Damien Thomine

A spectral gap for transfer operators of piecewise expanding maps

Introduction

An important task in dynamical systems, and the whole point of ergodic theory, is to study the measures invariant under some transformation. We will focus here on physical measures, which are empirical measures for a set of initial conditions of non-zero Lebesgue measure (see Subsection 1.2 for a definition of empirical mesures). To obtain non-trivial results, some assumptions must be made on those transformations, for instance on their smoothness. We shall study here maps which are piecewise differentiable, with Hölder derivative, and uniformly expanding (that is, with positive and bounded away from zero Lyapunov exponents) on some compact space.

In one dimension, a most interesting result was found in 1973 by A. Lasota and J.A. Yorke [START_REF] Lasota | On the existence of invariant measures for piecewise monotonic transformations[END_REF]: a piecewise twice differentiable uniformly expanding map on an interval admits a finite number of physical invariant measures, whose densities have bounded variation, and (up to taking an iterate of the transformation) they are mixing. This is obtained via the study of the Perron-Frobenius operator on the space of functions with bounded variation: the fixed points of this operator are densities of invariant measures, and the existence of a spectral gap provides the property of mixing at an exponential rate. To sum it up, the quasi-compactness of the Perron-Frobenius operator, along with some classical arguments, is enough to deduce many valuable ergodic properties of the system.

Problems arise when we weaken those assumptions. A possible generalization is to obtain results in higher dimensions, where general answers were obtained only recently, despite some early results on particular systems [START_REF] Keller | Ergodicité et mesures invariantes pour les transformations dilatantes par morceau d'une région bornée du plan[END_REF]. Actually, a naive generalization of the theorem proved by A. Lasota and J.A. Yorke does not hold in dimension 2 or higher, as is shown by the counter-examples of M. Tsujii [START_REF] Tsujii | Piecewise expanding maps on the plane with singular ergodic properties[END_REF] and J. Buzzi [START_REF] Buzzi | No or infinitely many A.C.I.P. for piecewise expanding C r maps in higher dimensions[END_REF]: some additional assumptions must be made. A first strategy is to require a lower bound on the angles made by the discontinuities; P. Góra and A. Boyarski [START_REF] Góra | Absolutely continuous invariant measures for piecewise expanding C 2 transformations in R n[END_REF] made a successful attempt in 1989, although the sufficient condition they found on the expanding rate is far from optimum. Another strategy is to study the combinatorial complexity, i.e. the way the space is cut by the discontinuities and the number of times they overlap; it was used for instance in 2000 by J.W. Cowieson [Cow00] and B. Saussol [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional expanding maps[END_REF], and we will follow this approach.

In all those studies, a major problem is to find a suitable space of functions on which the Perron-Frobenius operator may act. For piecewise smooth maps, the function spaces we use should show some regularity (otherwise, no spectral gap will be found), but not too much, since discontinuities must be allowed. For piecewise twice differentiable maps, the space of functions with bounded variation is suitable, as was shown by J.W. Cowieson [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF] (although he proved only the existence of invariant measures, and not stronger properties). For less smooth maps, B. Saussol worked with spaces of functions with bounded oscillation [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional expanding maps[END_REF]. Following the method used in [START_REF] Baladi | Good Banach spaces for piecewise hyperbolic maps via interpolation[END_REF], we will work here mainly with Sobolev spaces.

The approach followed in this article has many advantages. We work with usual and well-known spaces, instead of ad hoc function spaces. Moreover, the dynamic does not need to be studied in detail: this gives a very robust method, which can be more or less easily adapted to prove similar results for other function spaces or dynamical systems, since it relies only on the core properties of the function spaces. This paper is the outcome of a master's thesis, supervised by V. Baladi. The main goal is to simplify the proofs of the previous paper from V. Baladi and S. Gouëzel [START_REF] Baladi | Good Banach spaces for piecewise hyperbolic maps via interpolation[END_REF], restricting ourselves to the study of expanding maps. Most of the setting and proofs in sections 1 to 6 are directly adapted from that work (a notable exception being Lemma 4.1, corresponding to Lemma 31 in [START_REF] Baladi | Good Banach spaces for piecewise hyperbolic maps via interpolation[END_REF] but whose proof is adapted from another article [START_REF] Baladi | Banach spaces for piecewise cone hyperbolic maps[END_REF]). There are multiple gains: the setting is simpler, the function spaces used more standard (classical Sobolev spaces) and most of the proofs much shorter (some of which are even found in the literature). We will also deal with a limit case, when the transformation is piecewise differentiable with a Lipschitz derivative, and prove a new result involving functions with bounded variationbasically a stronger version of J.W. Cowieson's theorem [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF].

The definitions and setting are explained in Section 1, as well as the main results: a bound on the essential spectral radius of transfer operators, Theorem 1.3, and its consequence on the existence of finitely many mixing physical measures with densities in appropriate Sobolev spaces, and on the rate of mixing for Hölder test functions (Corollary 1.5 and Theorem 1.6). We shall also present an application of those result on a class of piecewise affine maps. Section 2 presents extensively the Sobolev spaces and their basic properties, as well as the space of functions with bounded variation. Section 3 contains the main lemmas, which exploit the properties of the Sobolev spaces; they are used in Section 4 to prove Theorem 1.3. Section 5 contains the proof of Theorem 1.6. Finally, in Section 6 we will deal with a limit case, where the transformation is piecewise differentiable with Lipschitz derivative and one works on functions with bounded variation, and a discussion on the previous similar results by J.W. Cowieson [Cow00] and B. Saussol [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional expanding maps[END_REF]. This last section also highlights the fact that our main theorems can be quite easily adapted to non-Sobolev function spaces.

Setting and results

If B is a Banach space, we denote the norm of an element f of B by f B . In this paper, a map defined on a closed subset of a manifold is said to be C k or C ∞ if it admits an extension to a neighborhood of this closed subset, which is C k or C ∞ in the usual sense. For α ∈ (0, 1), a map is said to be

C α if it is α-Hölder, C 1+α if it is C 1 with α-Hölder derivative, and C 1+Lip if it is C 1 with Lipschitz derivative.
Let X be a Riemannian manifold of dimension d, and let X 0 be a compact subset of X. We call C 1 hypersurface with boundary a codimension-one C 1 submanifold of X with boundary (i.e., every point of this set has a neighborhood diffeomorphic either to R d-1 or R d-2 × [0, +∞)).

Definition 1.1 (Piecewise C 1+α and C 1+Lip expanding maps).

For α > 0, we say that a map T : X 0 → X 0 is a piecewise C 1+α expanding map (respectively piecewise C 1+Lip expanding map) if:

• There exists a finite number of disjoint open subsets O 1 , . . . , O I of X 0 , covering Lebesgue-almost all X 0 , whose boundaries are unions of finitely many compact C 1 hypersurfaces with boundary.

• For 1 ≤ i ≤ I, there exists a C 1+α (respectively C 1+Lip ) map T i defined on a neighborhood of O i , which is a diffeomorphism onto its image, such that T coincides with T i on O i .

• For any x ∈ X 0 and whenever

T i (x) is defined, λ(x) := inf v∈Tx |DT i (x)v| |v| > 1.
Since we choose to study the asymptotical combinatorial complexity, we have to quantify it. It will be done the following way.

Let choose a non-zero integer n.

Let i = (i 0 , . . . , i n-1 ) ∈ {1, . . . , I} n . We define inductively sets O i by

O (i 0 ) = O i , and
O (i 0 ,...,i n-1 ) = {x ∈ O i 0 | T i 0 x ∈ O (i 1 ,...,i n-1 ) }. (1.1)
Let also

T n i = T i n-1 • • • • • T i 0 ;
it is defined on a neighborhood of O i . For any x ∈ X 0 and whenever T n i (x) is defined, we denote :

λ n (x) := inf v∈Tx |DT n i (x)v| |v| > 1.
We define the complexity at the beginning

D b n = max x∈X 0 Card{i = (i 0 , . . . , i n-1 ) | x ∈ O i }, (1.2)
and the complexity at the end

D e n = max x∈X 0 Card{i = (i 0 , . . . , i n-1 ) | x ∈ T n (O i )}. (1.3)
For T (x) = 2x mod 1 on [0, 1] we have D e n ≥ 2 n , but fortunately this quantity plays no role when T is piecewise C 1+Lip , where we can take p as close to 1 as needed in Corollary 1.5 (cf. Corollary 6.1), or work in the space of functions with bounded variation -see Theorem 1.4 or Cowieson's Theorem (referenced here as Theorem 6.2).

For generic piecewise expanding maps, the complexity D b n increases subexponentially, and therefore does not play an important role in the spectral formula (1.5) below (see [START_REF] Cowieson | Absolutely continuous invariant measures for most piecewise smooth expanding maps[END_REF]). However, some pathologic cases may arise when the iterates of the map T cut some open set Ω too often, and then map it onto itself. In spite of the expansion, these numerous "cut-and-fold" may make the images of Ω arbitrarily small, so that the physical measure starting from any point of Ω can be, for instance, a Dirac measure. This phenomenon is central in the examples given by M. Tsuji [START_REF] Tsujii | Piecewise expanding maps on the plane with singular ergodic properties[END_REF] and J. Buzzi [START_REF] Buzzi | No or infinitely many A.C.I.P. for piecewise expanding C r maps in higher dimensions[END_REF]. A control on the combinatorial complexities ensures that the expansion do beat the cuts.

Spectral results

The results about the physical measures will be obtained through the study of transfer operators (or Perron-Frobenius operators); we now define them.

Definition 1.2 (Transfer operator).

Let α > 0. For all g : X 0 → C such that the restriction of g to any O i is C α (in the sense that it admits a C α extension to some neighborhood of O i ), we define the transfer operator L g on L ∞ (X 0 ) by:

L g u(x) = y∈T -1 ({x})
g(y)u(y).

(1.4) These operators will act on Sobolev spaces H t p , for 1 < p < ∞ and 0 < t < min(1/p, α), or on functions with bounded variation. As for now, we will just recall that the Sobolev spaces H t p on R d are defined as {u ∈ L p :

F -1 ((1+|ξ| 2 ) t/2 (F u)(ξ)) ∈ L p }, with u H t p = F -1 ((1 + |ξ| 2 ) t/2 (F u)(ξ)) L p
, where F denotes the Fourier transform. Using charts, one can define a Sobolev space H t p on X. Here, t is an index for regularity; as t increases, H t p contains more regular functions. The Sobolev and "bounded variation" spaces will be defined precisely in Section 2.

The first result is about the essential spectral radius of L g when acting on H t p ; then, an application of this result to L 1/| det DT | , the Perron-Frobenius operator, will lead to the proof (under some conditions) of the existence of finitely many mixing physical measures.

Theorem 1.3 (Spectral theorem for piecewise C 1+α expanding maps).

Let α ∈ (0, 1], and let T be a piecewise C 1+α uniformly expanding map. Choose p ∈ (1, +∞), and 0 < t < min(1/p, α). Let g : X 0 → C be a function such that the restriction of g to any O i admits a C α extension to O i . Then L g acts continuously on H t p , where its essential spectral radius is at most

lim n→+∞ (D b n ) 1 p 1 n (D e n ) (1-1 p ) 1 n g (n) | det DT n | 1 p λ -t n 1 n L ∞ , (1.5)
where [START_REF] Triebel | General function spaces. III. (Spaces B g(x) p,q and F g(x) p,q , 1 < p < ∞: basic properties[END_REF]), the operator L g can be extended to a continuous operator on H t p . The restriction 0 < t < 1/p is exactly designed so that the space H t p is stable under multiplication by characteristic functions of nice sets (see Lemma 3.2). We also provide a version of this theorem for piecewise C 1+Lip maps, including piecewise C 2 maps. We cannot use the function space H 1 1 (which for instance contains only continuous functions in dimension 1); we will instead use the space of functions with bounded variation BV, which will be defined rigorously in Section 2: Theorem 1.4 (Spectral theorem for piecewise C 1+Lip expanding maps).

g (n) = n-1 i=0 g • T i ,
Let be T a piecewise C 1+Lip expanding map. Let g : X 0 → C be a function such that the restriction of g to any O i admits a Lipschitz extension to O i . Then L g acts continuously on BV(X 0 ), where its essential spectral radius is at most

lim n→+∞ (D b n ) 1 n g (n) | det DT n |λ -1 n 1 n L ∞ .
(1.6)

Physical measures

The empirical measure of T with initial condition x ∈ X 0 is the weak limit, if it exists, of 1/n n-1 k=0 δ T k x ; such a measure is always T -invariant. A physical measure of T is an invariant probability measures µ which is an empirical measure for a set of initial conditions of non-zero Lebesgue measure; the basin of a physical measure µ is the set of all initial conditions whose corresponding empirical measure is µ.

In the following and for any finite measure µ and integrable function f , the expression µ, f will be used for the integral of f against µ. We may see any integrable function u as the density (with respect to Lebesgue measure Leb) of some finite measure, and use u, f instead of u dLeb, f . For all f ∈ L ∞ and u ∈ H t p , we have u, f • T = L 1/| det DT | u, f ; a nonnegative and nontrivial fixed point u of L 1/| det DT | in H t p then corresponds (up to normalization) to an invariant probability measure µ u = u dLeb whose density with respect to the Lebesgue measure is in H t p . By Birkhoff's theorem, when such a measure µ u is ergodic, it is physical, since 1/n n-1 k=0 δ T k x converges weakly to µ u for Lebesgue-almost every point in the support of u.

Theorem 1.3 implies in this setting:

Corollary 1.5.

Under the hypotheses of Theorem 1.3, assume that

lim n→+∞ (D b n ) 1 p 1 n • (D e n ) (1-1 p ) 1 n • λ -t n | det DT n | 1 p -1 1 n L ∞ < 1. (1.7)
Then the essential spectral radius of L 1/| det DT | acting on H t p is strictly smaller than 1.

If (1.7) holds, we will be able to prove the existence of a spectral gap, i.e. of some τ < 1 such that {z ∈ Spec(L 1/| det DT | ) : τ < |z| < 1} = ∅, together with a nice repartition of the eigenvalues on the unit cicle. The number τ can be larger the essential spectral radius, as is shown for instance by G. Keller and H.H. Rugh in [START_REF] Keller | Eigenfunctions for smooth expanding circle maps[END_REF] in a 1-dimensional setting. This will imply the following: Theorem 1.6.

Under the assumptions of Theorem 1.3, if (1.7) holds, then T has a finite number of physical measures whose densities are in H t p , which are ergodic, and whose basins cover Lebesgue almost all X 0 . Moreover, if µ is one of these measures, there exist an integer k and a decomposition µ = µ 1 + • • • + µ k such that T sends µ j to µ j+1 for j ∈ Z/kZ, and the probability measures kµ j are mixing at an exponential rate for T k and α-Hölder test functions.

The above will be proved in Section 5, although the arguments are classical. We shall also obtain its bounded variation counterpart:

Theorem 1.7.
Under the assumptions of Theorem 1.4, assume that:

lim n→+∞ (D b n ) 1 n • λ -1 n 1 n L ∞ < 1. (1.8)
Then T has a finite number of physical measures whose densities are in BV(X 0 ), which are ergodic, and whose basins cover Lebesgue almost all X 0 . Moreover, if µ is one of these measures, there exist an integer k and a decomposition µ = µ 1 + • • •+ µ k such that T sends µ j to µ j+1 for j ∈ Z/kZ, and the probability measures kµ j are mixing at an exponential rate for T k and α-Hölder test functions.

Piecewise affine maps

We now describe an explicit class of maps for which the assumptions of the previous theorems are satisfied. Let A 1 , ..., A N be d×d matrices with no eigenvalue of modulus smaller or equal to 1, such that any two of these matrices commute. Let X 0 be a (non necessarily connected) polyhedral region of R

Proof.

Let K be the total number of the sides of the polyhedra O i . Around any point x, the boundaries of the sets O (i 0 ,...,i n-1 ) are preimages of theses sides by one of the maps

A i 0 , . . . , n-1 l=0 A i l . Hence, there are at most J(n) = K n k=1 Card{ k-1 l=0 A i l : {i 0 , ..., i k-1 } ∈ {1, .
., N} k } such preimages. Since all the A i commute, there are at most k N different maps which can be written as This proposition can also be deduced from Cowieson's theorem [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF] whenever the polyhedral domain is connected.

Sobolev and bounded variation spaces

It is now time to define precisely the function spaces we work with. First, we will present the Sobolev spaces H t p (p ∈ (0, 1) and 0 ≤ t) and some of their properties. Since they have been thoroughly studied in the 1960s-1970s, we may exploit many ready-to-use results. They are spaces of L p functions which satisfy some regularity condition; since the transfer operator of expanding dynamics tends to improve the regularity, this may imply the existence of a spectral gap for these operators. We show two equivalent ways to define these spaces; the first is via the Fourier transform, the second via interpolation theory. Then, we will present the space of functions with bounded variation, which is nicer than the Sobolev space H 1 1 in our setting.

Definition via Fourier transform

Definition 2.1 (Local spaces H t p ). For 1 < p < ∞, t ≥ 0, ξ ∈ R d , put a t (ξ) = (1 + |ξ| 2 ) t/2
. We define the space H t p of functions in R d as the subspace of functions u ∈ L p such that F -1 (a t F u) ∈ L p with its canonical norm, i.e., the L p norm of the expression above.

Since a t increases at infinity, the condition F -1 (a t F u) ∈ L p can be understood as a condition on the decay at infinity of F u, hence on the regularity of u.

If 0 ≤ t < α, we shall see that H t p is invariant under composition by C 1+α diffeomorphisms: this is Lemma 3.3. Hence, we can glue such spaces locally together in appropriate coordinate patches, to define a space H t p of functions on the manifold:

Definition 2.2 (Sobolev spaces on X 0 ). Let 0 ≤ t < α.
Fix a finite number of C 1+α charts κ 1 , . . . , κ J whose domains of definition cover a compact neighborhood of X 0 , and a partition of unity ρ 1 , . . . , ρ J , such that the support of ρ j is compactly contained in the domain of definition of κ j , and ρ j = 1 on X. The space H t p is then the space of functions u supported on X 0 such that (ρ j u) • κ -1 j belongs to H t p for all j, endowed with the norm

u H t p = J j=1 (ρ j u) • κ -1 j H t p .
(2.1)

Changing the charts and the partition of unity gives an equivalent norm on the same space of functions by Lemma 3.1. To fix ideas, we shall view the charts and partition of unity as fixed.

The Fourier transform approach also provides some effective theorems, for instance Fourier multipliers theorems (the following is e.g. Theorem 2.4/2 in [START_REF] Triebel | General function spaces. III. (Spaces B g(x) p,q and F g(x) p,q , 1 < p < ∞: basic properties[END_REF]):

Theorem 2.3 (Marcinkiewicz multiplier theorem). Let b ∈ C d (R d ) satisfy |ζ γ D γ b(ζ)| ≤ K for all multi-indices γ ∈ {0, 1} d ,
and all ζ ∈ R d . For all p ∈ (1, ∞), there exists a constant C p,d which depends only on p and d such that, for any u ∈ L p ,

F -1 (bF u) L p ≤ C p,d K u L p .
(2.2)

Definition via interpolation theory

Now, let us present the complex interpolation theory, developped by J.L. Lions, A.P. Calderón and S.G. Krejn (see e.g. §1.9 in [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF]).

A pair (B 0 , B 1 ) of Banach spaces is called an interpolation couple if they are both continuously embedded in a linear Hausdorff space B. For any interpolation couple (B 0 , B 1 ), we let L(B 0 , B 1 ) be the space of all linear operators L mapping B 0 + B 1 to itself so that L |B j is continuous from B j to itself for j = 0, 1. For an interpolation couple (B 0 , B 1 ) and 0 < θ < 1, we next define [B 0 , B 1 ] θ the complex interpolation space of parameter θ.

Set S = {z ∈ C | 0 < ℜz < 1}, and introduce the vector space

F (B 0 , B 1 ) = {f : S → B 0 + B 1 , analytic, extending continuously to S, with sup z∈S f (z) B 0 +B 1 < ∞, t → f (it) ∈ C b (R, B 0 ), and t → f (1 + it) ∈ C b (R, B 1 )}.
Then, we define the following norm on this space:

f F (B 0 ,B 1 ) := max( f (it) B 0 ∞ , f (1 + it) B 1 ∞ )
The complex interpolation space is defined for θ ∈ (0, 1) by

[B 0 , B 1 ] θ = {u ∈ B 0 + B 1 | ∃f ∈ F (B 0 , B 1 ) with f (θ) = u}, normed by u [B 0 ,B 1 ] θ = inf f (θ)=u f F (B 0 ,B 1 ) .
The main idea is that the [B 0 , B 1 ] θ spaces are intermediates between B 0 and B 1 , "close to B 0 " for low parameters θ and "close to B 1 " for high parameters. Usually, the embedding can be done for example in S ′ , the dual space of the space of C ∞ rapidly decaying functions. We mention here the following key result (Theorem 1.(c) of §2.4.2 in [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF]):

Proposition 2.4 (Interpolation of Sobolev spaces).

For any t 0 ,

t 1 ∈ R + , p 0 , p 1 ∈ (1, ∞) and θ ∈ (0, 1), the interpolation space [H t 0 p 0 , H t 1 p 1 ] θ is equal to H t p for t = t 0 (1 -θ) + t 1 θ and 1/p = (1 -θ)/p 0 + θ/p 1 . In particular, for all t ∈ (0, 1) and p ∈ (1, ∞), H t p = [L p , H 1 p ] t .
Remark 2.5 (Alternative definition of Sobolev spaces). By Proposition 2.4, we have another way to define the H t p spaces on a (compact) Riemannian manifold X, this time intrinsically. Since there exists a Lebesgue measure Leb on X, the spaces L p are well defined. Moreover, the space H 1 p is none other than {u ∈ L p : ∇u ∈ L p } with the norm u H 1 p = u L p + ∇u L p , the derivative being taken in a weak sense, and defined via the Levi-Civita connection (see Theorem 1.(b) of §2.3.3 in [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF]). It is not difficult to see that H t p = [L p (X 0 ), H 1 p (X 0 )] t , with a norm equivalent to the one previously defined (Remark 2 p.321 in [START_REF] Triebel | Theory of function spaces II[END_REF]).

Here, the main interest of interpolation theory will be to derive inequalities for operators in the H t p spaces from inequalities in the L p and H 1 p spaces, where the manipulations are much easier. This will be made possible thanks to the following property (see e.g. §1.9 in [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF]): for any interpolation couple (B 0 , B 1 ), θ ∈ (0, 1) and L ∈ L(B 0 , B 1 ), we have

L [B 0 ,B 1 ] θ →[B 0 ,B 1 ] θ ≤ L 1-θ B 0 →B 0 L θ B 1 →B 1 (2.

Functions with bounded variation

In order to define the space of functions with bounded variation, we proceed in the same way as in Subsection 2.1: we first define this space on open sets of R d , and then use the charts to define such a space on a Riemannian manifold.

Definition 2.7 (Functions with bounded variation).

Let us denote the Lebesgue measure on R d by Leb. Let Ω be an open set of R d . For u ∈ L 1 (Ω), we define the variation of u, V (u), by

V (u) = d -1 sup ϕ∈C ∞ c (Ω,R d ) ϕ ∞ ≤1 Ω u.div(ϕ) dLeb .
The space of functions with bounded variation on Ω, denoted BV(Ω), is the subspace of functions u of L 1 (Ω) such that V (u) < +∞, endowed with the norm u

BV(Ω) = u L 1 + V (u).
Remark 2.8. Formally, we have (see e.g. Remark 1.8 in [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]):

V (u) = d -1 Ω |∇(u)|.
The derivative being taken in a weak sense, |∇(u)| may be a measure with no density with respect to the Lebesgue measure. For C 1 functions, it can be taken literally, with

V (u) = d -1 Ω d i=1 ∂u ∂x i dLeb .
As for the Sobolev spaces, we may define the space of functions with bounded variation on a compact set X 0 of a Riemannian manifold X starting from its definition on R d (this space also behaves well under composition by C 1+Lip diffeomorphisms, as is shown by Lemma 6.6) and transporting by the charts. Definition 2.9 (Functions with bounded variation on X 0 ).

Fix a finite number of C 1+Lip charts κ 1 , . . . , κ J whose domains of definition cover a compact neighborhood of X 0 , and a partition of unity ρ 1 , . . . , ρ J , such that the support of ρ j is compactly contained in the domain of definition of κ j , and ρ j = 1 on X. The space BV of functions with bounded variation on X 0 (sometimes also denoted BV(X 0 )) is the space of functions u supported on X 0 such that (ρ j u) • κ -1 j belongs to BV(R d ) for all j, endowed with the norm

u BV = J j=1 (ρ j u) • κ -1 j BV(R d ) .
(2.4)

Since we can neither use the Fourier transform nor the interpolation theory here, we will need different tools for the part of our proof relying on functions with bounded variation. We choose to use two theorems dealing with the approximation of functions with bounded variation by C ∞ functions, and then to work with smooth functions (for which the computations can be done explicitly). They are respectively a variation on Theorem 1.17 and Theorem 1.9 in [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]. The first is obtained via convolution with smooth kernels (say, gaussian or C ∞ with compact support), the second follows directly from the definition of the variation.

Theorem 2.10 (Approximation of functions with bounded variation).

For any function u in BV(R d ), there exists a sequence of C ∞ functions (u n ) n∈N which converges in L 1 to u, and such that (

u n BV(R d ) ) n∈N converges to u BV(R d ) .
Theorem 2.11 (Control of the approximation).

Let u ∈ BV(R d ), and (u n ) n∈N a sequence of functions with bounded variation which converges to u in

L 1 . Then u BV(R d ) ≤ lim inf n→+∞ u n BV(R d ) .

Elementary inequalities

Our goal in this section is to prove several continuity results for operations in the H t p spaces, for instance multiplication by C α functions or by characteristic functions of intervals. They show the convenient properties of the spaces we are working with, provided we take good values of t and p, and are virtually sufficient to get the main theorems.

Hence, obtaining versions of the main theorem for other parameters or function spaces can be reduced to the obtention of those results of continuity; see for instance Section 6 for the corresponding results in the BV space.

First inequalities

The first inequalities are continuity results for some linear operations; they were proven in the 60s and 70s.

We first deal with the multiplication by smooth enough functions, which will be necessary primarily when we multiply u and g, and also when it comes to study what happens at a small scale (we will multiply the functions u with smooth functions with small supports). A proof of the following lemma can be found in §4.2.2 of [START_REF] Triebel | Theory of function spaces II[END_REF].

Lemma 3.1 (Multiplication by C α functions).
Let 0 < t < α be real numbers. There exists C t,α such that, for all p ∈ (1, +∞),

for all u ∈ H t p (R d ) and g ∈ C α (R d ), gu H t p ≤ C t,α g C α u H t p .
The next inequality (Corollary I.4.2 from [START_REF] Strichartz | Multipliers on fractional Sobolev spaces[END_REF]) is central for our study, since it tells us that H t p (R d ) behaves well in a "piecewise setting" up to constraints on the parameters t and p. We note that one has to combine Corollary I.4.2 from [START_REF] Strichartz | Multipliers on fractional Sobolev spaces[END_REF] with its Corollary I.3.7 to show that the constant C t,p in the following lemma does not depend on Ω otherwise than via L.

Lemma 3.2 (Multiplication by characteristic functions of nice sets).

Let 0 < t < 1/p < 1 be real numbers; let L ≥ 1. There exists C t,p such that for each measurable subset Ω of R d whose intersection with almost every line parallel to some coordinate axis has at most L connected components, for all u ∈ H t p (R d ),

1 Ω u H t p ≤ C t,p L u H t p .

Composition with C 1+α diffeomorphisms

The injection from H t p into L p is compact, t being positive and X 0 compact (in the case of compact subset of some R d , this is for instance Lemma 2.2 in [Bal]; the corresponding result for compact subset of Riemannian manifolds follows immediatly). Thus, Lemma 3.3 (a version of Lemma 24 in [START_REF] Baladi | Good Banach spaces for piecewise hyperbolic maps via interpolation[END_REF]) is essential in the way it gives the decomposition of L g u H t p into a part bounded by u H t p , and a part bounded by u L p . By Hennion's theorem [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipchitziens[END_REF], the part bounded by u L p will not affect our estimation of the essential spectral radius of L g , so that only the part bounded by u H t p matters.

When we apply Lemma 3.3, F will be the local inverse of an iterate of T , i.e. some T -n i (or rather the maps on some open set of R d obtained by transporting T -n i via the charts), and A will be a local approximation of DT -n i . If A is such an approximation in a neighborhood of some x, then, T being expanding, we can write A ≤ λn -1 where λn is the essential infimum of λ n on this neighborhood of x.

Lemma 3.3 (Composition with smooth diffeomorphisms).

Let F ∈ C 1 (R d , R d ) be a diffeomorphism and A ∈ GL d (R) such that, for all z ∈ R d , A -1 • DF (z) ≤ 2 and DF (z) -1 • A ≤ 2.
Then, for all t ∈ [0, 1], for all p ∈ (1, +∞), there exist constants C t,p and C ′ t,p , which do not depend on F nor on A, such that, for all u ∈ H t p ,

u • F H t p ≤ C t,p | det A| -1 p A t u H t p + C ′ t,p | det A| -1 p u L p
Proof.

We will write u

• F = u • A • A -1
• F , and put F = A -1 • F . First we deal with u • A, and then with u • F .

First step:

u • A We want to estimate u • A H t p = F -1 (a t F (u • A)) L p , where a t (ξ) = (1 + |ξ| 2 ) t/2 . A change of variables gives F -1 (a t F (u • A)) = F -1 (a t • t AF (u)) • A. If |ξ| > 1/ A , we have (a t • t A)(ξ) = (1 + | t Aξ| 2 ) t/2 ≤ (1 + A 2 |ξ| 2 ) t/2 ≤ 2 t/2 A t a t (ξ).
On the other hand, if |ξ| ≤ 1/ A , we have

(a t • t A)(ξ) ≤ (1 + A 2 |ξ| 2 ) t/2 ≤ 2 t/2 .
Finally, we get a t • t A ≤ 2 t/2 A t a t + 2 t/2 , and by the same means the same kind of upper bound for the ξ γ D γ a t • t A A t a t + 1 for all γ ∈ {0, 1} d . By Theorem 2.3, there exist C t,p and C ′ t,p such that

F -1 (a t • t AF (u)) L p ≤ C t,p | det A| -1/p A t F -1 (a t F (u)) L p + C ′ t,p | det A| -1/p F -1 (F (u)) L p ≤ C t,p | det A| -1/p A t u H t p + C ′ t,p | det A| -1/p u L p .
The following estimate ensues: The lemma follows immediately from (3.1) and (3.2).

u • A H t p ≤ C t,p | det A| -1/p A t u H t p + C ′ t,p | det A| -1/p u L p . ( 3
Since the injection H t p → L p is continuous (t being non-negative), a consequence of (3.1) is that, for all p ∈ (1, +∞) and t ∈ [0, 1], there exists a constant C t,p such that, for all u ∈ H t p (R d ), for all A ∈ GL d (R),

u • A H t p ≤ C t,p sup{ A , 1}| det A| -1/p u H t p .
(3.3)

Localization

For the proof of Theorem 1.3, we work locally, on small open sets, and the compactness of X 0 allows us to work globally with a finite number of such sets. However, we need to control the intersection multiplicity of those sets. This control will be obtained with a partition of the unity and a "zoom" to get some regularity when working at small scales. This lemma is Theorem 2.4.7 from [START_REF] Triebel | Theory of function spaces II[END_REF].

Lemma 3.4 (Localization principle). Let η ∈ C ∞ c (R d , R),
The constant C t,p,η depends on t, p, on the support of η and on its C k norm for some large enough k. The proof we give of a localization principle in Section 6 will make explicit a kind of dependence of C BV,η (the corresponding bound for the space of functions with bounded variation) in η.

Lemma 3.5.

Let 1 < p < +∞ and t ∈ R + . There exists a constant C t,p such that, for any functions v 1 , . . . , v l with compact support in R d , belonging to H t p , there exists a constant C t,p,{v} , depending only on the supports of the functions v i , with

l i=1 v i p H t p ≤ C t,p m p-1 l i=1 v i p H t p + C t,p,{v} l i=1 v i p L p ,
where m is the intersection multiplicity of the supports of the v i 's, i.e. m = sup x∈R d Card{i | x ∈ Supp(v i )}.

Proof.

Let B be the operator acting on functions by

Bv = F -1 ((1 + |ξ| 2 ) t/2 F v), so that v H t p = Bv L p .
By Lemma 2.7 in [Bal], for any compact K and any neighborhood K ′ of K, there exist C K,K ′ ,t,p > 0 and a function Ψ : R d → [0, 1] equal to 1 on K and vanishing on the complement of K ′ , such that for any v ∈ H t p with support in K,

ΨBv -Bv Lp ≤ C K,K ′ ,t,p v H t-1 p . (3.5)
Let v 1 , . . . , v l be functions with compact supports whose intersection multiplicity is m. Choose neighborhoods K ′ 1 , . . . , K ′ l of the supports of the v i s whose intersection multiplicity is also m, and functions Ψ 1 , . . . , Ψ l as above. Then

i v i p H t p = i Bv i p L p ≤ i Ψ i Bv i p L p + C K,K ′ ,t,p i v i p H t-1 p . (3.6) Since t ≤ 1, the inclusion L p → H t-1 p is continuous, and v i p H t-1 p ≤ C p v i p L p . By convexity, the inequality (x 1 + • • • + x m ) p ≤ m p-1 x p
i holds for any nonnegative numbers x 1 , . . . , x m . Since the multiplicity of the

K ′ i is at most m, this yields i Ψ i Bv i p ≤ m p-1 i |Bv i | p .
Integrating this inequality and using (3.6), we get the lemma.

Proof of the main theorem

Before proving Theorem 1.3, we need to show that T does not "hack" too much the functions, or in other words that its discontinuities are not so bad that functions in H t p do not stay in this space when composed by T . The following lemma will allow us to apply Lemma 3.2. For once, its proof is adapted from [BG10] instead of [START_REF] Baladi | Good Banach spaces for piecewise hyperbolic maps via interpolation[END_REF].

Lemma 4.1.

For 1 ≤ i ≤ I, let L i be the number of smooth boundary components of O i , and L = max i L i .

For any n ≥ 1, for any i = (i 0 , . . . , i n-1 ), for any x ∈ O i , for any j such that x ∈ Supp ρ j , there exists a neighborhood O ′ of x and an orthogonal matrix M such that the intersection of Mκ j (O ′ ∩ O i ) with almost any line parallel to a coordinate axis has at most Ln components.

Proof.

If x is in the interior of some O i , we may just take for O ′ a ball small enough so that O ′ ⊂ O i , and M = Id. Let assume that x belongs to a backwards image of a C 1 hypersurface of some ∂O i k by T k i , with k ≤ n. On a small neighbourhood of x, the smooth boundary components of O i are close to their tangent hyperplanes in x. We can choose d orthogonal vectors of norm 1 all transverse to all those hyperplanes. The smooth boundary components are C 1 , so that the vectors we chose are transverse to the boundary components (an not only to the hyperplanes) on a small enough ball around x, that we shall denote O ′ .

We have O

i = n-1 k=0 T -k O i k , each of the T -k O i k
being an open set bounded by at most L hypersurfaces; hence, O i is bounded by at most Ln preimages of those hypersurfaces under some T -k . By construction, each line parallel to any coordinate axis in the new basis intersects any of these backward images of hypersurfaces in at most one point, so that their intersection with O i ∩ O ′ has at most Ln connected components. All we need is to take for M the change-of-basis natrix from the new orthogonal basis to the canonical one.

We now have all the tools we need to prove the main theorem.

Proof of Theorem 1.3.

Let p and t be as in the assumptions of the theorem. Let n > 0, and r n > 1 (the precise value of r n will be chosen later). We define a dilation R n on R d by R n (z) = r n z. Let u n be another norm on H t p , given by

u n = J j=1 (ρ j u) • κ -1 j • R -1 n H t p . (4.1)
The norm u n is of course equivalent to the usual norm on H t p , but we look at the space X 0 at a smaller scale. Functions are much flatter at this new scale, so that we have no problem using Lemma 3.1, which involves their C α norm: we will replace it by their L ∞ norm. This will also enable us to use partitions of unity (η m ) with very small supports without spoiling the estimates. The use of this "zooming" norm is similar to the good choice of ǫ 0 in [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional expanding maps[END_REF].

We will prove that there exists C t,p,η,α such that, if n is fixed and r n is large enough, then there exists C t,p,n such that:

L n g u p n ≤ C t,p,n u p L p + C t,p,η,α D b n (D e n ) p-1 (Ln) p | det DT n |λ -tp n |g (n) | p L ∞ u p n .
(4.2) We have already observed that, since X 0 is compact and t > 0, the injection of H t p into L p is compact (see e.g. Lemma 2.2 in [Bal]). Hence, by Hennion's theorem (actually Corollary 1 in [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipchitziens[END_REF]), the essential spectral radius of L n g acting on H t p (for either u H t p or u n , since these norms are equivalent) is at most

C t,p,η,α (Ln) p D b n (D e n ) p-1 | det DT n |λ -tp n |g (n) | p L ∞ 1 p . (4.3)
Taking the power 1/n and letting n tend to +∞, we obtain Theorem 1.3 since the quantity (C 1/p t,p,η,α (Ln) p ) 1/n converges to 1. It remains to prove (4.2), for large enough r n . The estimate will be subdivided into three steps:

1. Decomposing u into a sum of functions v j,m with small supports and well controlled • n norms.

Estimating each term (1

O i g (n) v j,m ) • T -n i , for i of length n.
3. Adding all terms to obtain L n g u.

First step:

For

1 ≤ j ≤ J and m ∈ Z d , let ṽj,m = η m • (ρ j u) • κ -1 j • R -1 n , where η m (x) = η(x + m), with η : R d → [0, 1] a compactly supported C ∞ function so that m∈Z d η m = 1.
Since the intersection multiplicity of the supports of the functions η m is bounded, this is also the case for the ṽj,m . Moreover, if j is fixed, we get, using Lemma 3.4,

m∈Z d ṽj,m p H t p = m∈Z d η m • (ρ j u) • κ -1 j • R -1 n p H t p (4.4) ≤ C t,p,η (ρ j u) • κ -1 j • R -1 n p H t p ≤ C t,p,η u p n .
Since R n expands the distances by a factor r n while the size of the supports of the functions η m is uniformly bounded, the supports of the functions

v j,m = ṽj,m • R n • κ j = η m • R n • κ j • (ρ j u) are arbitrarily small if r n is large enough. Finally u = j ρ j u = j,m v j,m .
(4.5)

Second step: Fix j, k ∈ {1, . . . , J}, m ∈ Z d and i = (i 0 , . . . , i n-1 ). We will prove that

(ρ k (g (n) 1 O i v j,m ) • T -n i ) • κ -1 k • R -1 n H t p (4.6) ≤ C t,p,n u L p + C t,p (Ln) | det DT n | 1 p g (n) λ -t n L ∞ ṽj,m H t p .
First, if the support of v j,m is small enough (which can be ensured by taking r n large enough), there exists a neighborhood O of this support and a matrix M satisfying the conclusion of Lemma 4.1 for all x in O i . Therefore, the intersection of R n (M(κ j (O ∩ O i ))) with almost any line parallel to a coordinate axis has at most Ln connected components. Hence, Lemma 3.2 implies that the multiplication by

1 O∩O i • κ -1 j • M -1 • R -1
n sends H t p into itself, with a norm bounded by C t,p Ln. Using the fact that M and R n commute, the properties of M, and (3.3), we get

1 O i • κ -1 j • R -1 n • ṽj,m H t p ≤ C t,p Ln ṽj,m H t p . (4.7) Next, let ṽj,k,m = ((ρ k • T n i )1 O i ) • κ -1 j • R -1 n
• ṽj,m (we suppress i from the notation for simplicity). Let also χ be a C ∞ function supported in the neighborhood O of the support of v j,m with χ ≡ 1 on this support. Up to taking larger r n we may ensure that (χ(

ρ k • T n i )) • κ -1 j • R -1 n C α ≤ 2.
G = κ j • T -n i • κ -1 k , we have ((ρ k • T n i )1 O i v j,m ) • T -n i • κ -1 k • R -1 n = ṽj,k,m • R n • κ j • T -n i • κ -1 k • R -1 n (4.9) = ṽj,k,m • R n • G • R -1 n . Applying Lemma 3.3 to F = R n • G • R -1
n , we get (for some point x in the support of v j,m , and some matrix A of the form DG(R -1 n (x)) for some x, so that

A ≤ λ -1 n (x)) ṽj,k,m • R n • G • R -1 n H t p ≤ C t,p,n u L p (4.10) + C t,p | det A| -1 p λ n (x) -t ṽj,k,m H t p .
The constant C t,p,n also depends on the choice of A. Let χ be a C ∞ function supported in O ′ with χ ≡ 1 on the support of v j,m •T -n i . For δ > 0, we can ensure by increasing r n that the C α norm of (χg

(n) ) • T -n i • κ -1 k • R -1 n is bounded by |g (n) (x)
| + δ for some x in the support of v j,m . Choosing δ > 0 small enough, we deduce from (4.10), Lemma 3.1 and (4.8)

(ρ k (g (n) 1 O i v j,m ) • T -n i ) • κ -1 k • R -1 n H t p ≤ C t,p,n u L p + C t,p,α Ln | det DT n | 1 p g (n) λ -t n L∞
ṽj,m H t p . This proves (4.6).

Third step:

We have

L n g u = j,m i (1 O i g (n) v j,m ) • T -n i .
(Note that only finitely many terms in this sum are nonzero by compactness of the support of each ρ j .) We claim that the intersection multiplicity of the supports of the functions (1

O i g (n) v j,m ) • T -n i is bounded by C η D e n .
Indeed, this follows from the fact that any point x ∈ X 0 belongs to at most D e n sets T n i (O i ), and that the intersection multiplicity of the supports of the functions v j,m is bounded.

To estimate L n g u n , we have to bound each term (

ρ k L n g u) • κ -1 k • R -1 n H t p , for 1 ≤ k ≤ J.
Let us fix such a k. By Lemma 3.5, we have

(ρ k L n g u) • κ -1 k • R -1 n p H t p ≤ C t,p,n u p L p + C t,p,α (C t,p,η D e n ) p-1 j,m,i (ρ k (1 O i g (n) v j,m ) • T -n i ) • κ -1 k • R -1 n p H t p .
We can bound each term in the sum using (4.6) and the convexity inequality (a + b) p ≤ 2 p-1 (a p + b p ). Moreover, for any (j, m), the number of parameters i for which the corresponding term is nonzero is bounded by the number of sets O i intersecting the support of v j,m . Choosing r n large enough, we can ensure that the supports of the v j,m are small enough so that this number is bounded by D b n . Together with (4.4), this concludes the proof of (4.2), and of Theorem 1.3.

Existence of finitely many physical measures

In this section, we prove Theorem 1.6. In view of Theorem 1.4, and with a few minor adaptations, the same proof leads to Theorem 1.7. This proof is a simplified version of Appendix B in [START_REF] Baladi | Good Banach spaces for piecewise hyperbolic maps via interpolation[END_REF], which is itself adapted from [START_REF] Blank | Ruelle-Perron-Frobenius spectrum for Anosov maps[END_REF].

Proof.

First step: the spectrum First, note that the Lebesgue measure Leb on the manifold X 0 is in (H t p ) * and is a fixed point for L * , so that 1 is an eigenvalue of L * and thus is in the spectrum of L. By Corollary 1.5 the essential spectral radius ρ ess of L is strictly smaller than 1, and 1 is an eigenvalue of L.

We also have the inclusions H t p ⊂ L p ⊂ L 1 , so that L ∞ ⊂ H -t p * and any essentially bounded function on X 0 can be seen as a linear functional on H t p . In the following, when γ denotes an eigenvalue of L of modulus strictly bigger than the essential spectral radius, E γ will denote the corresponding finitedimensional eigenspace and π γ : H t p → E γ the corresponding canonical projection. Consider such an eigenvalue γ; since L ∞ ∩ H t p is dense is H t p , its image by π γ in E γ is a dense subspace of E γ , and therefore is E γ itself.

Let us first prove that L has no eigenvalue of modulus strictly bigger than 1, nor a nontrivial Jordan block for an eigenvalue of modulus 1. Otherwise, let γ be an eigenvalue of L of maximal modulus, with a Jordan block of maximal size d. Therefore, there exists a bounded function f such that n -d n-1 i=0 γ -i L i f converges to a nonzero limit u. For any g ∈ L ∞ , u, g :=

X 0 ug dLeb = lim n→+∞ 1 n d n-1 i=0 γ -i L i f, g = lim n→+∞ 1 n d n-1 i=0 γ -i f •g•T i dLeb .
If |γ| > 1 or d ≥ 2, this quantity converges to 0 when n → +∞ since f • g • T i dLeb is uniformly bounded. This contradicts the fact that u is nonzero.

Let us take any eigenvalue γ of modulus 1. Since γ is of maximal modulus and ρ ess < 1, the eigenprojection is given by

π γ f = lim n→+∞ 1 n n-1 i=0 γ -i L i f, (5.1)
where the convergence holds in

H t p . Let u ∈ E γ . Obviously, E γ ⊂ H t p ⊂ L p ⊂ L 1
and u can be seen as the density of a finite complex measure with respect to the Lebesgue measure.

For any f ∈ L ∞ ∩ H t p and any non-negative measurable bounded function g,

| π γ f, g | ≤ f L ∞ lim n→+∞ 1 n n-1 i=0 g • T i dLeb ≤ f L ∞ lim n→+∞ 1 n n-1 i=0 gL i 1 dLeb ≤ f L ∞ | π 1 1, g | .
This means that the measures u dLeb, where u belongs to some E γ with |γ| = 1, are all absolutely continuous with respect to the reference measure µ = π 1 1 dLeb, with bounded density.

For any u ∈ E γ , we write u dLeb = ϕ u µ, where ϕ u ∈ L ∞ (µ) is defined µ-almost everywhere. The equation Lu = γu translates into T * (ϕ u µ) = γϕ u µ. Hence, since µ is invariant,

|ϕ u • T -γ -1 ϕ u | 2 dµ = |ϕ u | 2 • T dµ + |γ -1 ϕ u | 2 dµ -2ℜ ϕ u • T γ -1 ϕ u dµ = 2 |ϕ u | 2 dµ -2ℜ γ -1 ϕ u dT * (ϕ u µ) = 0. Let F γ = {ϕ ∈ L ∞ (µ) | ϕ • T = γ -1 ϕ}. The map Φ γ : u → ϕ u is an injective morphism from E γ to F γ .
We now show that Φ γ is also surjective, or in other words that the functions ϕ u π 1 1 always belong to H t p . Let be ϕ ∈ F γ . Since the continuous functions are dense in L 1 (µ), and any continuous function can be uniformly approximated (thus approximated in L 1 (µ), all our measures being finite) by a sequence of C 1 functions, C α ⊂ H t p is dense in L 1 (µ). We choose an approximation (ϕ m ) of ϕ by C α functions in L 1 (µ), and put

u m = π γ (ϕ m π 1 1).
The functions (u m ) belong to H p t (Leb) by Lemma 3.1. Moreover, |Lu| ≤ L|u| for any u ∈ H p t , so that Lu L 1 (Leb) ≤ u L 1 (Leb) . For any f ∈ C α , we have:

π γ (f π 1 1) L 1 (Leb) ≤ lim inf n→+∞ 1 n n-1 i=0 L i (f π 1 1) L 1 (Leb) ≤ f L 1 (µ)
Hence, f → π γ (f π 1 1) is continuous from C α (endowed with the semi-norm • L 1 (µ) ) to E γ (endowed with the norm • L 1 (Leb) ). As a consequence, (u m ) is a Cauchy sequence in E γ , which is of finite dimension and thus complete. The sequence (u m ) converges to some u ∈ E γ . Then, we have Φ γ (u) = ϕ, and Φ γ is an isomorphism.

The eigenvalues of L of modulus 1 are exactly those γ such that F γ is not reduced to 0. We have ϕ n γ ∈ F γ n for all n whenever ϕ γ ∈ F γ , so that this set is an union of groups. Since L only has a finite number of eigenvalues of modulus 1, this implies that these eigenvalues are roots of unity. In particular, there exists N > 0 such that γ N = 1 for any eigenvalue γ.

We now have a good description of the spectrum of L: its radius is 1, its essential radius strictly smaller than 1, and the eigenvalues of modulus 1 form a finite group, none of them having a nontrivial Jordan block. The measures corresponding to those eigenvalues are all absolutely continuous with bounded density with respect to a reference invariant measure, π 1 1 dLeb, and are all absolutely continuous with respect to the Lebesgue measure.

Second step: mixing physical measures Let us now assume that 1 is the only eigenvalue of L of modulus 1 (in the general case, this will be true for L N , so we will be able to deduce the general case from this particular case). Under this assumption, L n u converges to π 1 u for any u ∈ H t p . Consider the subset of F 1 (the T -invariant functions of L(µ) ∞ ) given by the nonnegative functions whose integral with respect to the measure µ is 1. It is nonempty, since it contains the function µ(X 0 ) -1 . It is a bounded convex subset of F 1 , whose extremal points are of the form 1 B for some minimal invariant set B defined µ-almost everywhere. Such extremal points are automatically linearly independent; we can also assume that they are subsets of Supp(µ). Since F 1 is finite-dimensional, there is only a finite number of them, say 1 B 1 , . . . , 1 B l , and a function belongs to F 1 if and only if it can be written as ϕ = α i 1 B i for some scalars α 1 , . . . , α l . The decomposition of the function 1 ∈ F 1 is given by 1 = 1 B i , hence the sets B i cover the whole space up to a set of zero measure for µ. Moreover, since B i is minimal, the measure

µ i := 1 B i µ µ(B i )
is an invariant ergodic probability measure.

Let

u i = Φ -1 1 (1 B i ) = 1 B i π 1 1 ∈ H t p
, then any element of E 1 is a linear combination of the u i . In particular, this applies to π 1 (f u i ) for any f ∈ C α (we recall that f u i ∈ H t p by Lemma 3.1, and that 1

B i is T -invariant in L ∞ (µ)); we now compute π 1 (f u i ). π 1 (f u i ), 1 B j = lim n→+∞ 1 n n-1 k=0 L k (f u i ) • 1 B j dLeb = lim n→+∞ 1 n n-1 k=0 f • 1 B i • 1 B j • T k dµ = B i ∩B j f dµ = µ(B i )δ i,j f dµ i
Hence, we get:

π 1 (f u i ) = f dµ i u i (5.2)
This enables us to deduce that each measure µ i is exponentially mixing, as follows. Let δ < 1 be the spectral radius of Lπ 1 ; we have L nπ 1 = O(δ n ). Then, if f, g are C α functions, and following (5.2),

f • g • T n dµ i = 1 µ(B i ) L n (f u i ), g = 1 µ(B i ) π 1 (f u i ), g + O(δ n ) = f dµ i 1 µ(B i ) u i , g + O(δ n ) = f dµ i g dµ i + O(δ n ).
Since C α is dense L 1 , every µ i is mixing and thus ergodic. Those µ i are ergodic measures, and absolutely continuous with respect to the Lebesgue measure, so that they are also physical measures.

Third step: basins We now turn to the relationship between Lebesgue measure and the measures µ i . For any function f ∈ L ∞ (Leb) ∩ H t p , let us write

π 1 (f ) = l i=1 a i (f )u i .
(5.3)

We shall need to describe the coefficients a i (f ). Let be 1 ≤ i ≤ l, and f ∈ L ∞ (Leb)∩ H t p . We recall that the sets B i are T -invariant and disjoint for the Lebesgue measure (except perhaps for a set a zero Lebesgue measure).

lim n→+∞ X 0 f • 1 n n-1 k=0 1 B i • T k dLeb = lim n→+∞ 1 n n-1 k=0 1 B i , L k f = 1 B i , π 1 f = a i (f )µ(B i )
Moreover, all the 1/n n-1 k=0 1 B i • T k are bounded by 1 ; let h i be one of their weak L 2 (Leb) limits. The function h i is T -invariant Leb-almost everywhere, and satisfies for all f ∈ L ∞ (Leb) ∩ H t p :

a i (f ) = 1 µ(B i ) f h i dLeb Since a i (1) = 1, we have h i dLeb = µ(B i ).
Let us now compute h i h j dLeb. To begin with, h j being invariant,

a j (1 B i • T n ) = 1 B i • T n • h j dLeb = 1 B i • T n • h j • T n dLeb = a j (1 B i L n 1).
The computation now gives :

h i h j dLeb = µ(B j )a j (h i ) = µ(B j ) lim n→+∞ 1 n n-1 k=0 a j (1 B i • T k ) (5.4) = µ(B j ) lim n→+∞ a j 1 B i 1 n n-1 k=0 L k 1 = µ(B j )a j (u i ) = µ(B i )δ i,j
Taking i = j, we get h 2 i dLeb = µ(B i ) = h i dLeb. Since h i takes its values in [0, 1], this shows that there exists a subset C i of X 0 such that h i = 1 C i , with 1 C i dLeb = µ(B i ). Moreover, (5.4) shows that Leb(

C i ∩ C j ) = 0 if i = j. For any function f ∈ L ∞ (Leb) ∩ H t p , a i (f ) = 1 µ(B i ) C i f dLeb
Moreover,

X 0 i 1 C i dLeb = i µ(B i ) = µ(X 0 ) = Leb(X 0 )
This shows that the sets C i form a partition of the space modulo a set of zero Lebesgue measure. We have proved that

π 1 (f ) = l i=1 1 µ(B i ) C i f dLeb u i , (5.5)
which means that the sets C i form a partition of almost all X 0 into ergodic basins for the physical measures.

6 Piecewise C 1+Lip maps and functions with bounded variation

For now on, we will assume that the transformation T is a piecewise C 1+Lip uniformly expanding map, and work with the space of functions with bounded variation. We prove the existence of a spectral gap for the operator L 1/|DetDT | in this space under simple assumptions, and compare our results with previous works by J.W. Cowieson [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF] and B. Saussol [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional expanding maps[END_REF].

In order to simplify the notations, in this section λ n denotes the essential infimum of the λ n (x).

As an introduction and a last tribute to our work with Sobolev spaces, we first prove a corollary to Corollary 1.5. Corollary 6.1.

If T is a piecewise C 2 expanding map with piecewise C 1 boundaries and if:

lim n→+∞ (D b n ) 1 n λ -1 n n < 1,
then there exist 0 < t 0 < 1/p 0 < 1 such that, for all 0 < t < 1/p < 1 such that t 0 ≤ t and 1/p 0 ≤ 1/p, the essential spectral radius of L 1/|DetDT | acting on H t p is strictly smaller than 1.

Proof.

Let us put ǫ = 1lim

n→+∞ (D b n ) 1 n lim n→+∞ λ -1 n n .
We first choose 0 < t 0 < 1 such that lim

n→+∞ (D b n ) 1 n lim n→+∞ λ - t 0 n n < 1 -ǫ/2.
Then, we choose 1 < p 0 < 1/t 0 such that lim n→+∞ (D e n )

(1-

1 p 0 ) 1 n < 1 + ǫ/2.
Let 1 > t ≥ t 0 and 1 < p ≤ p 0 such that t < 1/p < 1. The essential spectral radius of L 1/|DetDT | acting on H t p is at most

lim n→+∞ (D b n ) 1 p 1 n • (D e n ) 1 n (1-1 p ) • λ -t n | det DT n | 1 p -1 1 n L ∞ ≤ lim n→+∞ (D b n ) 1 n • (D e n ) (1-1 p 0 ) 1 n • λ - t 0 n n ≤ (1 -ǫ/2)(1 + ǫ/2) < 1
Corollary 6.1 can be seen as an asymptotic result; even if we work with parameters t and p close to 1, we do not work on H 1 1 . Indeed, this result does not hold when replacing H t p by H 1 1 . Since T is piecewise continuous, we must allow discontinuities in the function spaces we let L 1/|DetDT | act on, and for instance in dimension one every function of H 1 1 has a continuous version (because its derivative is a function in L 1 ). Here, the problem comes from the fact that the characteristic function of an interval is in general not in H 1 1 (R), since its derivative -in general a sum of atomic measures -is not in L 1 . Therefore, we have to work in a broader space, for instance the space of functions with bounded variation; this will lead us to an alternative proof of a theorem shown by J.W. Cowieson in 2000 [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF]. We will also have a closer look at the results proved by B. Saussol in 2000 [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional expanding maps[END_REF], which may be linked to this work in a similar fashion.

Cowieson's theorem

We recall here J.W. Cowieson's results [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF], and compare them to what we prove later. Theorem 6.2 [START_REF] Cowieson | Stochastic stability for piecewise expanding maps in R d[END_REF].

Let X be a bounded and connected open set of R d with piecewise C 2 boundaries, and T : X → X be a piecewise C 2 and uniformly expanding map with piecewise C 2 boundaries.

If there exists n ∈ N such that λ n > D b n , then T admits an absolutely continuous invariant probability measure whose density is in BV(X).

This theorem is indeed a consequence of Theorem 1.7. First, notice that (λ n ) n∈N and (D b n ) n∈N are respectively supermultiplicative and submultiplicative. Thus, the limits lim The differences between Cowieson's theorem and our results are the following. To begin with, our setting is less restrictive: the domain does not need to be the closure of a bounded, connected open set of some R d , but a compact subset of a Riemannian manifold, and the boundaries do not need to be piecewise C 2 but piecewise C 1 . The transformation T also only needs to be piecewise C 1+Lip , not piecewise C 2 . Then, the condition on the expansion rate and the combinatorial complexity is the same. At last, we shall prove that the essential spectral radius of L 1/| det DT | acting on BV is strictly smaller than 1. Moreover, once the spectral gap is proven, we will get not only the existence of an absolutely continuous invariant measure with bounded variation, but far stronger results, as in Theorem 1.7.

We recall once more that, for generic piecewise expanding maps, the complexity D b n increases subexponentially (Theorem 1.1 in [START_REF] Cowieson | Absolutely continuous invariant measures for most piecewise smooth expanding maps[END_REF]), so that the consequences of Theorem 1.7 hold generically.

Basic lemmas

We need to adapt the lemmas from Sections 3 and 4 to this new setting in order to get Theorem 1.4 and subsequently Theorem 1.7. They are:

• A Fubini-type property, which allows us to work in a 1-dimensional setting: Lemma 6.3.

• The continuity of the multiplication by Lipschitz functions (equivalent of Lemma 3.1): Lemma 6.4.

• The continuity of the multiplication by characteristic functions of nice sets (equivalent of Lemma 3.2): Lemma 6.5.

• The effect of composition with C 1+Lip diffeomorphisms (equivalent of Lemma 3.3): Lemma 6.6.

• A localization lemma (equivalent of Lemma 3.4): Lemma 6.7.

gu BV(R) = R |gu| dLeb + R |D(gu)| dLeb ≤ R |gu| dLeb + R |gDu| dLeb + R |uDg| ≤ g L ∞ R |u| dLeb + g L ∞ R |Du| dLeb + Dg L ∞ R |u| dLeb ≤ g Lip u BV(R)
For any u ∈ BV(R), let be a sequence of C 1 functions (u n ) n∈N approaching u in the sense of Theorem 2.10.

Then, since

gu n -gu L 1 ≤ g L ∞ u n -u L 1 ,
we see that (gu n ) is a sequence of BV(R) functions which converges to gu in L 1 . Theorem 2.11 tells us that:

gu BV(R) ≤ lim inf n→+∞ gu n BV(R) ≤ g Lip lim inf n→+∞ u n BV(R) ≤ g Lip u BV(R)
Lemma 6.5 (Multiplication by characteristic functions of nice sets).

Let O be a measurable subset of R d whose intersection with almost every line parallel to a coordinate axis has at most L connected components. Then, for all u ∈ BV,

1 O u BV ≤ L u BV .

Proof.

Assume that u ∈ C 1 . Using again the Fubini-type property of Lemma 6.3, we just have to show that 1 O u BV(R) ≤ L u BV(R) if O is an union of intervals, or even that 1 O u BV(R) ≤ u BV(R) if O is an interval, and then sum the contributions of the different intervals. Let us put O = [a, b] (the case O = R + is even easier). Then:

1 O u L 1 ≤ u L 1 , V (1 O u) = |u(a)| + [a,b] |u ′ (x)| dx + |u(b)| = (-∞,a] u ′ (x) dx + [a,b] |u ′ (x)| dx + [b,+∞) u ′ (x) dx ≤ V (u).
Summing those inequalities ends this proof for C 1 functions. For a general function in BV(R), and using the same trick as in the end of the proof of Lemma 6.4, we just have to show that, if (u n ) converges to u in L 1 , then (1 O u n ) converges to 1 O u in L 1 , which is trivial.

The next lemma deals with the composition with C 1+Lip diffeomorphisms. k = (k 1 , ..., k d ) ∈ Z d such that Supp(η) ⊂ O k . We also note, for λ ∈ Z d , by λk the vector (λ 1 k 1 , .., λ d k d ).

We point out that, for any u in BV(R d ) and λ in R d , the sum

λ∈Z d u BV(λk+l+O k )
is equal to u BV( λ∈Z d λk+l+O k ) , which is at most u BV(R d ) (the L 1 norms are the same, and the inequality of the variations comes directly from Definition 2.7) We now split the sum, and then apply Lemma 6.4:

m∈Z d η m u BV(R d ) = λ∈Z d l∈Z d 0≤l<k η λk+l u BV(λk+l+O k ) ≤ η C 1 l∈Z d 0≤l<k λ∈Z d u BV(λk+l+O k ) ≤ η C 1 l∈Z d 0≤l<k u BV(R d ) = η C 1 d i=1 k i u BV(R d ) .
This is the lemma, with

C η = η C 1 d i=1 k i .
Since the space BV make us work morally with p = 1 and t = 1, an equivalent of Lemma 3.5 for functions with bounded variation is that, for any functions v 1 , . . . , v l with compact support in R d and belonging to BV(R d ),

l i=1 v i BV(R d ) ≤ l i=1 v i BV(R d ) .
This is just the triangular inequality for the BV(R d ) norm, so that there is nothing to prove here.

Finally, it is a well-known fact that, for any open and bounded set with nice boundaries (for instance piecewise C 1 boundaries) Ω of some R d , the canonical immersion of BV(Ω) into L 1 (Ω) is compact; see for instance Theorem 1.19 in [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]. The compactness of the canonical immersion of BV(X 0 ) into L 1 (X 0 ) ensues.

Saussol's theorem

In sections 6.1 and 6.2, we have seen how to obtain Theorem 1.7, that is, a version of Theorem 1.6 when T is a piecewise C 1+Lip uniformly expanding map, in the limit p = 1 and t = 1. This involves the space of functions with bounded variation, since the Sobolev space H 1 1 is not suitable for this task (for instance, it is not large enough in dimension 1). One might want to know suitable spaces when one works instead with a piecewise C 1+α uniformly expanding map, and wants to get to the limit t = α (in this setting, Theorem 1.3 tells nothing about the essential spectral radius of the Perron-Frobenius operator on the H α p spaces).

The last part of this article is a study of a previous result by B. Saussol [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional expanding maps[END_REF], which is very close to our own results and gives us another example of function spaces to which our method could be applied, with the parameters t = α and p = 1. B. Saussol proved the existence of a spectral gap of L 1/| det DT | when acting on spaces of functions with bounded oscillation V α , when T is a piecewise C 1+α uniformly expanding map.

First, let us present the functions with bounded oscillation. Let X 0 be a compact set of some R d . For any Borel set S ∈ B(R d ) such that Leb(S) > 0, and any f ∈ L 1 (R d ), we denote the essential infimum of f on S by Einf S f , and its essential supremum by Esup S f . Next, we choose some ǫ 0 > 0 and α ∈ (0, 1], and define for all f ∈ L 1 (R d ): osc(f, S) = Esup S f -Einf S f (6.3)

|f | α = sup 0<ǫ≤ǫ 0 ǫ -α R d
osc(f, B(x, ǫ)) dx (6.4)

Then, we define

V α (R d ) = {f ∈ L 1 (R d ) : |f | α < +∞} and V α = V α (X 0 ) = {f ∈ V α (R d )) : Supp f ⊂ X 0 }, both endowed with the norm f Vα = f L 1 + |f | α .
Remark 6.8.

Different choices of ǫ 0 lead to different norms, although they are all equivalent: the space V α does not depend on ǫ 0 .

We have already encountered V 1 (R): it is the same space as BV(R) [START_REF] Keller | Generalized bounded variation and applications to piecewise monotonic transformations[END_REF]. Hence, the results of Saussol are a generalization of the previous theorem by A. Lasota and J.A. Yorke [START_REF] Lasota | On the existence of invariant measures for piecewise monotonic transformations[END_REF]. Remark 6.9.

We endow C α , the set of α-Hölder functions on X 0 , with the norm

f α = f L 1 + |f | ′
α , where:

|f | ′ α = sup x,y∈X 0 |f (x) -f (y)| |x -y| α
For f ∈ C α , x ∈ X 0 and ǫ > 0, we get easily the following inequalities:

|f | α = sup 0<ǫ≤ǫ 0 ǫ -α R d osc(f, B(x, ǫ)) dx ≤ sup 0<ǫ≤ǫ 0 ǫ -α (2ǫ) α Leb(X 0 )|f | ′ α ≤ 2 α Leb(X 0 )|f | ′ α
Hence, for all α ∈ (0, 1], we have a continuous inclusion from C α into V α . However, V α is much larger, so that functions belonging to this space may present discontinuities.

The setting studied by B. Saussol is more general than the one we presented in Section 1: for instance, it allows under some conditions boundaries whose Hausdorff dimension is strictly larger than d-1, or maps whose sets of continuity are countably many. However, one of these conditions is very abstract, and the use of more flexible conditions leads naturally to our setting.

In the following, we consider that α ∈ (0, 1] is fixed, and we put γ d = Leb(B(0, 1)). Here is the main theorem, an adaptation of Theorem 5.1 and Lemma 2.1 in [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional expanding maps[END_REF] with the additional use of Hennion's theorem [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipchitziens[END_REF]): Theorem 6.10 (Saussol's theorem).

Let T be a piecewise C 1+α uniformly expanding map. L 1/| det DT | acts continuously on V α , and its essential spectral radius is at most:

λ -α + 4γ d D b 1 (λ -1)γ d-1 . (6.5)
This theorem naturally leads to a result of existence of physical measures (stated in a different way in [START_REF] Saussol | Absolutely continuous invariant measures for multidimensional expanding maps[END_REF]) similar to Theorem 1.6: Theorem 6.11.

Let t be a piecewise C 1+α uniformly expanding map such that the bound 6.5 for the essential spectral radius is smaller than 1. Then T has a finite number of physical measures whose densities are in V α , which are ergodic, and whose basins cover Lebesgue almost all X 0 . Moreover, if µ is one of these measures, there exist an integer k and a decomposition µ = µ 1 + • • • + µ k such that T sends µ j to µ j+1 for j ∈ Z/kZ, and the probability measures kµ j are mixing at an exponential rate for T k and α-Hölder test functions.

Since the setting and conclusion are already familiar to the reader, we will look more closely at the upper bound of the essential spectral radius. The estimate 6.5 is rather rough, and for α = 1 and small d is worse than the one given in Theorem 1.7. However, just like in the proof of Theorem 1.3, one may iterate the transformation to get better estimates of the essential spectral radius ρ ess of the Perron-Frobenius operator, and then take the limit as the number of iterations grows to infinity: Proof. The expression 6.5 appears in a Lasota-Yorke type inequality in the proof of Saussol's theorem. When using Hennions's theorem [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipchitziens[END_REF], one can get a better bound on the essential spectral radius of an operator by iterating this operator. In other words, we have for all positive integer n: Since this is true for all ǫ > 0, inequality 6.6 ensues.

A

  i l , and J(n) grows polynomially. Following the claim p.105 in [Buz97], D b n ≤ 2J(n) d . This quantity grows subexponentially. By Theorem 1.4, the essential spectral radius of L 1/| det DT | acting on BV is bounded by λ -1 .

n

  < 1 if and only if there exists some n ∈ N such that λ n > D b n , and the condition stated in Cowieson's theorem and condition (1.8) are equivalent.

  d D b n (λ n -1)γ d-1 1 n (C + ǫ) -1 ≤ 1.

  d , and define a map T on X 0 by cutting it into finitely many polyhedral subregions O 1 , . . . , O N , applying A i to each open set O i , and then mapping A 1 O 1 , . . . , A N O N back into X 0 by translations. It is obviously piecewise C ∞ and uniformly expanding. Let λ be the lowest modulus of all the eigenvalues of all the matrices A i ; we get easily λ n ≥ λ n .

	Proposition 1.8.
	Under those assumptions, the essential spectral radius of L 1/| det DT | acting on BV
	is at most λ -1 , thus strictly smaller than 1. Therefore, T satisfies the conclusions
	of Theorem 1.7.

  .1) This inequality ends the first part of the proof. Second step: u • F We will use interpolation. Notice that, since D F ≥ 1/2 everywhere, | det F -1 | ≥ 2 d and u • F L p ≤ 2 d/p u L p for all u ∈ L p . We also get the same way u • F Du L p (see for instance §2.3.3 in [Tri78]).Hence, by Proposition 2.4, there exists C t,p such that, for all t ∈ [0, 1], for all u ∈ H t p ,

	u • F	H t p	≤ C t,p 2 d/p u H t p .	(3.2)
				H 1

p ≤ C p 2 d/p+1 u H 1 p for some C p , since the H 1 p norm is equivalent to u L p +

  Then, there exists a constant C t,p,η such that, for all u ∈ H t p ,

	1
	p
	η m u p H t p	≤ C t,p,η u H t p .	(3.4)
	m∈Z d

and write, for all x ∈ R d and m ∈ Z d , η m (x) = η(x + m).

• A summation lemma (equivalent to Lemma 3.5).

• The compactness of the injection from BV(X 0 ) to L 1 (X 0 ).

In may arguments below, we will work in fact with C 1 functions, and then proceed by approximations to get the results for functions with bounded variation, the approximation being done with Theorems 2.10 and 2.10. We hope that the quite explicit computations involved will help to explain the properties described by the many lemmas in Section 3 and Section 4.

We start with the Fubini-like theorem, which allows us to reduce the problem to dimension 1 in the proofs of some following lemmas. Let

We put x = (x 1 , ..., x i-1 , x i+1 , ..., x d ). We will denote by u x,i the map t → u(x 1 , ..., x i-1 , t, x i+1 , ..., x d ), and by u i the map x → u x,i BV(R) .

Lemma 6.3 (Fubini-type property).

For all u ∈ C 1 ∩ BV(R d ) and i ∈ 1..d, then u i ∈ L 1 and

On the other hand,

Then, all is left is to sum the equalities.

For any Lipschitz function u ∈ Lip(R d , C), we put u Lip = u L ∞ + L(u), where L(u) is the best Lipschitz constant for u. With this lemma, we can easily deduce the following:

Proof.

Assume first that u ∈ C 1 . Thanks to the Fubini-type property of Lemma 6.3, we just have to show that gu BV(R) ≤ g Lip u BV(R) . Since g is continuous, this will be an application of the Leibniz formula (see e.g. Proposition 3.2 in [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF]; the derivative of g has to be taken in a weak sense):

Proof.

Assume first that u is C 1 . We put, as in the proof of Lemma 3.3,

We now deal with u • A -1 • F ; we write F = A -1 • F . Since the derivative of F is everywhere bounded between 1/2 and 2, | det D( F -1 )| ≤ 2 d and a change of

Together, (6.1) and (6.2) give

Now, let us take any function u in BV(R d ), and use the same trick as in the end of the proof of Lemma 6.4. Since u

Hence, inequality (6.2) holds for any function in BV(R d ).

Inequality (6.1) can be obtained with a slighly different method: u • A L 1 ≤ | det A| -1 u L 1 holds obviously for any u in L 1 . We just have to prove that V (u • A) ≤ | det A| -1 A V (u) for any u in BV(R d ); the approximation by C 1 function still works (the reader can easily check that Theorem 2.11 is still true if one writes V (•) instead of • BV ).

Therefore, Lemma 6.6 holds for any function in BV(R d ).

A consequence of (6.1) is that, for all u ∈ BV(R d ) and

The next result to prove is the equivalent of the first localization principle, Lemma 3.4.

Lemma 6.7 (Localization principle).

Let be η ∈ C 1 c (R d , R), and write, for all x ∈ R d and m ∈ Z d , η m (x) = η(x + m). Then, there exists a constant C η such that, for all u ∈ BV(R d ),

Proof.

We define on R d a relation by x < y if and only if x i < y i for all i ∈ {1, ..., d}, and another relation ≤ the same way. For k ∈ Z d , 0 < k, we define O k = {x ∈ R d : 0 < x < k}. Up to a translation, we may assume that there exists some Clearly, a sufficient condition for the conclusions of Theorem 6.11 to hold is lim n→+∞ λ

1 n < 1. The V α spaces were constructed to satisfy the same properties that we needed (they include the characteristic functions of nice enough sets, the multiplication by α-Hölder functions behaves nicely, and the injection into L 1 is compact), and one should be able to adapt the different lemmas as we did for the space of functions with bounded variation. Finally, our method would probably give a different (and worse) estimate of the essential spectral radius of the Perron-Frobenius operator, such as (we take t = α and p = 1 in Theorem 1.3):

Saussol also gives a lower bound on the spectral gap via the study of the cones in V α (for another example of cone contraction method, see e.g. [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF]), and an upper bound on the number of ergodic physical measures. Such features could perhaps be adapted to our current setting, with Sobolev spaces or the space of functions with bounded variation.


M J Baptist
email: martin.baptist@wur.nl

Baptist, H. M. J M F Leopold

Effectiviteit Van Vooroeversuppleties

The effects of shoreface nourishments on Spisula and scoters in The Netherlands

Keywords: Shoreface nourishment, Spisula subtruncata, Melanitta nigra, North Sea, Seabirds, Benthos, Sediments

The coast of The Netherlands is protected by nourishing sand. Generally, two different techniques are used, beach nourishment and shoreface nourishment. The latter technique supplies sand at a water depth of about 5 to 8 metres in the surf zone, and has been used on a regular basis since 1997 with increasing volumes since 2001. Observations on the bivalve mollusc Spisula subtruncata that was abundant before 1997 and a key food species for wintering seaduck show a decline since 2001. This coincided with a decrease in the abundance of the Common Scoter Melanitta nigra, the most numerous wintering seaduck off the Dutch coast. These observations raised concern about shoreface nourishments. This study analyses the timing and locations of shoreface nourishments in combination with S. subtruncata abundance and spatial distribution. Against the expectation, no causal relationship was found between the decline of S. subtruncata and shoreface nourishments. Other causes, such as climate change, fisheries, unsuccessful settlement or predation of spatfall are more likely behind the decline of Spisula along the Dutch coast.

12 Sand nourishments can affect the ecosystem in various ways (Van Dalfsen and 13 Essink, 1997;Speybroeck et al., 2006). Direct effects are the burial of ben-14 thic species under a layer of sand. In the direct vicinity suffocation of benthos 15 can occur due to the settling of a plume of suspended particles. A plume of 16 fine particles may also affect primary production and the foraging success of 17 filter-feeding benthos, fish and birds. In addition there might be disturbance 18 by noise and ship manoeuvering. Indirect effects are habitat change, such as 19 altered morphology and sedimentology. This may also change habitat char-20 acteristics such as penetrability, compactness, and organic matter and silt 21 content.

22

In 1993 a shoreface nourishment was applied as large-scale pilot experiment 23 along Terschelling. Possible ecological effects were extensively monitored (Ty-24 deman, 1994; Van Dalfsen and Oosterbaan, 1996;Van Dalfsen and Essink, 25 1997). It was concluded that there were mainly short-term, local effects as 26 a result of burial of benthic species. A recovery of the benthic fauna was 27 reached after two years, with the exception of long-lived species of molluscs 28 and echinoderms. Particular attention was paid to Spisula subtruncata, a bi-29 valve that formed the staple food source for the Common Scoter (Melanitta 30 nigra) (Leopold et al., 1995;Degraer, 1999;Degraer et al., 2007). Spisula sub-31 truncata is found primarily in fine sands of about 200 µm in the lower shoreface 32 at water depths of about 5 to 20 metres (Baptist et al., 2006;Craeymeersch 33 and Perdon, 2006;Degraer et al., 2007). In their study, Van Dalfsen and Es-34 sink (1997) gave a warning: "[...] shoreface nourishment may very well strike 35 important Spisula banks, and then pose a serious risk to the survival of win-36 tering seaducks".

37

Spisula surveys in Dutch coastal waters have been conducted since the 1990s 38 (Leopold, 1996;Craeymeersch and Perdon, 2006). Spisula was very abundant

47

There is no statistically significant correlation between declining Spisula abun-48 dance and increasing nourishment volumes (for untransformed data: r = -0.33, 49 two tailed probability = 0.29). However, the coincidence implies a possible were interpolated, and extrapolated to the line of mean tide height, to yield 91 complete coverage of Spisula density (numbers per m 2 ) for the entire Dutch 92 coastal zone. The amount of overlap between nourished areas and Spisula 93 banks proved to be highly variable in space and time. Spisula banks occur at 94 least down to the -20 m NAP depth contour and up to the surf zone where 95 nourishments take place. Therefore, the Spisula density was determined for:

96

(1) The location of the nourishment itself (see Fig. 3). Sampling did not in all 97 cases take place close enough inshore to cover the nourishment locations.

98

This implies that the resulting distribution is based on extrapolation. The highest densities (>1000 m -2 ) were found at approximately 2 to 4 km 172 from the coast, at depths of -9 to -15 m NAP (Leopold, 1996). The shoreface 173 nourishment was, therefore, not applied in the area with the highest densities, , 1995;Armonies et al., 2001;Ellien et al., 2004). If spatfall takes place in [START_REF] Weinberg | Bathymetric shift in the distribution of Atlantic sur-459 fclams: response to warmer ocean temperature[END_REF].

263

We observed that successful spatfall of Spisula does not occur yearly, Fig. 6. 0-year adult 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

  , coastal protection offered by beaches and dunes is main-2 tained by nourishing sand. At first, this was achieved by supplying sand di-3 rectly onto the beach but later shoreface nourishment became the preferred 4 technique (Roelse, 2002; Grunnet and Ruessink, 2005; Klein, 2005; Steijn, 5 1.

39

  along large parts of the Dutch coast in the 1990s, but during the period 2000 40 -2005 numbers declined. The total biomass in 2005 was the lowest found 41 since 1995, and has only slightly increased in later years (Craeymeersch and 42 Perdon, 2006; Goudswaard et al., 2008). The decline in total Spisula abun-43 dance coincided with the increase in shoreface nourishments, Fig 1. Moreover, 44 the numbers of overwintering Common Scoters also showed a decline in recent 45 years, Fig. 2. This raised concern about the ecological effects of shoreface nour-46 ishments, particularly after the warning of Van Dalfsen and Essink (1997).

99( 2 )Figure 4

 24 Figure4shows the biomass of adult Spisula and the coincidence with nour-112

Figure 5 Fig. 3 .

 53 Figure5presents the Spisula densities for 0-year and adult specimens for a 159

  171

174

  and has only buried Spisula at the landward fringe of the bank. 175 Ter Heijde 1997. Spisula did occur here in the early 1990s, but was no longer 176 present at the time of the nourishment. 177 Noordwijk 1998. This 1.3 Mm 3 nourishment was applied on top of a Spisula-178 bank according to our interpolated data. Exact data on this are not available, 179 since direct monitoring of Spisula on-site has not been carried out. Spisula did 180 not recover in the years following this nourishment. 181 Ameland oost 1998. This 2.5 Mm 3 nourishment was applied at a depth 182 of -5 to -7 m NAP, just outside the surf zone (Steijn, 2005). The surveys in 183 1998 show highest densities of Spisula at depths of -6 m NAP and deeper. 184 The nourishment was located on the landward side of the Spisula habitat, 185 but must have covered part of it. A recovery has been observed, based on a 186 successful spatfall in 1999. However, since 2003, the Spisula bank no longer 187 exists. Our results furthermore show that in the 1990s, Spisula banks were 188 located further away from the coast than in later years. 189 Bergen aan Zee 2000. This 1 Mm 3 nourishment was located in an area with 190 high densities of 0-year Spisula, which stretched further offshore as well, ac-191 cording to our interpolated data. In 2001 high densities of adult Spisula were 192 observed, but these gradually disappeared in later years. 193 Groote Keeten -Zwanenwater 2003. This 2.57 Mm 3 nourishment was lo-194 cated in an area where high densities of Spisula were found until 2001. Spisula 195 therefore had disappeared before the nourishment was applied. 196 Texel Molengat 2003. This area contained very rich Spisula banks in 1998 197 and 1999. These were heavily fished (Leopold, 1999) and have disappeared by 198 2000, before the nourishment was applied. 199 Texel De Slufter 2004. At this location rich Spisula banks in 1998 and 1999 200 were found. Spisula had disappeared well before the nourishment was applied.

  to Common Scoters204There has been a general increase of numbers of Common Scoters wintering 205 along the east Atlantic coast in the early 1990s, followed by a decline(ICES, 206 2005). This decline was marked in the Netherlands and Belgium where the diet 207 was mainly Spisula subtruncata, which also declined in abundance(Degraer 208 et al., 2007; Goudswaard et al., 2008). Numbers of Common Scoters did not 209 decline in France, where Spisula probably was far less important, but num-210 bers did decline sharply in northwest Portugal, where their diet is presently 211 unknown(ICES, 2005). The decreasing trend in numbers of Common Scot-212 ers in The Netherlands since the 1990s (Fig.2) is attributed to the declining 213 availability of their staple food source, Spisula(Leopold et al., 1995; Leopold, 214 1996 Leopold, 214 , 1999;; Fox, 2003; ICES, 2005). The distribution of Common Scoters in 215 the coastal zone of The Netherlands used to be determined by the presence 216 of dense banks of Spisula. In 1987 the majority of Scoters was present along 217 the Zuid-Holland coast north of Noordwijk. From 1991 to 1993 large groups 218 were first found along Schiermonnikoog, but gradually they shifted towards 219 Ameland and Terschelling (Leopold, 1996). Here, Spisula-banks were present 220 but they were heavily fished, which resulted in Scoters leaving for the Noord-221 Holland coast, off Petten (Leopold, 1996). In the following years, 1994 and 222 1995, Common Scoters flew back and forth between Ameland/Terschelling 223 and Noord-Holland, but now both areas were fished. The severe winters of 224 1995/1996 and 1996/1997 brought ice-cover in the coastal waters north of the 225 Wadden-islands and a large mortality of Spisula and other benthic macrofauna 226 (Armonies et al., 2001). The scoters moved to Noord-Holland again. In 1999 227 the densest bank of Spisula was located offshore of southwest Texel. This bank 228 was fished away in May 1999 (Leopold, 1999) and did not recover. The scot-229 ers retreated to their favorite areas Terschelling and Ameland, where the last 230 banks of Spisula were present. Nowadays, Spisula has (almost) disappeared 231 and this clearly affects the coastal population of Common Scoters. Although 232 scoters can feed on Ensis americanus as an alternative food source, these bi-233 valve molluscs are apparently an inferior prey (Leopold and Wolf, 2003; Wolf 234 and Meininger[START_REF] Van Vooroeversuppleties | Nourishing the shoreface: observations and hindcasting of 457 the Egmond case, The Netherlands[END_REF] Skov et al., 2008), because of their awkward shape and 235 their longer handling time.

  236 4.2 The decline of Spisula 237 If shoreface nourishments have not caused the decline of Spisula, and subse-238 quently scoters, what then was the cause? As with many other benthic species 239 in the marine environment, the spatial and temporal distribution and abun-240 dance of Spisula is highly variable (Hagmeier, 1930; Degraer, 1999; Degraer 241 et al., 2007). Cardoso et al. (2007) suggest that the current low population size 242 is due to unsuccessful settlement and/or severe predation by benthic predators 243 during the first months of life. One of the factors that influences the settlement 244 success is the wind direction and force during the pelagic larval phase, which 245 for Spisula lasts one month (Cardoso et al., 2007). The resulting hydrographic 246 patterns determine where the postlarval stage ends up, so where the spatfall 247 (settlement of shellfish larvae on or in the substrate) takes place (Belgrano 248 et al.

  249 a favourable environment, a new Spisula bank may be formed. Craeymeersch 250 and Perdon (2006) tried to link data on wind direction and wind force to the 251 reproduction success of North Sea Spisula subtruncata, but did not find a good 252 correlation. 253 During the life cycle of bivalves, many other factors play a role as well. The 254 presence, or rather absence, of dense banks of adult specimens in the neigh-255 bourhood may be a limiting factor, as was found for Mya truncata in the 256 Frisian Front (Amaro et al., 2003). Macoma balthica in the adjacent Wadden 257 Sea, has been hit by climate change that led to a decrease in reproductive 258 success, due to a shortage of food for larvae and an increased predation by 259 shrimp after mild winters (Philippart et al., 2003; Strasser et al., 2003; Bos 260 et al., 2006). Climate change can also lead to adult mortality as a result of 261 thermal stress and poor physiological condition, as was observed for Spisula 262 solidissima solidissima

264

  The last successful recruitment in the Voordelta occurred in 1995. Along the 265 Zuid-Holland coast, recruitment in 1996 was followed by a smaller recruitment 266 in 2000. Recruitment in 2000 was better along the Noord-Holland and best 267 along the Wadden coasts. But 2000 was the last successful recruitment for 268 Spisula anywhere in The Netherlands. 269 Since Spisula is a short-lived species (approximately 4 years), the absence of 270 a successful recruitment after 2000 is disturbing. It might be that spatfall has 271 occurred in the meantime, but was preyed upon by the Brown shrimp Cran-272 gon crangon (Beukema and Dekker, 2005). Data on shrimp densities along 273 the Dutch coast show no trend for the Holland coast and indeed an increasing 274 trend for the Wadden coast (Henk Heessen -IMARES, pers. com.). In recent 275 years spatfall of Ensis americanus has been very successful, despite shrimp 276 predation, but there is not enough knowledge on the biology of shellfish larvae 277 to explain possible differences. Based on Fig. 6 one might suspect a northern 278 shift in the distribution of Spisula. Notably the 0-year distribution in 2000 279 shows a distinct northern direction. The mean sea water temperature in the 280 North Sea has risen. Possibly Spisula subtruncata does not show a bathymetric 281 shift in response to warming, as does Spisula solidissima solidissima (Wein-282 berg, 2005), but a northerly shift. Unfortunately, recent data on the presence 283 of Spisula subtruncata in German waters is lacking (Laursen and Essink, 2005; 284 ICES, 2005), but scoters off the Danish coast apparently still feed on Spisula 285 (Skov et al., 2008). 286 Since 2004, large adult Spisula-banks have been absent in the Dutch coastal 287 zone. Spisula does not suffer from reproductive failure (Cardoso et al., 2007) 288 and there might still be a viable, but low density population left. In com-289 bination with correct meteorological and hydrodynamical circumstances, a 290 successful spatfall might re-occur, but there is no telling when, in the near 291 or distant future, this might happen. An interesting observation is that the 292 numbers of European Otter Clam Lutraria lutraria in Dutch coastal waters 293 is increasing since the beginning of this century. The estimated biomass (in 294 fresh weight) for Lutraria in 2008 was approximately ten times higher than 295 for Spisula (Goudswaard et al., 2008). Their biomass should would need to 296 increase by 10-fold to reach the biomass of Spisula of the 1990s. Future studies 297 need to establish whether Lutraria can serve as an alternative food source for 298 wintering scoters.

Figure 1 .

 1 Figure 1. Total number of adult Spisula subtruncata (solid line, left axis) and the nourished volume of sand (bars, right axis) for the entire Dutch coast. Number of nourishments per year above the bars. Data courtesy of the National Institute for Coastal and Marine Management..

Figure 2 .

 2 Figure 2. Winter maxima of abundance of Common Scoter (Melanitta nigra) along the entire Dutch coast. Data courtesy of the National Institute for for Coastal and Marine Management / RIKZ, IMARES, NIOZ and Dutch Seabird Group.

Figure 3 .

 3 Figure 3. Locations of the shoreface nourishments along the Dutch Coast. Large labels denote the eight nourishment locations discussed in the text and small labels (underlined) denote regional names used in the text. NAP is Dutch Ordnance Level.

Figure 4 .

 4 Figure 4. Biomass of adult Spisula combined with nourished volumes in four sections of the Dutch coast. Numbers denote nourished volumes in Mm 3 . The surface area of the circles scales with the size of the volume or biomass. .

Figure 5 .

 5 Figure 5. Estimated Spisula density for 0-year and adult age classes in nourishment locations. A white, thin bar represents the year of the shoreface nourishment, 'n.d.' means 'no data'. The Terschelling nourishment took place in 1993 (off scale). These eight examples are scattered over the total time-frame of this study and over a large part of the Dutch coast.

Figure 6 .

 6 Figure 6. Biomass of 0-year and adult age classes of Spisula in four main sections of the Dutch coast. Numbers given denote biomass of fresh weight of adults in ktons (10 6 kg). Table 1. Shoreface Shoreface nourishments in The Netherlands. Data from Roelse (2002), Steijn (2005) and Ministry of Transport, Public Works and Water Management. Beach Pole denotes the section along the beach where the nourishment took place. NAP is Dutch Ordnance Level. Table 2. Spisula abundance at nourishment locations with area in hectares, as calculated from interpolated data. Nourishment locations that were used in multiple years and overlap almost completely were combined to one (n.d. -no data).Table 3. Spisula abundance at extended (down to -20 m depth contour if possible) nourishment locations, with area in hectares, as calculated from interpolated data (n.d. -no data).

  Figure5e

  Figure5g

  Figure 6

Table 2 ACCEPTED MANUSCRIPT

 2

		ad.	0.1	0.3	0	0.4	0.1	0.3	0	0	0	0	0
	Vlieland oost	30.4 0-yr	0	2 n.d.		0	0	0	0	0 n.d.	n.d.	n.d.	n.d.
		ad.	0	0 n.d.		0	0	0	0	0 n.d.	n.d.	n.d.	n.d.
	Terschelling midden	212.5 0-yr	0.3	150.1	49.2	148.8	2.9	130.8	51.9	37.6	0.1	0	0.5
		ad.	19.3	103.7	11.3	1.6	17.3	6.8	0.7	16.6	25.1	3.5	2.2	0.7
	Ameland midden	231.6 0-yr	0.1	18.2	13.8	341.2	0.2	493.3	14.6	2	0	0	0
		ad.	71.3	4.4	4	8.5	35.3	0.3	0.7	85.1	4.7	2.2	0.2
	Ameland oost	368.9 0-yr	0	14.5	0.5	582.4	0	66.8	418.7	0.7	0.2	0	0
		ad.	0.4	1.3	2.3	2.2	143.5	1.6	3.2	56.2	67.3	6	2.2	0.2

Table 3

 3

Spisula survey of the National Institute of Fisheries Research This project was committed and supervised by Ilse Jonker, Ministry 332 of Transport, Public Works and Water Management, National Institute for