks, its C r resolution entropy is

h R,C r (f, D) := sup φ∈D h R (f, φ)
and its C r uniform resolution entropy is:

H R,C r (f, D) := lim sup n→∞ 1 n log R f (C r , n, D)
where R f (C r , n, D) := max φ∈D R f (C r , n, φ). We set:

h k,C r R (f ) = h R,C r (f, D k r ) and H k,C r R (f ) = H R,C r (f, D k r ).
The following is immediate but very important:

Fact 22. Let 1 ≤ r < ∞ and 0 ≤ k ≤ d. Let f : M → M be a C r self-map of a compact d-dimensional manifold. The sequence n → R f (C r , n, D k r ) is sub- multiplicative: R f (C r , n + m, D k r ) ≤ R f (C r , n, D k r )R f (C r , m, D k r ).
The key technical result of Yomdin's theory can be formulated as follows:

Proposition 23. Let 1 ≤ r < ∞ and α > 0. Let f : M → M be a C r self-map of a compact manifold. There exist constants C ′ , C(r, α), ǫ 0 (r, α) with the following property. For any C r singular disk φ, any number 0 < ǫ < ǫ 0 (r, α) and any integer n ≥ 1,

C ′ ǫ k r f (ǫ, n, φ) ≤ R f (C r , n, φ) ≤ C(r, α)Lip(f ) ( k r +α)n r f (ǫ, n, φ
). Remark that the above constants depend on f . The first inequality follows from Fact 20. The second is the core of Yomdin theory, we refer to [START_REF] Buzzi | Représentation markovienne des applications régulières de l'intervalle[END_REF] for details.

Properties of Dimensional Entropies

We turn to various properties of dimensional entropies, most of which can be shown using resolution entropy and its submultiplicativity.

5.1. Link between Topological and Resolution Entropies. We start by observing that Proposition 23 links the topological and resolution entropies.

Corollary 24. For all positive integers r, k, any collection of C r k-disks D and any C r self-map f on a manifold equipped with a C r size:

h top (f, D) ≤ h R,C r (f, D) ≤ h top (f, D) + k r log Lip(f ) H top (f, D) ≤ H R,C r (f, D) ≤ H top (f, D) + k r log Lip(f ). If the disks in D are C ∞ , then, for r ≤ s < ∞, h R,C r (f, D) ≤ h R,C s (f, D) ≤ h top (f, D) + k s log Lip(f ).
Letting s → ∞, we get:

Corollary 25. If f is C ∞ , then, for all 1 ≤ r < ∞, h R,C r (f, D k ∞ ) = h top (f, D k ∞ )
. The same holds for uniform topological entropy.

Gap between Uniform and Ordinary Dimensional Entropies. Yomdin theory gives the following relation:

Proposition 26. Let 1 ≤ r < ∞ and f : M → M be a C r self-map of a compact d-dimensional manifold. For each 0 ≤ k ≤ d, h k,C r top (f ) ≤ H k,C r top (f ) ≤ h k,C r top (f ) + k r lip(f )
and the same holds for the resolution entropies

h k,C r R (f ) and H k,C r R (f ).
In particular, in the C ∞ smooth case, all the versions of the dimensional entropies agree:

h k top (f ) = H k top (f ) = h k R (f ) = H k R (f ) = lim r→∞ H k,C r top (f ). Proof. It is obvious that the uniform entropies dominate ordinary ones. By Fact 20, h k,C r top (f ) ≤ h k,C r R (f ) and H k,C r top (f ) ≤ H k,C r R (f ). Therefore it is enough to show: (4) H k,C r R (f ) ≤ h k,C r top (f ) + k r lip(f ).
Let α > 0. Let ǫ 0 > 0 as in Proposition 23. This proposition defines a number C(r, α). By definition, for every φ ∈ D k r , there exists n φ < ∞ such that

r f (ǫ 0 , n φ , φ) ≤ e (h k,C r top (f )+α)n φ .
We can arrange it so that this holds for all k-disks ψ in some C 0 neighborhood U φ of φ. We also assume n φ so large that C(r, α) ≤ e αn φ . By Proposition 23, each such ψ admits a resolution with size at most

r f (ǫ, n φ , ψ) × C(r, α)Lip(f ) k r n φ ≤ e (h k,C r R (f )+ k r lip(f )+2α)n φ .
D k r is relatively compact in the C 0 topology, hence there is a finite cover

D k r ⊂ U φ1 ∪ • • • ∪ U φK . Let N := max n φj .
It is now easy to build, for each n ≥ 0 and each ψ ∈ D k r a C r resolution R of order n with:

|R| ≤ exp(h k,C r top (f ) + k r lip(f ) + 2α)(n + N ).
(4) follows by letting α go to zero.

Continuity properties.

Proposition 27. We have the following upper semicontinuity properties:

(1) f → H k,C r R (f ) is upper semicontinuous in the C r topology for all 1 ≤ r < ∞; (2) f → H k top (f ) is upper semicontinuous in the C ∞ topology; (3) the defect in upper semi-continuity of f → H k,C r top (f ) at f = f 0 is at most k r lip(f 0 ): lim sup f →f0 H k,C r top (f ) ≤ H k,C r top (f 0 ) + k r lip(f 0 ).
Proof. We prove [START_REF] Bonatti | Dynamics beyond uniform hyperbolicity[END_REF]. The sub-multiplicativity of resolution numbers observed in Fact 22 implies that:

H k,C r R (f ) = inf n≥1 1 n log R f (C r , n, D k r ). For each fixed posi- tive integer n, R g (C r , n, D k r ) ≤ 2 k R f (C r , n, D k r )
for any g C r -close to f (use a linear subdivision). Thus f → H k,C r R (f ) is upper semi-continuous in the C r topology. We deduce (3) from [START_REF] Bonatti | Dynamics beyond uniform hyperbolicity[END_REF]. Let f n → f in the C r topology. By the preceding,

H k,C r R (f ) ≥ lim sup n→∞ H k,C r R (f n ). By Proposition 26, H k top (f ) ≥ H k,C r R (f ) - k r lip(f ).
(2) follows from (3) using Lemma 10.

On the other hand, f → H k top (f ) fails to be lower semi-continuous except for interval maps for which topological entropy is lower semi-continuous in the C 0 topology by a result of Misiurewicz. In every case there are counter examples:

Example 4. For any d ≥ 2 and 1 ≤ k ≤ d, there is a self-map of a compact manifold of dimension d at which h k top (f ) fails to be lower semi-continuous. Let h : R → [0, 1] be a C ∞ function such that h(t) = 1 if and only if t = 0. Let F λ : [0, 1] d → [0, 1] d be defined by F λ (x 1 , . . . , x d ) = (h(λ)x 1 , 4x 1 x 2 (1 -x 2 ), x 3 , . . . , x d ) Observe that if λ = 0, then h(λ) ∈ [0, 1) and F n λ (x 1 , . . . , x d ) approaches {(0, 0)} × [0, 1] d-2 on which F λ is the identity. Therefore h top (F λ ) = 0. On the other hand h top (F 0 ) = h top (x → 4x(1 -x)) = log 2. Now, H k top (F λ ) ≤ h top (F λ ) = 0 for any λ = 0 and H k top (F 0 ) ≥ h k top (f ) ≥ h top (F 0 , {1} × [0, 1] × {(0, . . . , 0)}) = log 2 for any k ≥ 1.

Hyperbolicity from Entropies

We now explain how the dimensional entropies can yield dynamical consequences. We start by recalling an inequality which will yield hyperbolicity at the level of measures. Then we give the definition and main results for entropy-expanding maps. Finally we explain the new notion of entropy-hyperbolicity for diffeomorphisms. 6.1. A Ruelle-Newhouse type inequality. One of the key uses of dimensional entropies is to give bounds on the exponents using the following estimate. This will give hyperbolicity of large entropy measure from assumptions on these dimensional entropies.

Theorem 8. [START_REF] Buzzi | On entropy-expanding maps[END_REF] Let f : M → M be a C r self-map of a compact manifold with r > 1. Let µ be an ergodic, invariant probability measure with Lyapunov exponents

λ 1 (µ) ≥ λ 2 (µ) ≥ • • • ≥ λ d (µ) repeated according to multiplicity. Recall that H k top (f ) is the uniform k-dimensional entropy of f . Then: h(f, µ) ≤ H k top (f ) + λ k+1 (µ) + + • • • + λ d (µ) + .
Remark 28. For k = 0 this reduces to Ruelle's inequality. For k equal to the number of nonnegative exponents, this is close to Newhouse inequality (with H k top (f ) replacing γ k (f )). The proof is similar to Newhouse's and relies on Pesin theory. 6.2. Entropy-expanding Maps. We require that the full topological entropy only appear at the full dimension.

Definition 29. A C r self-map f : M → M of a compact manifold is entropy- expanding if: H d-1 top (f ) < h top (f
). An immediate class of examples is provided by the interval maps with non-zero topological entropy.

The first consequence of this condition is that ergodic invariant probability measures with entropy > H d-1 top (f ) have only Lyapunov exponents bounded away from zero. This follows immediately from Theorem 8.

This also allows the application of (a non-invertible version of) Katok's theorem, proving a logarithmic lower bound on the number of periodic points.

Katok's proof gives horseshoes with topological entropy approaching h top (f ). In particular these maps are points of lower semi-continuity of f → h top (f ) in any C r topology, r ≥ 0. Combining with the upper semi-continuity from Yomdin theory we get:

Proposition 30. The entropy-expansion property is open in the C ∞ topology.

Thus we can use the following estimate Proposition 31. [START_REF] Buzzi | Ergodicité intrinsèque de produits fibrés d'applications chaotiques unidimensionelles[END_REF] The Cartesian product of a finite number of C ∞ smooth interval maps, each with nonzero topological entropy is entropy-expanding.

To get dynamically interesting examples:

Example 5. For |ǫ| small enough, the plane map F ǫ : (x, y) → (1 -1.8x 2 -ǫy 2 , 1 -1.9y 2 -ǫx 2 ) preserves [-1, 1] 2 and its restriction to this set is entropy-expanding.

A sufficient condition, considered in a different approach by Oliveira and Viana [START_REF] Oliveira | Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms[END_REF][START_REF] Oliveira | Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps[END_REF] is the following: Lemma 32. Let f : M → M be a diffeomorphism of a compact Riemanian manifold. Let Λ k T f be the maximum over all 1 ≤ ℓ ≤ k and all x ∈ M of the Jacobian of the restrictions of the differential T x f to any k-dimensional subspace of T x M . Then H k top (f ) ≤ log Λ k T f . In particular, log Λ k T < h top (f ) implies that f is entropy-expanding. An even stronger condition is (d -1)lip(f ) < h top (f ).

The proof of this lemma is a variation on the classical proof of Ruelle's inequality.

We are able to analyze the dynamics of entropy-expanding maps with respect to large entropy measures rather completely.

Theorem 9. Let f : M → M be a C ∞ self-map of a compact manifold. Assume that f is entropy-expanding. Then:

• f has finitely many maximum measures;

• its periodic points satisfies a multiplicative lower bound.

This can be understood as generalization of the Markov property which corresponds to partition having boundaries with essentially finite forward or backward orbits. The proof of the theorem involves a partition whose boundaries are pieces of smooth submanifolds, therefore of entropy bounded by H d-1 top (f ). In [START_REF] Buzzi | Puzzles of Quasi-Finite Type, Symbolic Dynamics and Zeta Functions[END_REF], we are able to define a nice class of symbolic systems, called puzzles of quasi-finite type, which contains the suitably defined symbolic representations of entropy-expanding maps satisfying a technical condition and have the above properties. Moreover, their periodic points define zeta functions with meromorphic extensions and their natural extensions can be classified up to entropy-conjugacy in the same way as interval maps.

6.3. Entropy-Hyperbolicity. Entropy-expanding maps are never diffeomorphisms. Indeed, they have ergodic invariant measures which have nonzero entropy and only positive Lyapunov exponents. Wrt the inverse diffeomorphism these measures have the same nonzero entropy but only negative Lyapunov exponents, contradicting Ruelle's inequality. Thus we need a different notion for diffeomorphism.

Definition 33. The unstable (entropy) dimension is:

d u (f ) := min{0 ≤ k ≤ d : H k top (f ) = h top (f )}. If f is a diffeomorphism, then the stable dimension is: d s (f ) := d u (f -1 ) (if f not a diffeomorphism we set d s (f ) = 0).
Observe that f is entropy-expanding if and only if d u (f ) coincides with the dimension of the manifold.

Lemma 34. Let f : M → M be a C r self-map of a compact d-dimensional manifold with r > 1. Then: d u (f ) + d s (f ) ≤ d.
Proof. Theorem 8 implies that measures with entropy > H du(f )-1 top (f ) have at least d u (f ) positive exponents. The same reasoning applied to f -1 shows that such measures have at least d s (f ) negative exponents. By the variational principle such measures exist. Hence d u (f ) + d s (f ) ≤ d. 7.2. Dimensional Entropies of Examples. Let f i : M i → M i are smooth maps for i = 1, . . . , n and consider the following formula:

H k top (f 1 × • • • × f n ) = max ℓ1+•••+ℓn=k H ℓ1 top (f ) + • • • + H ℓn top (f ).
This is only known in special cases -see [START_REF] Burguet | Entropie des systèmes dynamiques différentiables[END_REF]. It would imply that product of entropyexpanding maps are again entropy-expanding.

If f : M → M is an expanding map of a compact manifold, is it true that H d-1 top (f ) < h top (f ). Note that this fails for piecewise expanding maps (think of a limit set containing an isolated invariant curve with maximum entropy).

Likewise is an Anosov diffeomorphism, even far from linear, entropy-hyperbolic?

Find examples where h k,C r top (f ) < H k,C r top (f ). 7.3. Other types of dimensional complexity. Other "dimensional complexities" have been investigated from growth rates of multi(co)vectors for the Kozlovski entropy formula [START_REF] Kozlovski | An integral formula for topological entropy of C ∞ maps Ergodic Theory Dynam[END_REF] to the currents which are fundamental to multidimensional complex dynamics [START_REF] Gromov | On the entropy of holomorphic maps[END_REF] and the references therein.

How do they relate to the above dimensional entropies? 7.4. Necessity of Topological Assumptions. We have seen in Sect. 2.1 that, for maps, the assumption of no zero Lyapunov exponent for the large entropy measure, (or even that these exponents are bounded away from zero) is not enough for our purposes (e.g., finiteness of the number of maximum measures). Such results seem to require more uniform assumptions, like the one we make on dimensional entropies. Is it the same for diffeomorphisms? That is, can one find diffeomorphisms with infinitely many maximum measures, all with exponents bounded away from zero?

Let f be a C r self-map of a compact manifold. Assume that there are numbers h < h top (f ), λ > 0, such that the Lyapunov exponents of any ergodic invariant measure with entropy at least h fall outside of [-λ, λ]. Assume also that the set of invariant probability measures with entropy ≥ h is compact. Does it follow that there are only finitely many maximum measures? 7.5. Entropy-Hyperbolicity. In a work in progress with T. Fisher, we show that the condition of Lemma 38 is satisfied by a version of a well-known example of robustly transitive, non-uniformly hyperbolic diffeomorphism of T 4 due to Bonatti and Viana [START_REF] Bonatti | Dynamics beyond uniform hyperbolicity[END_REF]. Building nice center-stable and center-unstable invariant laminations, we expect be able to show the same properties as in Theorem 9.

I however conjecture that the finite number of maximum measures, etc. should in fact hold for every C ∞ entropy-expanding diffeomorphism, even when there is no such nice laminations. Of course this contains the case of surface diffeomorphisms which is still open (see Conjecture 1), despite the result on a toy model [START_REF] Buzzi | Measures of Maximum Entropy for Piecewise Affine Surface Homeomorphisms[END_REF]. 7.6. Generalized Entropy-Hyperbolicity. It would be interesting to have a more general notion of entropy-hyperbolicity. For instance, if a hyperbolic toral automorphism is entropy-hyperbolic, this is not the case for the disjoint union of two such systems of the same dimenion if they have distinct stable dimensions. It may be possible to "localize" the definition either near points or near invariant measures to avoid these stupid obstructions (this is one motivation for the above question on variational principles for dimensional entropies).

If one could remove such obstructions, the remaining ones could reflect basic dynamical phenomena opening the door to a speculative "entropic Palis program". where the above partial derivatives are computed at t = χ -1 i (x) and π j (u 1 , . . . , u d ) = u j .

Fact 39. If • C r and • ′ C r are two C r size defined by the above procedure, there exists a constant K such that, for any C r k-disk φ : Q k → M :

φ C r ≤ K • φ ′ C r .
Fact 40. Let φ : Q k → M be a C r disk for some finite r ≥ 1. Then, for any t 0 ∈ Q k , there exists a C ∞ approximation φ ∞ : Q k → M such that:

∀t ∈ Q k d(φ ∞ (t), φ(t)) ≤ φ C r t -t 0 r . and φ ∞ C ∞ ≤ 2 φ C r .
This is easily shown by considering a neighborhood of φ(t 0 ) contained in a single chart of A and approximating φ by its Taylor expansion in that chart.

  Appendix A. C r sizes We explain how to measure the C r size of singular disks of a compact manifold M of dimension d. Here 1 ≤ r ≤ ∞.We select a finite atlas A made of charts χ i :U i ⊂ R d → M such that changes of coordinates χ -1 i • χ j are C r -diffeomorphisms of open subsets of R d . Then we define, for any singular k-disk φ:φ C r := sup x∈M inf Ui∋x max s1+•••+s k ≤r max 1≤j≤d |∂ s1 t1 . . . ∂ s k t k (π j • χ i • φ)(t 1 , . . . , t k )|

  Volume growth. Entropy is a growth rate under iteration. Equipping M with a Riemannian structure allows the definition of volume growth of submanifolds.

	The claimed equality follow in
	turn by symmetry.
	4. Other growth rates of submanifolds
	4.1.

Definition 12. Let φ : Q k → M be a singular k-disk. Its (upper) growth rate is:

That is, each φ j 2 = φ 1 • L j with L j : Q k → Q k linear and K j=1 L j (Q k ) = Q k .

We can now propose our definition: Definition 35. A diffeomorphism such that d u (f )+d s (f ) = d is entropy-hyperbolic.

Obviously surface diffeomorphisms with non-zero topological entropy are entropyhyperbolic.

Exactly as above, we obtain from Theorems 8 and 4: Theorem 10. Let f : M → M be a C r diffeomorphism of some compact manifold with r > 1. Assume that f is entropy-hyperbolic. Then:

Then f is entropy-hyperbolic.

Further directions and Questions

We discuss some developping directions and ask some questions.

7.1. Variational Principles. It seems reasonable to conjecture the following topological variational principle for dimensional entropies, at least for C ∞ self-maps and diffeomorphisms:

In each dimension, there is a C ∞ disk with maximum topological entropy, i.e., h k top (f ). Does it fail for finite smoothness?

A probably more interesting but more delicate direction would be an ergodic variational principle. Even its formulation is not completely clear. A possibility would be as follows:

For each dimension k, h k top (f ) is the supremum of the entropies of k-disks contained in unstable manifolds of points in any set of full measure with respect to all invariant probability measures.


Sofiane Khadraoui
email: sofiane.khadraoui@femto-st.fr

Member, IEEE Micky Rakotondrabe

Member, IEEE Philippe Lutz
email: plutz@femto-st.fr

Robust control for a class of interval model: application to the force control of piezoelectric cantilevers

Keywords: Robust performances, Interval models, controller design, piezocantilevers, manipulation force

A method to design robust performances controllers is proposed in this paper. The method is valuable for a class of uncertain parametric systems: systems with zero-order numerator.

In this work, interval model that accounts parametric uncertainties is described using interval analysis. Robust controller is derived by combining the interval arithmetic with the classical direct synthesis. As the derived controller is also an interval, we prove using numerical analysis that the midpoint can be taken as the final controller to be implemented. The proposed design method is applied to the control of manipulation force in piezocantilevers where the compliance of the manipulated objects is uncertain. The experimental results show the efficiency of the proposed method.

I. INTRODUCTION

Smart materials are widely used in the development of microsystems and microrobots. Piezoelectric materials are among the most appreciated because of their high resolution and rapidity [START_REF] Devasia | A survey of control issues in nanopositioning[END_REF]. One of the main applications of piezoelectric materials in microsystems is piezoelectric microgrippers dedicated to micromanipulation and microassembly tasks [START_REF] Haddab | A microgripper using smart piezoelectric actuators[END_REF] [START_REF] Agnus | A smart microrobot on chip: design, identification and modelingŠ[END_REF]. A piezoelectric microgripper is composed of two piezocantilevers Fig. 1. They are used to pick, transport and place micro-objects with high positioning accuracy. During the manipulation of micro-objects, especially biological and optical materials that are usually fragile, pure position control is usually not adequate. Force control is often needed to prevent damage of the manipulated microobjects and also to well keep them in the microgripper. For that, the first piezocantilever of the microgripper must be controlled on position while the second one on force [START_REF] Rakotondrabe | Modelling and H∞ force control of a nonlinear piezoelectric cantilever[END_REF] [START_REF] Rakotondrabe | Modelling and robust position/force control of a piezoelectric microgripper[END_REF].

In order to synthesize a controller for the manipulation force, a model is necessary. However, it is known that the model linking this manipulation force and the input control is dependent on the micro-object characteristics [START_REF] Rakotondrabe | Modelling and H∞ force control of a nonlinear piezoelectric cantilever[END_REF]. Therefore, it is necessary to have a model that accounts the microobjects characteristics. In many cases, such characteristics are known only in a range of variation. A controller must be therefore computed to ensure robust performances for all micro-objects characterized inside this range.

To ensure robust perfomances for uncertain systems, robust controllers synthesis which accounts model uncertainties is needed. Among these controllers, we find the robust control laws H 2 , µ-synthesis and H ∞ that have been used to control microsystems, examples are given in [START_REF] Sebastian | Achieving Subnanometer Precision in a MEMS-Based Storage Device During Self-Servo Write Process[END_REF] [START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF]. The efficiency of these control methods is proved in several applications while its major disadvantage is the derivation of high-order controllers which are time consuming and limit the embedding possibilities. A possible alternative to these classical robust control laws is to combine classical techniques of linear control design with interval analysis to construct simple robust controllers. Indeed, contrarily to scalar points, intervals are values that represent a set of real numbers that can be used to bound parametric uncertainties.

The first idea on interval arithmetic has appeared in 1924 by Burkill and 1931 by Young, then later in 1966 with R.E. Moore's works [START_REF] Moore | Interval Analysis[END_REF]. Since, several applications on interval analysis have been raised. Many of them relates to guaranteed estimation and robust stability while few concerns the design of controllers ensuring performances. In [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF], Jaulin and Walter have proposed guaranteed parameters estimation based on the SIVIA algorithm (Set Inversion Via Interval Analysis). The works in [START_REF] Braems | Guaranteed estimation of electrochemical parameters by set inversion using interval analysis[END_REF] [START_REF] Jaulin | Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis[END_REF][12] utilize a linear model characterized by a bounded error equation (interval) in order to define the values set of parameters compatible with the measurements, model and error bounds. In [START_REF] Jaulin | Applied Interval Analysis[END_REF][13], the stability analysis of the closed-loop with a given controller was proposed using the Routh's criteria and the Kharithonov's theorem. Concerning the design of controller, [START_REF] Ye | Using interval arithmetic for robust state feedback design[END_REF] proposed an approach of state feedback control combined with the intervals for the parameters model to synthesize a controller that ensures the stability. In [START_REF] Bondia | Guaranteed tuning of PID controllers for parametric uncertain systems[END_REF], an approach of robust controller design based on the inclusion of the interval closed-loop transfer inside an interval transfer defining the wanted performances have been proposed. The computation of the controller is formulated as a set-inclusion problem, where a check of the inclusions satisfaction for a set of frequencies is needed. In [START_REF] Chen | A two-degrees-of-freedom design methodology for interval process systems[END_REF], robust controller synthesis approach guaranteeing both the robust stability and performances for an interval system is addressed. In the latter work, the computation of two controllers is required, a robust stabilizing feedback controller is firstly computed. Subsequently, a pre-filter is constructed to ensure the wanted performances. [START_REF] Li | Interval Model Control of Consumable Double-Electrode Gas Metal Arc Welding Process[END_REF] proposed a control algorithm predictionbased interval model and its application to a welding process.

The main objective of this paper is the design of a robust controller ensuring performances and valuable for systems with multiple uncertain parameters. The proposed method is valuable for a class of uncertain systems: systems with zeroorder numerator. In addition to its principle simplicity, the main advantage is the derivation of a low order controller. For that, we combine direct controller synthesis with the interval analysis.

The paper is organized as follows. First the interval computation and its basic terms are presented. Afterwards, we introduce the direct synthesis method to compute an interval controller using the wanted performances and the interval system model. To prove the efficiency of the proposed method, an application on the force control of piezocantilevers has been presented. Finally, the experimental results end the paper.

II. BASIC TERMS AND CONCEPTS ON INTERVALS

In this section, we remind interval arithmetic (see [START_REF] Moore | Interval Analysis[END_REF] for more details).

A. Definitions

A closed interval, denoted by [x] , is the set of real numbers given by:

[x] = [x -, x + ] = x ∈ R/x -≤ x ≤ x + (1)
The endpoints x -and x + are respectively the left and right endpoint of [x].

The width of an interval [x] is given by:

w([x]) = x + -x - (2)
The midpoint of [x] is given by:

mid([x]) = x + + x - 2 (3)
The radius of [x] is defined by:

rad([x]) = x + -x - 2 (4)

B. Operations on intervals

The elementary mathematical operations are extended to intervals, the operation result between two intervals is an interval containing all the operations results of all pairs of numbers in the two intervals.

Given two intervals [x] = [x -, x + ] , [y] = [y -, y + ] and • ∈ {+, -, ., /} , we can write that:

[x] • [y] = {x • y |x ∈ [x], y ∈ [y] } (5)

C. Interval system

Parametric Uncertain systems can be modeled by interval systems. An interval system denoted [G](s, [p], [q]) is a system where [p] and [q] are two boxes of interval numbers:

[G](s, [p], [q]) = m j=0 [qj ]s j n i=0 [pi]s i =          m j=0 qj s j n i=0 pis i q j ∈ [q - j , q + j ] p i ∈ [p - i , p + i ]          (6) with: [q] = [[q 1 ], ..., [q n ]] and [p] = [[p 1 ], ..., [p n ]].

III. COMPUTATION OF THE CONTROLLER

The objectif of this section is to derive robust control law by combining the previous interval analysis with a linear theory of control called direct synthesis of controller. The proposed method is suitable for single-input single-output interval systems.

A. Closed-loop control

Given

B. Definition of the system and of the wanted closed-loop model

In this work, we aim to design robust control law for a class of interval plants. Define the following interval system with zero-order numerator:

[G](s, [a]) = 1 n i=0 [a i ]s i (7)
Such as:

[a] = [[a 0 ], ..., [a n ]] is a box of uncertain parame- ters.
Remark 1: The effect of disturbances can be assumed to be in the range of interval in which parameter lies.

From given specifications (generally given in terms of settling time, static error, small overshoot,...etc), an interval closed-loop model denoted [H](s, [b]) is also derived. It is known as a reference interval model and is given by:

C. Direct synthesis of the controller

The main idea of this method consists to compute a controller [C](s) from the wanted closed-loop transfer The closed-loop transfer is defined by:

[H](s, [b]) = [C](s)[G](s, [a]) 1 + [C](s)[G](s, [a]) (9)
From ( 9), we derive the interval controller [C](s):

[C](s) = 1 [G](s, [a]) 1 [H](s,[b]) -1 (10)
As the controller contains 1/[G](s, [a]), the method can be classified as a compensation technique.

Based on the interval arithmetic presented in section. II, the following interval controller is derived after introducing (8) in [START_REF] Braems | Guaranteed estimation of electrochemical parameters by set inversion using interval analysis[END_REF] and replacing [G](s, [a]) by ( 7):

[C](s, [a], [b]) = n i=0 [a i ]s i m j=0 [b j ]s j -1 (11)
The causality of the controller is obtained when:

m ≥ n (12)
Note that, there exist a set of controllers inside the interval controller defined in [START_REF] Jaulin | Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis[END_REF] that guarantee the required performances. The robustness of such controller to ensure performances depends on the wanted specifications and the interval width of the parameters of the uncertain system.

In the last section, using the SIVIA algorithm, we demonstrate that the midpoint controller is one among the possible solutions.

IV. APPLICATION TO PIEZOCANTILEVERS

The aim of this section is to apply the proposed method for controlling the manipulation force of piezocantilevers on different micro-objects. Instead of manipulating microobjects, we manipulate beams (cantilevers) with known compliances. A piezocantilever is a cantilever based on piezoelectric layers. Let (Fig. 3) shows a piezocantilever with rectangular cross-section and based one passive layer and one piezoelectric layer. It is also called unimorph piezocantilever. When a voltage U is applied to the piezolayer, it contracts or expands. As a result, the whole cantilever bends with a deflection δ.

A. presentation of the setup

The experimental setup Fig. 4 is composed of:

• A unimorph piezocantilever with the dimensions L × b × h = 15mm × 2mm × 0.3mm,
• Two beams having different compliances, the first beam is rigid while the second one is flexible. The principle is shown in Fig. 4-a,

• a Keyence optical sensor with 10nm of resolution is used to measure the piezocantilever deflection. We utilize a computer-DSpace hardware combined with the Matlab-Simulink software for the implementation of the controller. B. Definition of the interval model 1) Modelling: According to the works in [START_REF] Rakotondrabe | Modelling and H∞ force control of a nonlinear piezoelectric cantilever[END_REF][5], the linear relation that relates the input voltage U applied to a piezocantilever, the force applied to the manipulated microobject and the resulting deflection δ (see Fig. 5) can be written as follows: where s p > 0 is the elastic constant of the piezocantilever (in our case s p = 2µm/mN ), d p > 0 is the piezoelectric constant and D(s) is the transfer function representing the dynamic part of the piezocantilever (with D(0) = 1). D(s) and d p have to be identified.

δ = (d p .U -s p .F m ).D(s) (13)
Note that ( 13) is correct when the manipulator (piezocantilever) is in contact with the micro-object.

As demonstrated in [START_REF] Rakotondrabe | Modelling and H∞ force control of a nonlinear piezoelectric cantilever[END_REF], the relation between the manipulation force, the deflection and the characteristics of the micro-object can be written as follows:

δ = s o .F m .D o (s) (14)
where s o > 0 represents the micro-object compliance and D o (s) is its dynamic part.

In our work, we neglect the dynamic part of the microobject, i.e D o (s) = 1. This assumption is valuable when the compliance of the manipulated objects are not too high, and when its mass is negligible which is the case for microobjects. Therefore we have:

δ = s o .F m (15)
Finally, after replacing the deflection δ in (13) by that of (15), we obtain the linear transfert voltage-force that takes into account the micro-object characteristics:

G(s) = F m U = d p D(s) s o + s p D(s) (16)
2) Identification of D(s) and of piezoelectric constant d p : In order to identify the dynamic part D(s) and the piezoelectric constant d p , a step voltage with 20V of amplitude is applied to the piezocantilever without contact with any objects. A second order model can sufficiently model a lot of system particularly piezocantilevered structures. Using the output error method and the matlab software, the model d p .D(s) was identified and plotted with the experimental result (Fig. 6). This shows the adequacy between both. D(s) = 1 6.627 × 10 -8 s 2 + 5.268 × 10 -6 s + 1 [START_REF] Li | Interval Model Control of Consumable Double-Electrode Gas Metal Arc Welding Process[END_REF] and d p = 0.6533µm/V . Fig. 6.

Step response of the piezocantilever: the measured deflection compared with the simulation of dp.D(s).

The main objective is the control of the manipulation force for multiple micro-objects having different characteristics. In this work, the experiment will be done with a flexible and a rigid beams with compliances s of = 6.4237µm/mN and s or = 1.7123µm/mN for the flexible and rigid beams respectively. From this, an interval compliance is derived: [1.7123, 6.4237]µm/mN . The synthesized controller has to ensure some required performances for any object having a compliance inside this interval.

[s o ] =
The interval model describing the transfer voltage-force is given by: In order to increase the stability margin of the closed-loop system, we propose to extend the intervals of the model (19). When the interval width of the parameters in the model is too large, it is difficult to find a controller that respects both the stability and performances of the closed-loop. After some experiences, we choose to expand the interval width of each parameter of (19) by 10%. 10% represents the maximal value allowed in this application. Finally, the extended parameters of the interval model which will be used to compute the controller are given as follows:

[G](s, [s o ]) = d p D(s) [s o ] + s p D(s) (18
[G](s, [a]) = 1 [a 2 ]s 2 + [a 1 ]s + [a 0 ] (19)

C. Computation of the reference interval model

We want to satisfy the following specifications for the closed-loop:

• no overshoot,

• settling time 15ms ≤ tr 5% ≤ 80ms, • static error |ε| ≤ 1%. Based on the condition in [START_REF] Jaulin | Applied Interval Analysis[END_REF], we choose a second order model such as:

[H](s, [K p ], [τ ]) = [K p ] ([τ ]s + 1) [τ ] 10 s + 1 (21)
(21) can be rewritten as follows:

[H](s, [K p ], [τ ]) = 1 0.1 [τ ] 2 [Kp] s 2 + 1.1 [τ ] [Kp] s + 1 [Kp] (22)
Where the parameters [K p ] and [τ ] define the specified static error and settling time respectively:

• [K p ] = 1 + ε = [0.99, 1.01], • [τ ] = [tr 5% ] 3 = [5ms, 26.66ms].

D. Computation of the interval controller

Using [START_REF] Jaulin | Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis[END_REF], we obtain the following interval controller:

[C](s, [a], [K p ], [τ ]) = [a 2 ]s 2 + [a 1 ]s + [a 0 ] 0.1 [τ ] 2 [Kp] s 2 + 1.1 [τ ] [Kp] s + 1 [Kp] -1 (23)
After replacing each parameter in (23), we get:

[C](s, [c]) = [c 2 ]s 2 + [c 1 ]s + [c 0 ] [c 5 ]s 2 + [c 4 ]s + [c 3 ] (24)
with:

[c 5 ] = [0.247, 7.182] × 10 -5 , [c 4 ] = [0.544, 2.962] × 10 -2 , [c 3 ] = [-0.01, 0.01], [c 2 ] = [a 2 ], [c 1 ] = [a 1 ] and [c 0 ] = [a 1 ].

V. IMPLEMENTATION AND EXPERIMENTAL RESULTS

The midpoint of each interval parameters of [C](s) (24) has been taken for the implementation of the controller.

The closed-loop experiments are performed on the two beams (flexible and rigid). First, a step response analysis is performed. Fig. 7 shows the experimental results when a step reference of 20mN is applied. To check that the implemented controller ensures the specifications, the temporal envelope of the wanted interval model [H](s, [K p ], [τ ]) is also plotted in the same figure. Especially, we mean by the envelope of the wanted interval model, the step responses of two transfer functions, the first is with the minimum settling time tr = 15ms and maximum static error ε = 0.01, while the second is with the maximum settling time tr = 80ms and minimum static error ε = -0.01. As shown on Fig. 7 the controller has played its role and satisfied the specifications. Indeed, the settling times are: 15ms ≤ tr 1 = 36ms ≤ 80ms , 15ms ≤ tr 2 = 35ms ≤ 80ms respectively for the flexible and the rigid beams, the static errors are neglected and belong to the required interval |ε| ≤ 1%. Next, a harmonic analysis is performed. For that, a sine reference input of different frequencies is applied. The resulting magnitudes with the two beams, are plotted in Fig. 8. The magnitude envelope of the wanted closed-loop is also plotted in the same graph. It is clear that for any compliance of the object inside the interval [s o ] = [1.7123, 6.4237]µm/mN , the controller ensures the required performances. As seen on the figure, the different plots show that the wanted interval model includes the experimental results, except in high frequencies because of the neglected high frequency dynamics in the model.

VI. PERFORMANCES ANALYSIS

In this part, we prove the choice of the midpoint parameters of the interval controller (see 23 and 25). The idea consists to demonstrate that this midpoint controller ensures the specified performances for any system inside the interval model, i.e ∀G(s) ∈ [G] (s, [a]).

Consider an interval controller

[C](s, [c]) (see 23): [C](s, [c]) = [c 2 ]s 2 + [c 1 ]s + [c 0 ] [c 5 ]s 2 + [c 4 ]s + [c 3 ] (26)
Based on this controller (26) and on the interval model, we compute the closed-loop:

[H cl ](s, [a], [c]) = [C](s, [c]) • [G](s, [a]) 1 + [C](s, [c]) • [G](s, [a]) (27)
The objective is to compute the set parameters S c of the controller that garantee the specifications defined in section. IV

S c = {c ∈ C |[H cl ](s, [a], [c]) ⊆ [H](s) } ( 28
)
This problem is known as a set-inclusion problem which can be solved using the SIVIA algorithm (Set Inversion Via Interval Analysis) proposed by [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF]. SIVIA algorithm is limited to treat a set-inclusion problem with three parameters (time computation increases exponentially with the number of parameters). According to ( 26

VII. CONCLUSION

In this paper, a simple method to synthesize robust controllers by combining interval computation and direct synthesis control method has been presented. The proposed approach is based on a given interval model and a wanted interval closed-loop. An interval controller has been derived by applying interval arithmetic. We prove that the midpoint of this interval controller is one of the robust controllers ensuring the performances. The proposed control law was applied to the control of force in piezocantilevers. The experimental results demonstrate the efficiency of the proposed method.

Fig. 1 .

 1 Fig. 1. A microgripper manipulating a micro-object.

Fig. 2 .

 2 Fig. 2. A closed-loop system.

  [H](s, [b]) = 1 m j=0 [b j ]s j (8) Where [b] = [[b 0 ], ..., [b m ]] is a box of interval parameters.

  [H](s, [b]) and from the interval model [G](s, [a]), the transfer [H](s, [b]) being derived from given specifications.

Fig. 3 .

 3 Fig. 3. Piezocantilever principle.

Fig. 4 .

 4 Fig. 4. The experimental setup.

FFig. 5 .

 5 Fig.5. A piezocantilever in contact with a micro-object.

)

  After replacing s p , d p , D(s) and [s o ] computed previously, we obtain the interval model [G](s, [a]):

  such as: [a] = [[a 0 ], [a 1 ], [a 2 ]], [a 0 ] = [5.682, 12.894], [a 1 ] = [4.51, 16.92] × 10 -6 and [a 2 ] = [17.369, 65.161] × 10 -8 .

[a 0

 0 ] =[5.3217, 13.2547] [a 1 ] = [3.88, 17.55] × 10 -6 [a 2 ] = [14.97, 67.56] × 10 -8 (20) The interval model [G](s, [a]) with the interval parameters given in (20) is valuable for a set of beams having compliances in the range [so] = [1.7123, 6.4237]µm/mN .

C

  mid (s) = 0.4127s 2 + 10.72s + 9.288 × 10 6 s(25.07s + 1.742 × 10 4 )

Fig. 7 .

 7 Fig. 7. Step response of the closed-loop: envelope of the wanted interval model compared with the experimental results with C mid (s).

Fig. 8 .

 8 Fig. 8.Magnitudes of the wanted interval model compared with the experimental results with C mid (s).

  ) the controller [C](s, [c]) has six parameters. Since the parameters in the numerator are function of those of the interval model, we propose to set [c 2 ] = mid([a 2 ]), [c 1 ] = mid([a 1 ]), and [c 0 ] = mid([a 0 ]). Furthermore, we fix [c 3 ] = 0 in order to cancel the static error. Finally, we work with the remaining parameters [c 5 ] and [c 4 ]. After the application of the SIVIA algorithm with an accuracy of ε = 10 -7 and an initial box [c 50 ] × [c 40 ] = [2 × 10 -5 , 4.5 × 10 -5 ] × [0.015, 0.018], we obtained the subpaving given in the Fig. 9. The dark colored subpaving (S c ) corresponds to the set parameters [c 5 ] and [c 4 ] of the controller (26) with [c 2 ] = mid([a 2 ]), [c 1 ] = mid([a 1 ]), and [c 0 ] = mid([a 0 ]), and c 3 = 0 that ensure the performances for the interval model. As we can see in the figure, the controller C mid (s) (25) is inside the obtained set solution and is one of the controllers that meet the required specifications.

Fig. 9 .

 9 Fig. 9. Set solution of the parameters [c 5 ] and [c 4 ] ensuring performances with controller parameters [c 2 ] = mid([a 2 ]), [c 1 ] = mid([a 1 ]), [c 0 ] = mid([a 0 ]), and c 3 = 0.

  -C (22) for [G](s, [a]), i.e finding S c that meet the inclusion [H cl ](s, [a], [c]) ⊆ [H](s). Mathematically, this set solution S c can be written as follows: