Frédéric Campana

J.-P Demailly

Misha Verbitsky

Jean-Pierre Demailly

Kähler threefolds without nontrivial subvarieties Compact Kähler 3-manifolds without nontrivial subvarieties

Keywords: Mathematics Subject Classification: 32J17, 32J25, 32J27 Kähler manifold, complex threefold, holomorphic foliation, nef bundle, pseudoeffective bundle, hard Lefschetz theorem, Lelong number

We prove that any compact Kähler 3-dimensional manifold which has no nontrivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of so-called simple manifolds, central in the bimeromorphic classification of compact Kähler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of Siu and of the second author on the Lelong numbers of closed positive (1, 1)-currents, and with a version of the hard Lefschetz theorem for pseudo-effective line bundles, due to Takegoshi and Demailly-Peternell-Schneider. In a similar vein, we show that a normal compact and Kähler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple nonprojective torus if it carries no effective divisor. This is a crucial step towards completing the bimeromorphic classification of compact Kähler 3-folds.

Introduction

We consider here connected compact Kähler manifolds X of (complex) dimension n > 1. An irreducible compact analytic subset Z of X will be said to be a subvariety of X. It is said to be nontrivial if its (complex) dimension is neither 0 nor n.

The bimeromorphic classification of compact Kähler manifolds can be reduced, by means of suitable functorial fibrations, to the following two extreme particular cases: either X is projective or X is simple, which means that any general point x of X is not contained in any nontrivial subvariety of X (a point is general if it lies in the intersection of countably many dense Zariski open subsets). If X is a nonprojective torus, this new notion of simpleness coincides with the classical one, defined by the absence of nontrivial subtori. The functorial fibrations needed for this reduction are relative algebraic reductions and relative Albanese maps (see [START_REF] Campana | Réduction algébrique d'un morphisme faiblement Kählérien propre[END_REF], [START_REF] Campana | Réduction d'Albanese d'un morphisme propre et faiblement Kählérien et applications (I, II)[END_REF], [START_REF] Fujiki | On the structure of compact complex manifolds in C. Algebraic varieties and analytic varieties[END_REF]) in the category of connected compact Kähler manifolds, where the morphisms are the dominant rational maps with connected fibre.

The present text is concerned with the bimeromorphic classification of such simple X. Its difficulty, in contrast to the projective case, is not due to the abundance and complexity of the examples, but to their (expected) scarceness and simple structure, which makes essentially all usual invariants of the classification vanish.

For example, if X is simple, its algebraic dimension 1 a(X) vanishes, which implies, by [START_REF] Ueno | Classification theory of compact analytic spaces and algebraic varieties[END_REF], Lemma 13.1 and Lemma 13.6, that its Albanese map is surjective and has connected fibres. Because the fibres must then have dimension either 0 or n, we get: either q = 0, or X is bimeromorphic to its Albanese torus. Thus only the case where q = 0 needs to be considered.

The known 2 examples of simple compact Kähler manifolds are (up to bimeromorphic equivalence) general complex tori, K3 surfaces, or the general member of the deformation families of hyperkähler manifolds 3 . 'Simple' surfaces are thus classified: they are either tori (q > 0), or K3 (q = 0). But when n 3, the classification is open. The situation is however expected to be similar to the surface case in higher dimensions (see Conjecture 1.1), where the only known examples are constructed out of surfaces which are either K3, or tori.

The following converse was essentially formulated in [START_REF] Campana | Isotrivialité de certaines familles Kählériennes de variétés non projectives[END_REF], as Question 1.4.

Conjecture 1.1. Let X be a simple compact Kähler manifold. Then the following hold.

1. either X has a finite étale cover bimeromorphic to a complex torus, or H 0 (X, Ω 2 X ) is generated by some σ which is generically symplectic (that is, n = 2m is even, and m σ = 0).

We should thus have: κ(X) = 0. If dim(X) is odd, then X should be bimeromorphic to a complex torus, possibly after some finite étale cover. Such a cover will be needed, as shown by the Kummer quotient by the -1 involution.

2. When X does not contain any nontrivial subvarieties, X should be either a complex torus, or an irreducible hyperkähler manifold 4 .

Thus K X should be trivial in this case, the two cases being distinguished by q > 0, or q = 0.

1 Recall that a(X) ∈ {0, ..., n} is the transcendence degree of the field of global meromorphic functions on X, a(X) = n if and only if X is projective.

2 It is shown in [START_REF] Kaledin | Hyperholomorphic sheaves and new examples of hyperkähler manifolds[END_REF] that a stable bundle which does not degenerate to a direct sum of stable bundles on the generic deformation of a Hilbert scheme of K3 has a compact moduli space; this may lead to new examples of hyperkähler manifolds. The rigidity of the category of coherent sheaves on the general members of these families is established in [START_REF] Verbitsky | Coherent sheaves on general K3 surfaces and tori[END_REF], [START_REF] Verbitsky | Coherent sheaves on generic compact tori[END_REF].

3 Recall that X is hyperkähler it is Kähler and admits a holomorphic symplectic form. It is said to be irreducible if, moreover, π 1 (X) is finite, and h 2,0 (X) = 1.

4 Notice that a general deformation of a Hilbert scheme of a K3 surface has no subvarieties [START_REF] Verbitsky | Trianalytic subvarieties of the Hilbert scheme of points on a K3 surface[END_REF], while a general deformation of a generalized Kummer variety has some subvarieties, partially classified in [START_REF] Kaledin | Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties[END_REF].

3. If X is generically symplectic, then π 1 (X) should be finite, of cardinality at most 2 m , where 2m = n.

Assertion 3 follows from [START_REF] Campana | Réduction d'Albanese d'un morphisme propre et faiblement Kählérien et applications (I, II)[END_REF], if χ(O X ) = 0 for X simple and generically symplectic. In what follows, we prove the second assertion of the conjecture for n = 3. This conjecture can be motivated by the conjectural existence of minimal models in the bimeromorphic category of connected compact Kähler manifolds. Indeed, if such a theory exists, and if X is simple, we have a(X) = 0, which brings κ(X) 0. The possibility κ(X) = -∞ is excluded, since X should then be uniruled. This implies that κ(X) = 0.

In this case, a Kähler minimal model is a map X X ′ such that X ′ has terminal singularities and K X ′ ≡ 0). A (conjectural) Bogomolov-type decomposition ([Bog74], [START_REF] Beauville | Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF]) for manifolds with terminal singularities and K X ′ ≡ 0 could be used to represent a finite cover of X ′ as a product. However X is simple, hence either X ′ is covered by a simple torus (with finite locus of ramification), or X ′ carries a symplectic holomorphic 2-form, establishing the conjecture above.

The above conjecture can thus be reformulated in terms of conjectural minimal models for compact Kähler manifolds, defined as normal compact complex spaces bimeromorphic to compact Kähler manifolds, and having Qfactorial terminal singularities. The simple ones should then have a torsion canonical sheaf, and be either of the form T /G, quotients of simple nonprojective tori by a finite group of automorphisms, or carry a multiplicatively unique holomorphic 2-form, symplectic on the regular locus.

Theorem 1.2. Let X be a compact Kähler 3-dimensional manifold. Assume that X has no nontrivial subvariety. Then X is a torus.

The proof will be given as Corollary 4.4. Although it is an easy combination of known results, it seems worthwile to present it, because it indicates that the above conjecture might be accessible in dimension 3 with the actually existing techniques. The proof also gives some information on the higher dimensional case (see Corollary 4.5).

We also show the following result.

Theorem 1.3. Let X be a 3-dimensional Kähler normal connected compact complex space with terminal singularities. Assume that K X is nef, and that X does not contain any effective divisors5 . Then X = T /G is isomorphic to the quotient of a nonprojective simple torus by a finite cyclic group of automorphisms.

We will use the following simple criterion: Lemma 1.4. Let X be a simple compact Kähler threefold. If χ(O X ) = 0, then X is bimeromorphic to its Albanese torus.

Proof. By the above remarks, it is sufficient to show that q > 0. But 0 = χ(O X ) = 1 -q + h 2,0 -h 3,0 . By Kodaira's theorem, h 2,0 > 0 since X is not projective, and h 3,0 1 since a(X) = 0. Thus 0 = 1 -q + h 2,0 -h 3,0 1 -q + 1 -1 = 1 -q, and q > 0.

The proof of Theorem 1.2 is thus reduced to showing that χ(O X ) = 0, which is the objective of the following sections.

Finally, let us mention that compact (connected) complex manifolds, not necessarily Kähler, without nontrivial subvarieties are also quite important in model theory, giving examples of so-called trivial Zariski geometries ([Moo05], [START_REF] Moosa | Model theory and Kaehler geometry[END_REF] [MMT08]). Our arguments below show that compact 3-dimensional complex manifolds without nontrivial subvarieties are tori, provided their canonical bundle are pseudo-effective. Brunella's theorem and the hard Lefschetz theorem appear to be the only steps of the proof requiring the Kähler hypothesis in depth.

Pseudo-effective and nef line bundles

We refer to [START_REF] Demailly | Analytic methods in algebraic geometry[END_REF] for the basic notions and properties of currents on complex manifolds, and only recall briefly some of the facts needed here.

Let X be a compact, n-dimensional complex manifold and let L be a holomorphic line bundle on X. Assume that L is equipped with a singular hermitian metric h with local weights e -ϕ which are locally integrable. Its curvature current is then Θ(L, h) := i∂∂ϕ, and its first Chern class c 1 (L) is represented by the cohomology class 1 2π [Θ(L, h)] in the Bott-Chern cohomology group H 1,1 BC (X), where

H p,q BC (X) = {d-closed (p, q)-currents}/{∂∂-exact (p, q)-currents}.
If X is Kähler, H 1,1 BC (X) is isomorphic to the Dolbeault cohomology group H 1,1 (X) and can be viewed as a subspace of H 2 (X, C).

Definition 2.1. A (1, 1) cohomology class {α} ∈ H 1,1 BC (X) is said to be pseudoeffective if it contains a closed positive (1, 1)-current Θ. The class {α} is said to be nef if for a given smooth positive (1, 1)-form ω > 0 on X and every ε > 0, it contains a smooth closed (1, 1)-form α ε such that α ε ≥ -εω [or alternatively, in the Kähler case, if {α} is a limit of Kähler classes {α + εω}, where ω is Kähler]. A class {α} is said to be big if it contains a Kähler current, namely a closed (1, 1)-current Θ such that Θ ≥ εω for some smooth positive (1, 1)-form ω and some ε > 0. A line bundle L on X is said to be nef (resp. pseudo-effective, resp. big) if its first Chern class c 1 (L) ∈ H 1,1 BC (X) has the same property.

It follows from the Bochner-Kodaira technique that a big line bundle L → X has maximal Kodaira dimension κ(L) = dim X. Therefore, a compact complex manifold X carrying a big line bundle is Moishezon, that is, bimeromorphic to a projective algebraic manifold; in particular, X is not simple. Now, we have the following result.

Theorem 2.2. ([DPa04])Let X be a compact Kähler manifold. Let {α} ∈ H 1,1 (X) be a nef class. If X α n > 0 then {α} is big. As a consequence, if X carries a nef line bundle L such that c 1 (L) n > 0, then L is big and X cannot be simple.

This special case where {α} = c 1 (L) (the only one that we need here) can also be obtained as a consequence of holomorphic Morse inequalities [START_REF] Demailly | Holomorphic Morse inequalities, Several complex variables and complex geometry[END_REF]. It can be seen as a strengthening of the Grauert-Riemenschneider conjecture proved by Siu [START_REF] Siu | A vanishing theorem for semi-positive line bundles over non-Kähler manifolds[END_REF] (although the latter is also valid in the non-Kähler case).

Definition 2.3. The Lelong number ν x (Θ(L, h)) = ν(ϕ, x) of the (1, 1)-current Θ(L, h) at x ∈ X, is defined as lim inf z→x ϕ(z)
|z-x| , for a local metric on X near x. Definition 2.4. For a positive real number c > 0, the Lelong set F c of a (1,1)-current η is the set of points x ∈ M with ν(η, x) c.

By a well-known theorem of Siu ([Siu74]

) any Lelong set of a positive, closed current is a complex analytic subvariety of M (the proof of this difficult result has been considerably simplified, using regularization of currents and the Ohsawa-Takegoshi extension theorem, see [START_REF] Demailly | Regularization of closed positive currents and Intersection Theory[END_REF].) Theorem 2.5. ([Dem92]) Let X be a compact complex manifold. Let L be a pseudo-effective holomorphic line bundle on X, with singular hermitian metric h with positive curvature current Θ(L, h). Assume that the Lelong sets of Θ(L, h) are all zero-dimensional. Then L is nef, and big as soon as there is at least one nonzero Lelong number.

Proof. The first assertion is ([Dem92], Cor. 6.4). The second follows from ([Dem92], Cor. 7.6). See also [START_REF] Verbitsky | Parabolic nef currents on hyperkaehler manifolds[END_REF], Thm. 3.12.

Corollary 2.6. If X is a compact Kähler manifold without nontrivial subvarieties, and with pseudo-effective canonical bundle, then K X is nef, and for any singular hermitian metric h on K X , m > 0 with positive curvature current, the Lelong number vanish at any point. In particular, the associated multiplier ideal sheaves on K ⊗m X are all trivial for any m > 0 (that is, e -m ϕ is integrable for any m > 0).

Hard Lefschetz theorem for the cohomology of pseudo-effective line bundles

We recall the version of the Hard Lefschetz theorem which is going to be used here.

Theorem 3.1. ([Tak97], [START_REF] Demailly | Pseudo-effective line bundles on compact Kähler manifolds[END_REF])Let (X, ω) be a compact Kähler manifold, of dimension n with Kähler form ω, let K X be its canonical bundle, and let L be a pseudo-effective holomorphic line bundle on X equipped with a singular Hermitian metric h. Assume that the curvature Θ of (L, h) is a positive current on X and denote by I(h) the corresponding multiplier ideal sheaf. Then the wedge multiplication operator η -→ ω i ∧ η induces a surjective map

H 0 (X, Ω n-i X ⊗ L ⊗ I(h)) ω i ∧ • ----→ H i (X, K ⊗ L ⊗ I(h)).
Here ω is considered as an element in H 1 (X, Ω 1 X ) and multiplication by ω maps

H k (X, Ω n-l X ⊗ F ) to H k+1 (X, Ω n-l+1 X ⊗ F ).
This theorem was obtained under various hypotheses during the 1990s, see [START_REF] Enoki | Strong-Lefschetz-type theorem for semi-positive line bundles over compact Kähler manifolds[END_REF] and [START_REF] Ch | Théorèmes d'annulation génériques pour les fibrés vectoriels semi-négatifs[END_REF]. The most general form given here is due to [DPS01], Thm. 2.1.1. It was proved in [START_REF] Takegoshi | On cohomology groups of nef line bundles tensorized with multiplier ideal sheaves on compact Kähler manifolds[END_REF] when L is nef.

Corollary 3.2. Let X be a compact Kähler manifold of dimension n > 1 without any nontrivial subvarieties. Assume that K X is pseudo-effective. Then h i (X, mK X ) n i , for any i 0, and the polynomial P (m) := χ(X, mK X ) is constant, equal to χ(O X ).

Proof. Since n > 1 and X has no nontrivial subvarieties, the algebraic dimension of X vanishes: a(X) = 0. Therefore h 0 (X, E) rank(E), for any holomorphic vector bundle E on X. Combining with the Hard Lefschetz theorem (Theorem 3.1), this gives the first claim of Corollary 3.2. The second claim is clear because a polynomial function P (m) which remains bounded when m → +∞ is necessarily constant.

Corollary 3.3. Let X be a compact Kähler manifold of dimension 3 without any nontrivial subvariety. Assume that K X is pseudo-effective. The polynomial

P (m) := χ(X, mK X ) is constant, equal to χ(O X ) = 0.
Proof. The intersection number K 3 X vanishes (either by Theorem 2.5, or because it is the leading term of P (m), up to the factor 3!). The Riemann-Roch formula therefore gives P (m) :

= (1-12 m) 24 • c 1 (X) • c 2 (X). The boundedness of P (m) then implies that 24 • χ(O X ) = c 1 (X) • c 2 (X) = 0.
Remark 3.4. Arguments close to some of the ones presented here have already been used in the proof of [START_REF] Demailly | Pseudo-effective line bundles on compact Kähler manifolds[END_REF], theorem 2.7.3: if X is a compact Kähler manifold with pseudo-effective canonical bundle admitting a metric with weights ϕ having analytic singularities and positive curvature current, then either H 0 (X, Ω i X ⊗ O(mK X )) = 0 for infinitely many m > 0 and some i 0, or χ(X, O X ) = 0. 4 Brunella's pseudo-effectivity criterion.

The rest of our arguments is based on the following strong (and very difficult) theorem by Brunella.

Theorem 4.1. ( [START_REF] Brunella | Uniformization of foliations by curves, Holomorphic dynamical systems[END_REF])Let X be a compact Kähler manifold with a 1-dimensional holomorphic foliation F given by a nonzero morphism of vector bundle L → T X , where L is a line bundle on X, and T X is its holomorphic tangent bundle. If L -1 is not pseudo-effective, then the closures of the leaves of F are rational curves (and X is thus uniruled).

The following corollary had already been observed in [START_REF] Höring | Uniformisation in dimension four: toward a conjecture of Iitaka[END_REF], Prop. 4.2.

Corollary 4.2. If X is an n-dimensional compact Kähler manifold with H 0 (X, Ω n-1 X ) = 0, then K X is pseudo-effective if X is not uniruled.

Proof. The vector bundle Ω n-1 X is canonically isomorphic to K X ⊗ T X . Any nonzero section thus provides a nonzero map K -1 X → T X and an associated foliation.

Corollary 4.3. If X is a 3-dimensional nonprojective compact Kähler manifold, then either K X is pseudo-effective or X is uniruled. If X is simple, then K X is pseudo-effective.

Proof. Kodaira's theorem implies that any compact Kähler manifold with H 2,0 (X) = 0 is projective. Thus H 2,0 (X) = 0, and the preceding corollary applies (since 2 = (n -1), here) and gives the first claim. If X is uniruled, it is not simple, hence the second assertion.

By combining Corollary 4.3, Corollary 3.3, and Lemma 1.4, we get our main result: Corollary 4.4. If X is a 3-dimensional compact Kähler manifold without nontrivial subvarieties, then X is a complex torus.

In higher dimensions, we can replace the assumption from Corollary 3.2 that K X was pseudo-effective with the existence of a holomorphic (n -1)form, giving the following result.

Corollary 4.5. Let X be a compact Kähler manifold of dimension n > 1 without nontrivial subvarieties. Assume that H 0 (X, Ω n-1 X ) = 0. Then we have h i (X, mK X ) n i for any i 0 and the polynomial P (m) := χ(X, mK X ) is constant, equal to χ(O X ) (The intersection numbers K j X • Todd n-j (X) thus all vanish when j > 0, as expected since K X should then be trivial.).

The following result (announced as Theorem 1.3) strengthens Theorem 1.2. Theorem 4.6. Let X be a 3-dimensional Kähler normal connected compact complex space with terminal 1 singularities. Assume that K X is nef, and that X does not contain any effective divisors2 . Then X = T /G is isomorphic to the quotient of a nonprojective simple torus by a finite cyclic group of automorphisms.

Proof. By [START_REF] Demailly | A Kawamata-Viehweg Vanishing Theorem on compact Kähler manifolds[END_REF], Thm. 7.1, κ(X) = 0. Thus mK X is a trivial line bundle for some m > 0. Let X ′ → X be the cyclic cover of X defined by a nonzero section of mK X so that mK X is Cartier. The singularities of X ′ are thus terminal as well ([KM98], 5.20). They are also rational, since K X ′ is Cartier (by [START_REF] Kollár | Birational Geometry of Algebraic Varieties Cambridge Tracts in Mathematics[END_REF], 5.24). We thus have χ(O X ′ ) = χ(O X ′′ ) for any smooth model X ′′ of X ′ . But

χ(O X ′ ) = -K X ′ .c 2 (X ′ ) 24 = 0,
since K X ′ is locally free trivial (see for example [START_REF] Fletcher | Contributions to Riemann-Roch on projective threefolds with only canonical singularities[END_REF]). The conclusion now follows from Lemma 1.4

Remark 4.7. In order to show the conjecture for 'simple' Kähler threefolds, it would be sufficient: 1. to show the following statement (which is a form of the abundance conjecture in this context): If X is a simple (normal, terminal, Q-factorial, Kähler) threefold, and D an effective divisor on X, then D is not nef. A possible approach might consist in adapting the proof in the projective case. A more direct approach were however desirable.

2. to show the existence of minimal models such as the ones appearing in Theorem 1.3 for any compact Kähler threefold X. (This step has just been achieved in [START_REF] Höring | Minimal Models for Kähler Threefolds[END_REF], where a conjecture equivalent to the one above is formulated in dimension 3.) Remark 4.8. In [START_REF] Ch | Automorphism groups of 3-dimensional complex tori[END_REF], it is shown that the only group G that can act on a 3-dimensional torus without nontrivial analytic subvarieties is Z 2 act-ing by ± Id. In fact, from their classification (cf. the proof of Thm. 4.1 and Prop. 3.6 in [START_REF] Ch | Automorphism groups of 3-dimensional complex tori[END_REF]), any other nontrivial action corresponds to a decomposable torus or an abelian variety. Therefore G is trivial or G = Z 2 in Theorem 4.6.

revised version, January 13, 2014

The nonexistence of effective divisors linearly equivalent to a positive multiple of K X actually suffices for the conclusion.

we refer to[START_REF] Kollár | Birational Geometry of Algebraic Varieties Cambridge Tracts in Mathematics[END_REF] for this notion, which makes sense without change in the analytic category. In particular, the sheaf K X is then Q-Cartier.

Divisors linearly equivalent to a positive multiple of K X actually suffice. This second hypothesis should follow from the first if X is 'simple'.

Acknowledgements: We are grateful to the Tata Institute of Fundamental research in Mumbai, India, where this paper was conceived, and especially to I. Biswas and A.J. Parameswaran, organizers of our stay there.

1 Misha Verbitsky is partially supported by RFBR grants 12-01-00944-a, AG Laboratory NRI-HSE, RF government grant, ag. 11.G34.31.0023, and NRI-HSE Academic Fund Program in 2013-2014, research grant 12-01-0179. from Foundation Compositio Mathematica, accepted in final form on September 4, 2013. 2010


Franc ¸oise

P Ène

Î T Beno

Saussol

POISSON LAW FOR SOME NONUNIFORMLY HYPERBOLIC DYNAMICAL SYSTEMS WITH POLYNOMIAL RATE OF MIXING

Keywords:

We consider some nonuniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs-Markov-Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball B(x, r) converges to a Poisson distribution as the radius r → 0 and after suitable normalization.

Many dynamical systems with some hyperbolicity enjoy strong statistical properties. Let us mention a few of them: existence of physical measure, exponential decay of correlations, central limit theorem, large deviation principles, etc. That is, the probabilistic behavior of these systems mimics an i.i.d. process. Beyond uniform hyperbolicity the situation may be different. This has a visible consequence for example in the validity of the CLT, which is often related to a summable decay of correlation.

We will consider here a class of nonuniformly hyperbolic systems for which the polynomial decay of correlation can be arbitrarily slow. The setting, introduced by Alves and Pinheiro [START_REF] Alves | Slow rates of mixing for dynamical systems with hyperbolic structure[END_REF] and generalized by Alves and Azevedo [START_REF] Alves | Statistical properties of diffeomorfisms with weak invariant manifolds[END_REF] is a system modeled by a Young tower on which we have a uniform polynomial control on the contraction along stable manifolds and backward contraction along unstable manifolds. The full description of the setting is in Section 2.

Let f be an invertible map defined on a riemaniann manifold M giving rise to a metric d. Suppose that µ is an invariant measure for f . Given x in M, we are interested in the statistical behavior, with respect to µ, of the number of occurences of entrance times in the ball B(x, r). Namely, setting N (x, r)(y) = ♯ n ∈ N : d(f n y, x) < r, 1 ≤ n ≤ t µ(B(x, r)) we are interested in the limit distribution of N (x, r), as r goes to 0. The main result of the paper is that this limit is the Poisson distribution, for µ-a.e. x. This question has been adressed for many different dynamical systems. Our result is a generalization of the recent work by Collet and Chazottes [START_REF] Chazottes | Collet Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems[END_REF] who studied towers with exponential tail of return time to a setting where the tail is only polynomial (We refer to [START_REF] Chazottes | Collet Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems[END_REF] and references therein for details on previous works). Our work applies for example to stadium like billiards. During the preparation of this work Freitas, Haydn and Nicol [START_REF] Freitas | Convergence of rare events point processes to the poisson for billiards[END_REF] obtained a similar result on Poisson distribution for these billiards by a different method (inducing the billiard map on a suitable reference set where the induced map has a tower with exponential tail). We also mention Haydn and Wasilewska [START_REF] Wasilewska | Limiting distribution and error terms for the number of visits to balls in mixing dynamical systems[END_REF] whose approach needs a polynomial tail of sufficiently large order, in a nonuniformly expanding setting.

Our limit theorem relies on precise mixing estimates for sets defined with balls. For some systems the indicator functions of balls have a bounded norm in a good Banach space of functions for which one can use the decay of correlations directly. This leads to a Poisson distribution and even gives stronger results [START_REF] Aytac | Laws of rare events for deterministic and random dynamical systems[END_REF]. Here in our situation one cannot expect to implement this strategy and we need to approximate our balls to get mixing estimates. That is why we need a control on the measure of neighborhood of balls as in [START_REF] Aytac | Laws of rare events for deterministic and random dynamical systems[END_REF]. Outside absolutely continuous measure this leads to delicate questions. In [START_REF] Chazottes | Collet Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems[END_REF] a general result is obtained for SRB measure with onedimensional unstable manifold. The generalization to higher dimensional systems or polynomial tower being open, we left the condition (3) as an assumption. We emphasize that this condition is the weakest that one can ask in our setting.

A major step in our proof of the Poisson distribution is to bypass the lengthy and delicate study of short returns by a simple argument based on recurrence rates. Indeed we show that for our systems

min{n ≥ 1 : d(f n x, x) < r} ≈ r -dim H µ
for µ-a.e. x, where dim H µ stands for the Hausdorff dimension of µ (See Section 6 for precise statement). There are some systems with polynomial decay of correlation for which the above behavior does not hold and it turns out that the Poisson distribution cannot happen in these cases (See [START_REF] Galatolo | Skew products, quantitative recurrence, shrinking targets and decay of correlations[END_REF]).

Assumptions

We consider an invertible transformation f : M → M on a finitedimensional Riemannian manifold satisfying assumptions of [START_REF] Alves | Statistical properties of diffeomorfisms with weak invariant manifolds[END_REF]. These assumptions ensure the existence of a f -invariant SRB probability measure µ and that (M, f, µ) can be modeled by a Gibbs Markov Young tower with good properties as described briefly below for completeness.

Recall that a stable (resp. unstable) manifold is an embedded disk γ ⊂ M such that, for every x, y ∈ γ, dist(f n x, f n y) → 0 (resp. dist(f -n x, f -n y) → 0) as n goes to infinity.

Consider two continuous families Γ u and Γ s of respectively unstable and stable C 1 manifolds such that there exists α min > 0 so that, for every

(γ s , γ u ) ∈ Γ s × Γ u , we have dim γ s + dim γ u = dim M, #(γ s ∩ γ u ) = 1 and |∠(γ s , γ u )| ≥ α min .
We set Λ := ( γ u ∈Γ u γ u ) ∩ ( γ s ∈Γ s γ s ) and assume that the following properties hold (P1) Markov: there exists a family (Λ i ) i≥1 of pairwise disjoint subsets of the form

Λ i = ( γ u ∈Γ u γ u ) ∩ ( γ s ∈Γ s i γ s ) for some family (Γ s i ) i of pairwise disjoint subsets of Γ s such that (a) for some γ ∈ Γ u , we have Leb γ (Λ) > 0 and Leb γ (Λ \ i Λ i ) = 0; (b) For every i ≥ 1, there exists an integer R i ≥ 1 such that f R i (Λ i ) = ( γ u ∈Γ u i γ u ) ∩ ( γ s ∈Γ s γ s ) for some Γ u i ⊂ Γ u .
Moreover, for every γ s ∈ Γ s i , there exists γ s 0 ∈ Γ s such that f R i (γ s ) ⊂ γ s 0 and, for every γ u ∈ Γ u , there exists γ u 0 ∈ Γ u i such that γ u 0 ⊂ f R i (γ u ). This enables us to define a particular return time R : Λ → N and the associated return map f R : Λ → Λ by setting

R |Λ i ≡ R i and (f R ) |Λ i ≡ f R i .
We also define a separation time s : Λ × Λ → N ∪ {∞} for the return map as follows:

∀x, y ∈ Λ, s(x, y) := min{n ≥ 0 : ∃j ≥ 1, (f R ) n (x) ∈ Λ j and (f R ) n (y) ∈ Λ j }.
With these notations, we assume that there exist α > 0, β ∈ (0, 1) and C > 0 such that, for every γ u 0 , γ u 1 ∈ Γ u and every γ s ∈ Γ s , we have (P2) Polynomial contraction on stable leaves: for every x, y ∈ γ s and every n ≥ 1, we have dist(f n (x), f n (y)) ≤ Cn -α ; (P3) Backward polynomial contraction on unstable leaves: for every x, y ∈ γ u 0 and every n ≥ 1, we have dist(f -n (x), f -n (y)) ≤ Cn -α ; (P4) Bounded distortion: for every x, y ∈ γ u 0 ∩ Λ, we have

log det D(f R ) u (x) det D(f R ) u (y) ≤ Cβ s(f R (x),f R (y)) .
(P5) Regularity of the stable foliation: consider the map Θ γ u 0 ,γ u

1 : γ u 0 ∩Λ → γ u 1 ∩ Λ defined by Θ γ u 0 ,γ u 1 (x)
is the unique x ′ for which there exists γ ∈ Γ s such that x ∈ γ ∩ γ u 0 and x ′ ∈ γ ∩ γ u 1 . We assume that (a) Θ γ u 0 ,γ u 1 is absolutely continuous and

U := d((Θ γ u 0 ,γ u 1 ) * Leb γ u 0 ) d Leb γ u 1 = i≥0 det Df u • f i det Df u • f i • Θ -1 γ u 0 ,γ u 1 ;
(b) for every x, y ∈ γ u 1 , we have log(U (x)/U (y)) ≤ Cβ s(x,y) . We assume that gcd(R i , i ≥ 1) = 1. We consider here the case when the return time R has a polynomial tail distribution, more precisely we assume that Leb

γ (R > n) ≤ Cn -ζ , for some ζ > 1, (1)
which ensures the integrability of R with respect to Leb γ . Under these conditions, the systems admits a SRB measure µ: µ has no zero Lyapunov exponent and the conditional measures on local unstable manifolds are absolutely continuous with respect to the Lebesgue measure on these manifolds. We recall the definition of the Hausdorff dimension of the measure µ as

dim H µ = inf µ(Y )=1 dim H Y.
We make the standing assumption that

α > 1 dim H µ . (2)
We say that u v to express that there exists a constant C > 0 (depending only on the dynamical system) such that u ≤ Cv. We also write u ≈ v when u v and v u.

Poisson law for the number of entrance to balls

For any x ∈ M and r > 0, we write B(x, r) for the ball of center x and radius r for the Riemannian distance on M. The main result of the paper states that for typical centers x, the time spent into the ball B(x, r), up to time t/µ(B(x, r)), follows asymptotically the Poisson law with mean t.

We assume that, for µ-almost every x ∈ M, there exists δ

∈ (1, α dim H µ) such that µ(B(x, r + r δ ) \ B(x, r)) = o(µ(B(x, r))) (3)
(see appendix for discussion on (3)).

Theorem 3.1. Let (M, f, µ) be as above. For µ-a.e. x ∈ M such that (3) holds,

lim r→0 µ      y ∈ M : ⌊t/µ(B(x,r))⌋ j=0 1 B(x,r) (f j y) = k      = t k k! e -t (4)
for any k ≥ 0 and any t ≥ 0.

The condition (3) on the coronas is always satisfied for a subsequence: (see the appendix for details). Proposition 3.2. Let θ < 1. For every x ∈ M, there exists a sequence r n (x) ∈ (θ n+1 , θ n ) such that for any δ ∈ (1, α dim H µ), (3) holds for the sequence r = r n (x).

A quick look at the proof of Theorem 3.1 (especially Proposition 5.1) shows that the convergence (4) along the subsequence r n (x) still holds in this case, provided the other assumptions are satisfied.

3.1. An abstract Poisson approximation result. We use the method developed by Chazottes and Collet, Theorem 2.1 in [START_REF] Chazottes | Collet Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems[END_REF]. We recall that, for any probability measures P and Q on a same measurable space (E, E), the total variation distance d T V (P, Q) between P and Q is

d T V (P, Q) = sup A∈E P (A) -Q(A).
If Y and Z are random variables taking integer values, the total variation distance between their law is written d T V (Y, Z) (with a small abuse of notation) and is given by

d T V (Y, Z) = 1 2 ∞ k=0 |P(Y = k) -P(Z = k)|.
By Poisson(λ) we denote a Poisson random variable with mean λ > 0, namely

P(Poisson(λ) = k) = λ k k! e -k . Theorem 3.3 ([7]). Let (X n ) n∈N be a stationary {0, 1}-valued process and ε := P(X 1 = 1). Set S j i = X i + • • • + X j for any 0 ≤ i ≤ j.
Then, for all positive integers p, M, N such that M ≤ N -1, and 2 ≤ p < N , one has

d T V (X 1 + • • • + X N , Poisson(N ε)) ≤ R(ε, N, p, M ) with R(ε, N, p, M ) = 2N M [R 1 (ε, N, p) + R 2 (ε, p)] + R 3 (ε, N, p, M ) where R 1 (ε, N, p) := sup 0≤j≤N -p 0≤q≤N -j-p P(X 1 = 1, S N -j p+1 = q) -εP(S N -j p+1 = q) R 2 (ε, p) := P(X 1 = 1, S p 2 ≥ 1) R 3 (ε, N, p, M ) := 4 M pε(1 + N ε) + (εN ) M M ! e -N ε + N ε 2 .
The definition of R 2 given here differs slightly from the original statement, but an eye to the proof shows that the arguments go trough with this tiny change. However, this modification plays an essential role in the present paper since it allows a very efficient estimate of the term R 2 .

3.2.

Proof of the main theorem. For a fixed x ∈ M, we will apply Theorem 3.3 to the processes defined for y ∈ M by X n+1 (y) = 1 B(x,r) (f n y), n ∈ N, r > 0. For any integers p ≤ q, we write S q p (x, r) for the number of visits to B(x, r) of the orbit of f between times p and q, i.e. ∀y ∈ M, S q p (x, r)(y) := #{ℓ = p, ..., q : f ℓ (y) ∈ B(x, r)}.

We denote the error terms R 1 and R 2 for this process by R 1 (x, r, N, p) and R 2 (x, r, p). That is 3)). For µ-a.e. x, due to (2), by Proposition 5.1, we have with

R 1 (x, r, N, p) = sup 0≤j≤j+q≤N -p Cov µ (1 B(x,r) , 1 {S N -j p+1 (x,r)=q} ) , R 2 (x, r, p) = µ({y ∈ B(x, r) : S p 2 (x, r) ≥ 1}). Proof of Theorem 3.1. Let t > 0. Let σ < d := dim H µ be such that σα > δ (with δ as in (
p r = ⌊r -σ ⌋ and N r = ⌊t/µ(B(x, r))⌋, R 1 (x, r, N r , p r ) ≤ C ′ 0 µ(B(x, r + r δ ))p -ζ+1 r + µ(C r,r δ (x))(1 + N r µ(B(x, r + r δ ))) = O(µ(B(x, r)))r σ(ζ-1) + o(µ(B(x, r)))(1 + t + o(1)) = o(µ(B(x, r)),
due to (3) and since ζ > 1 (see ( 1)). By Proposition 7.1 we have

R 2 (x, r, p r ) = o(µ(B(x, r))).
The conclusion follows from Theorem 3.3 by taking M r → ∞ sufficiently slowly.

Correlation estimates on towers

We define the tower ∆ and the map F : ∆ → ∆ as follow

∆ := {(x, ℓ) ∈ Λ × Z : 0 ≤ ℓ < R(x)}, F (x, ℓ) = (x, ℓ + 1) if ℓ < R(x) -1 and F (x, R(x) -1) = (f R (x), 0
). We define a family of partitions (Q k ) k≥0 by

Q 0 := {Λ i × {ℓ}, i ≥ 1, ℓ < R i } and ∀k ≥ 1, Q k := k i=0 F -i Q 0 .

Now we define the projection

π : ∆ → M by π(x, ℓ) := f ℓ (x). Observe that f • π = π • F . Due to Lemma 3.1 in [1]
, there exists C 2 > 0 such that, for every non negative integer k and every

Q ∈ Q 2k , we have diam(πF k (Q)) ≤ C 2 k -α . (5)
Now we consider the quotient tower ∆ := ∆/ ∼ with (x, ℓ) ∼ (y, ℓ ′ ) if ℓ = ℓ ′ and if x and y are on a same γ s ∈ Γ s . We define π : ∆ → ∆ as the canonical projection and the map F : ∆ → ∆ such that F • π = π • F . We define Qk as the projection of the partition Q k on ∆. It will be also useful to consider the separation time s : ∆ × ∆ → Z + ∪ {∞} as follows:

s((x, ℓ), (y, ℓ ′ )) = s(x, y) if ℓ = ℓ ′ ; s((x, ℓ), (y, ℓ ′ )) = 0 if ℓ = ℓ ′ .
We fix a γ ∈ Γ u and consider the measure m on ∆ such that the measure on ∆ℓ corresponds to Leb γ . Recall that there exist probability measures µ, ν and ν on M, ∆ and ∆ respectively such that

f * µ = µ, F * ν = ν, F * ν = ν, ν = π * ν, µ = π * ν
and such that • the measure ν admits a density function ρ with respect to m;

• For every c ′ > 0, there exists C ′ such that, for every probability measure λ absolutely continuous with respect to m, with density probability ϕ : ∆ → R + satisfying

∀ Q ∈ Q0 , ∀x, y ∈ Q, |ϕ(x) -ϕ(y)| ≤ c ′ |ϕ(x)|β s(x,y) , we have ∀n ≥ 1, d T V ( F n * λ, ν) ≤ C ′ n -ζ+1 (6)
(see Theorem 3.5 in [START_REF] Alves | Statistical properties of diffeomorfisms with weak invariant manifolds[END_REF]).

This leads to the following decorrelation result at the core of our estimates.

Lemma 4.1. There exists C ′ > 0 such that, for every non negative integers k, n with n ≥ 2k, for every A union of atoms of Q k and every B in σ(∪ m≥0 Q m ), we have

| Cov ν (1 A , 1 B • F n )| ≤ C ′ n -ζ+1 ν(A).
Proof. Notice that A = π-1 Ā and B = π-1 B with Ā := πA and B := πB. Therefore, we have

|Cov ν (1 A , 1 B • F n )| = Cov ν (1 Ā, 1 B • F n )) = E m ρ(1 Ā -ν( Ā))1 B • F n = ν( Ā) E m (φ -ρ)1 B • F n-k ≤ ν(A)d T V ( F n-k * ( F k * λ), ν
), writing λ for the measure whose density with respect to m is ρ1 Ā/ν( Ā) and φ for the density function of F k * λ. Now, due to (20) in [START_REF] Alves | Slow rates of mixing for dynamical systems with hyperbolic structure[END_REF] (see also Lemma 4.5 in [START_REF] Alves | Statistical properties of diffeomorfisms with weak invariant manifolds[END_REF]), we observe that

∀ Q ∈ Q0 , ∀y, z ∈ Q, |φ(y) -φ(z)| ≤ φ(y)e Cρ+C F (C ρ + C F )β s(y,z)
and, due to [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF] and to n -k ≥ n/2, we obtain

|Cov ν (1 A , 1 B • F n )| ≤ C ′ ν(A)n -ζ+1 , (7)
for some C ′ > 0.

Decorrelations between successive returns

The aim of this section is to estimate R 1 .

Proposition 5.1. There exists C ′ 0 > 0 such that for any integer p we have

R 1 (x, r, N, p) ≤ C ′ 0 µ(B(x, r + s))p -ζ+1 + µ(C r,s (x))(1 + N µ(B(x, r + s))),
where C r,s (x) is the corona B(x, r + s) \ B(x, r) and s = C 2 ⌊p/4⌋ -α .

Proof. Let K := ⌊p/4⌋. We have to estimate the following quantity:

R 1 (x, r, N, p) = sup 0≤j≤j+q≤N -p Cov µ (1 B(x,r) , 1 {S N -j p+1 (x,r)=q} ) . Using successively f * µ = µ, µ = π * ν and f • π = π • F , we obtain Cov µ (1 B(x,r) , 1 {S N -j p+1 (x,r)=q} ) = = Cov µ 1 B(x,r) • f K , 1 {S N -j-p 1 (x,r)=q} • f K+p = Cov ν 1 B(x,r) • π • F K , 1 {S N -j-p 1 (x,r)=q} • π • F K+p = Cov ν 1 F -K π -1 (B(x,r)) , 1 F -K π -1 ({S N -j-p 1 (x,r)=q}) • F p .
First we approximate the sets appearing in this last formula with union of elements of the Q k 's. More precisely, we approximate

F -K π -1 (B(x, r)) by D (K)
x,r with

D (K)
x,r :=

Q∈Q 2K :Q∩F -K π -1 B(x,r) =∅ Q = Q∈Q 2K :πF K Q∩B(x,r) =∅ Q and F -K π -1 ({S N -j-p 1 (x, r) = q}) by {S N -j-p,(K) 1 (x, r) = q} where S k 0 ,(K) 1 (x, r)(y) := #{ℓ = 1, ..., k 0 : y ∈ D (K+ℓ)
x,r }.

Let us set

A ′ := D (K) x,r , A := F -K π -1 B(x, r), A ′′ := Q∈Q 2K :Q∩(D (K) x,r \F -K π -1 B(x,r)) =∅ Q, B ′ := {S N -j-p,(K) 1 (x, r)(y) = q}, B := F -K π -1 ({S N -j-p 1 (x, r) = q})
and

B ′′ := N -j-p ℓ=1 Q∈Q 2(K+ℓ) :Q∩(D (K+ℓ) x,r \F -K-ℓ π -1 B(x,r)) =∅ Q. Observe that A ⊂ A ′ ⊂ A ∪ A ′′ and B ⊂ B ′ ⊂ B ∪ B ′′ so that we obtain | Cov ν (1 A , 1 B • F p ) -Cov ν (1 A ′ , 1 B ′ • F p )| ≤ ≤ |E[(1 A ′ -1 A )1 B ′ • F p ] + E[1 A (1 B ′ -1 B ) • F p ] -(ν(A ′ ) -ν(A))ν(B ′ ) -ν(A)(ν(B ′ ) -ν(B))| ≤ | Cov ν (1 A ′′ , 1 B ′ • F p )| + | Cov ν (1 A ′ , 1 B ′′ • F p )| + +2ν(A ′′ )ν(B ′ ) + 2ν(A ′ )ν(B ′′ ). (8)
Now, due to ( 5) and (3), we observe that

ν(A ′′ ) ≤ ν(F -k π -1 (B(x, r + s) \ B(x, r))) ≤ µ(C r,s (x)) (9)
and that

ν(B ′′ ) ≤ N -j-p ℓ=1 ν(D (K+ℓ) x,r \ F -(K+ℓ) π -1 (B(x, r))) ≤ N -j-p ℓ=1 ν(F -(K+ℓ) π -1 (B(x, r + s) \ B(x, r)) ≤ N µ(C r,s (x)). ( 10
)
Proposition 5.1 follows from ( 8), ( 9), [START_REF] Freitas | Convergence of rare events point processes to the poisson for billiards[END_REF], ν(A ′ ) ≤ µ(B(x, r + s)), ν(B ′ ) ≤ 1 together with Lemma 4.1 applied three times (with (A ′ , B ′ ), (A ′ , B ′′ ) and (A ′′ , B ′ ) respectively).

Recurrence rate for tower systems

Recall the definition of the recurrence rate Rec(x) = lim r→0 log τ r (x) -log r , when the limit exists, where

τ r (x) = inf{n ≥ 1 : d(f n x, x) < r}. (11)
A large lower bound for Rec will allow a more efficient estimate of the second error term R 2 in the next section. For systems modeled by a Young tower as described in Section 2, we show below that the recurrence rate is as large as it can be.

Since the measure µ is hyperbolic, its pointwise dimension exists a.e.

[5] and we have lim

r→0 log µ(B(x, r)) log r = dim H µ µ-a.e.. (12)
In the case of super-polynomial decay of correlation, [START_REF] Rousseau | Poncaré recurrence for observations[END_REF] applies and gives Rec = dim H µ µ-a.e.. In our setting this decay might be only polynomial hence the general result above may not apply. However, thanks to the Markov-tower structure we can refine the argument in [START_REF] Rousseau | Poncaré recurrence for observations[END_REF]. Theorem 6.1. For a system (f, µ) modeled by a Young tower we have Rec = dim H µ µ-a.e.

First, we need the following decorrelation lemma (we use the notation d := dim H µ). Lemma 6.2. There exists C > 0 such that for all x and r > 0 we have

µ(B(x, r) ∩ f -n {S ℓ 1 (x, r) ≥ 1}) ≤ C[µ(B(x, r + s))n -ζ+1 + ℓµ(B(x, r + s)) 2 ] where s = ⌊n/4⌋ -α .
Proof. We define the set A ′ := D (K)

x,r as in the proof in Section 5, but here we take K = ⌊n/4⌋. We have F -K π -1 B(x, r) ⊂ A ′ . In the same way, the set

F -K π -1 {S ℓ 1 (x, r) ≥ 1} is contained in B ′ := ∪ ℓ j=1 F -j A ′ . Hence, due to Lemma 4.1, µ(B(x, r) ∩ f -n {S ℓ 1 (x, r) ≥ 1}) ≤ ν(A ′ ∩ F -n B ′ ) ≤ C ′ ν(A ′ )n -ζ+1 + ν(A ′ )ν(B ′ ).
We remark that ν(A ′ ) ≤ µ(B(x, r + K -α )) since π(F K A ′ ) ⊂ B(x, r + K -α ) due to [START_REF] Alves | Slow rates of mixing for dynamical systems with hyperbolic structure[END_REF]. Moreover, by invariance of ν, ν(B ′ ) ≤ ℓν(A ′ ), which finishes the proof.

The proof of Theorem 6.1 follows the lines of Lemma 16 in [START_REF] Rousseau | Poncaré recurrence for observations[END_REF] but takes advantage of the (Markov) tower structure, which allows to use the much more efficient decorrelation Lemma 6.2 above instead of an approximation by Lipschitz functions.

Proof of Theorem 6.1. Let ε > 0 and take β = 1 d-4ε with ε > 0 so small that β ≤ α and 1 + 2εβ < ζ. Set G(ε, r) = {x ∈ M : ∀r < r, r d+ε ≤ µ(B(x, r)) ≤ r d-ε } with r so small that that µ(G(ε, r)) > 1 -ε (see [START_REF] Hirata | Statistics of return times: a general framework and new applications[END_REF]).

Choose

q = ⌊1/(ζ -1 -2εβ)⌋. Let ℓ k = k q and set n k+1 = ℓ 1 + • • • + ℓ k . We have n k ≈ k q+1 . Set r k = n -β k . We have n -α k ≤ r k since β ≤ α.
By Lemma 6.2 we have for x ∈ G(ε, r) and k sufficiently large so that

r k + n -α k < r µ(B(x, r k ) ∩ f -n k {S ℓ k 1 (x, r k ) ≥ 1}) ≤ 2 2d C[r d-ε k n -ζ+1 k + ℓ k r 2d-2ε k ].
Conditioning on B(x, r k ) we get

µ(f -n k {S ℓ k 1 (x, r k ) ≥ 1}|B(x, r k )) ≤ 2 2d C[r -2ε k n -ζ+1 k + ℓ k r d-3ε k ]. Let T k = {y ∈ M : ∃j = n k + 1, . . . , n k+1 , d(f j (y), y) < r k /2}. Take a maximal r k /4-separated subset G k ⊂ G(ε, r). The balls B(x, r k /2), x ∈ G k , cover G(ε, r).
Moreover the balls B(x, r k ) cover it with a multiplicity bounded by some constant C ′′ depending only on the finite dimensional manifold M. Summing up over x ∈ G k we get

µ(G(ε, r) ∩ T k ) ≤ x∈G k µ(B(x, r k /2) ∩ T k ) ≤ x∈G k µ(B(x, r k ) ∩ f -n k {S ℓ k 1 (x, r k ) ≥ 1}) ≤ 2 2d CC ′′ [r -2ε k n -ζ+1 k + ℓ k r d-3ε k ] ≤ C ′′′ (k (q+1)(1-ζ+2εβ) + k q-(q+1)β(d-3ε) ).
This upper bound is summable in k by construction. An application of Borel-Cantelli lemma shows that for µ-a.e. x ∈ G(ε, r) there exists k(x) such that x ∈ T k for k ≥ k(x). For x non periodic we have τ r (x) → ∞. Hence there exists r(x) such that τ r(x) (x) > n k(x) . Let r < min(r(x), r k(x) )/2. Take k ≥ k(x) such that r k+1 < 2r ≤ r k . Since x does not belong to any of the

T j 's, k(x) ≤ j ≤ k, we get τ r (x) ≥ n k . Hence log τ r (x) -log r ≥ log n k -log r k+1 /2 → 1 β .
Since ε is arbitrary this proves that for µ-a.e. x

lim r→0 log τ r (x) -log r ≥ dim H µ.
On the other hand the limsup is always bounded by the dimension [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF], whence the convergence of the recurrence rate and its equality with the dimension of µ.

Estimate of short returns : R 2

We now provide an optimal estimate of R 2 in the following sense: By Kac's Lemma the mean return time into B(x, r) cannot be larger than 1/µ(B(x, r)). Thus it is clear that any σ > dim H µ would contradict (13). (13)

Proof. Let η > 0. Let r 1 > 0 and define

K σ,r 1 = {x ∈ M : ∀r < r 1 , τ 2r (x) > r -σ }.
Let ι > 0. By Theorem 6.1 there exists r 1 > 0 such that µ(M \ K σ,r 1 ) < ι.

Let Kσ,r 1 be the set of the Lebesgue density points of K σ,r 1 with respect to the measure µ. For x ∈ Kσ,r 1 we have

lim r→0 µ(B(x, r) ∩ K σ,r 1 ) µ(B(x, r)) = 1. Let r 2 < r 1 and set Kσ,r 1 ,r 2 = x ∈ M : ∀r < r 2 , µ(B(x, r) ∩ K σ,r 1 ) µ(B(x, r)) > 1 -η .
Take r 2 > 0 so small that µ(K σ,r

1 \ K σ,r 1 ,r 2 ) < ι. If x ∈ Kσ,r 1 ,r 2 and r < r 2 we get R 2 (B(x, r), ⌊r -σ ⌋) = µ({y ∈ B(x, r) : τ B(x,r) (y) ≤ r -σ }) ≤ µ({y ∈ B(x, r) : τ 2r (y) ≤ r -σ }) ≤ µ(B(x, r) \ K σ,r 1 ) ≤ ηµ(B(x, r)).
This concerns all points in M except a subset of measure 2ι arbitrary small, which proves the proposition.

Application to solenoid with intermittency

We can apply our main theorem to the example considered by [START_REF] Alves | Slow rates of mixing for dynamical systems with hyperbolic structure[END_REF]: Let g :

T 1 → T 1 (with T 1 = R/Z) be a continuous map of degree d ≥ 2 such that • g is C 2 on T 1 \ {0} and g ′ > 1 on T 1 \ {0},
• g(0) = 0, g ′ (0+) = 1 (right derivative at 0) and there is γ > 0 such that -xf ′′ (x) ∼ |x| γ in the right vicinity of 0, • g ′ (0-) > 1 (left derivative at 0). Let J 0 = (0, x 1 ), J 1 = (x 1 , x 2 ),...,J d-1 = (x d-1 , 0) be the successive intervals (from the left to the right) on which g defines a bijection onto T 1 \ {0} (the x i are the non zero preimages of 0 by g). Let D ⊂ C be the unit disk and consider the solid torus M = T 1 × D. We define the map

f (x, z) = (g(x), θz + 1 2 e 2iπx ),
where θ ∈ (0, 1) is such that θ g ′ ∞ < 1 -θ (i.e. 0 < θ < 1/(1 + g ′ ∞ )). It was already proven by Alves and Pinheiro that this map fits into the general scheme described in Section 2 with the parameters ζ = 1/γ > 1 and α = 1 + 1/γ. In particular the map f admits an SRB measure µ if and only

if γ < 1. Let d((x, z), (x ′ , z ′ )) = max(|x -x ′ |, |z -z ′ |).
Theorem 8.1. For any γ < √ 2/2, the conclusion of Theorem 3.1 holds for µ-almost every x ∈ M, i.e. for µ-almost every x ∈ M, the number of visits to B(x, r) up to time ⌊t/µ(B(x, r))⌋ converges in distribution (with respect to µ) to a Poisson random variable of mean t.

The only condition that is left to verify to prove Theorem 8.1 is that (3) about coronas. This condition is in principle highly dependent on the choice of the metric we put on M. We will prove it for d, which is the most natural metric on T 1 × C, but the proof could be adapted to the Euclidean metric as well. The measure µ is supported on the attractor Λ = n≥0 f n M. Let Z be the essential partition of M into K j = J j × D (j = 0, ..., d -1). This is a Markov partition for f . We denote by Z n m = n-1 j=m f -j Z. Its elements Z ∈ Z n m will be denoted indifferently by their code [a m , . . . , a n-1 ] meaning that f j Z ∈ K a j for any j = m, . . . , n -1.

Let μ be the marginal of µ on T 1 . It is indeed the SRB measure of g, we denote its density h. The partition (mod μ

) Z = {J 0 , • • • , J d-1 } is again a Markov partition for g. Let Zn = Z∨g -1 Z∨• • •∨g -n+1 Z.
We denote indifferently by [a 0 . . . a n-1 ] ∈ Zn the element Z ∈ Zn such that g j Z ⊂ J a j for any j = 0, . . . , n.

We collect below some elementary facts on the interval map g needed to study its statistical properties. Proposition 8.3 (e.g. [START_REF] Liverani | A probabilistic approach to intermittency[END_REF][START_REF] Young | Recurrence times and rates of mixing[END_REF]). (i) The density is uniformly controlled on cylinders:

H := sup n sup [0 n ] = Z∈ Zn max Z h min Z h < ∞. ( 14
)
(ii) For any n 0 there exists a constant D(n 0 ) > 0 such that: for any integer n ≥ n 0 , for any y, y ′ ∈ Z = [a 0 . . . a n-1 ] ∈ Zn such that a n-n 0 = 0 we have

(g n ) ′ (y ′ ) (g n ) ′ (y) ≤ D(n 0 )e D(n 0 )|g n (y ′ )-g n (y)| . ( 15
)
In particular for any subintervals I, J ⊂ Z

|g n I| |g n J| ≤ D(n 0 )e D(n 0 ) |I| |J| . ( 16
)
(iii) There exists a constant D 1 > 0 such that, for every nonnegative integers m and k and, for every cylinder Z = [0 k a k+1 • • • a k+m ] ∈ Zm+k with a k+1 = 0, and any z ∈ Z we have

k + 1 D 1 ≤ h(z) ≤ D 1 (k+1), (k + 1) -1-1/γ D 1 ≤ | Z| |[a k+1 • • • a k+m ]| ≤ D 1 (k+1) -1-1/γ (17) and so μ( Z) ≈ (k + 1) -1/γ |[a k+1 • • • a k+m ]|. ( 18
)
Lemma 8.4. Let p < min(2, 1/γ). For µ-a.e. ξ ∈ M we have ℓ j=0

1 K 0 (f -n+j (ξ)) ≤ µ(K 0 )ℓ + o(n 1/p ).
Proof. By Theorem 1.6 in [START_REF] Dedecker | Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains[END_REF] we have

∞ n=0 1 n μ      x : max 1≤k≤n k j=0 1 J 0 (g j x) -k μ(J 0 ) > n 1/p      < ∞
and using the invariance of µ we get that

∞ n=0 1 n µ      ξ : max 1≤k≤n k j=0 1 K 0 (f -n+j ξ) -kµ(K 0 ) > n 1/p      < ∞.
We now conclude accordingly that for µ-a.e. ξ

lim n | k j=0 1 K 0 (f -n+j ξ) -kµ(K 0 )| n 1/p < ∞.
The conclusion follows since p < 1/γ was arbitrary.

From now on we fix p < 1/γ and ξ = (x, z) ∈ Λ satisfying the conclusion of Lemma 8.4. The code

Z 0 -n (ξ) = [a -n , . . . , a -1 ] is such that ℓ n = o(n 1/p )
, where ℓ n is the minimal integer ℓ ≥ 0 such that a -n+ℓ = 0.

We also take n 0 ≥ 1 the minimal integer such that a -n 0 = 0. An immediate computation gives

f n (x, z) = (g n (x), θ n z + 1 2 n-1 j=0 θ n-1-j e 2iπg j (x) ). ( 19
)
We consider the natural projections π X : M → T 1 and π D : M → D.

Lemma 8.5. For any ξ = (x, z) ∈ M \ ∂Z and any r < |x -0| we have

f n+k 0 (M) ∩ B(ξ, r) ⊂ Z 0 -n (ξ). with n = ⌊log(r)/ log(θ)⌋ -k 0 , for some constant k 0 independent of ξ, r.
Observe that, with the notations of this lemma, we have

θ n+k 0 +1 < r ≤ θ n+k 0 . Proof of Lemma 8.5. Claim 8.6. Let ξ ′ ∈ f n (M) such that d(ξ, ξ ′ ) < r. There exists ξ ′′ ∈ Z 0 -n (ξ ′ ) such that π X ξ = π X ξ ′′ and |π D ξ -π D ξ ′′ | < 4r.
Proof. Taking y the preimage by g n of π X ξ lying in the same branch of

g n than π X f -n (ξ ′ ) gives a point ξ ′′ = f n (y, π D (f -n (ξ))) ∈ Z 0 -n (ξ ′ ) such that π X ξ = π X ξ ′′ and |π D ξ -π D ξ ′′ | < π 1-θ r < 4r. Claim 8.7. There exists a constant k 0 such that π X f -n+k 0 ξ = π X f -n+k 0 ξ ′′ , in particular ξ ′′ ∈ Z 0 -n+k 0 (ξ), where n is the smallest integer such that r < θ n . Proof. Set ξ ′′ = (x, z ′′ ) with |z -z ′′ | < 4r. Remark that if g(u) = g(v) and u = v we have |e 2iπu -e 2iπv | ≥ 2 sin(π|u -v|) ≥ 2 sin( π g ′ ∞ ). Set (x -n , z -n ) = f -n ξ and (x ′′ -n , z ′′ -n ) = f -n ξ ′′ . Let k denote the largest integer ≤ n -1 such that g k (x -n ) = g k (x ′′ -n ) (if any). We have 4r > |z -z ′′ | = θ n (z -n -z ′′ -n ) + 1 2 k j=0 θ n-1-j (e 2iπg j (x -n ) -e 2iπg j (x ′′ -n ) )
from which we get

θ n-1-k sin( π g ′ ∞ ) < 8r + 4θ n + θ n-k 1 -θ and thus 1 g ′ ∞ ≤ sin( π g ′ ∞ ) < 12θ k+1 + θ 1 -θ .
Taking k 0 the smallest integer k such that the above formula does not hold shows that the section {x} × D 4r (z) is contained in a unique cylinder Z 0 -n+k 0 (ξ). A change of indices finishes the proof of the lemma.

Let us write D(z, r)

:= {z ′ ∈ C : |z -z ′ | < r} for the open disc of center z and radius r in C. If Z 0 -j (ξ) = [a -j • • • a 0 ], we set (g -j ) ξ for the inverse branch of g j restricted to Zj 0 (π X f -j ξ). Lemma 8.8. The cylinder Z 0 -n (ξ) is the tube Z 0 -n (ξ) = {(x, z) ∈ T × D : x ∈]0, 1[, z ∈ D(γ ξ,n (x), θ n )}. around the curve γ ξ,n : ]0, 1[→ C defined by γ ξ,n (x) = 1 2 n j=1 θ j-1 e 2iπ(g -j ) ξ (x)
with slope γ ′ ξ,n (x) = u n (γ ξ,n (x)) where

u n (ξ ′ ) := iπ n j=1 θ j-1 1 (g j ) ′ (π X f -j ξ ′ ) e 2iππ X f -j ξ ′ .
Proof. A change of indices in Expression 19 shows the first assertion, the second follows by derivation.

For every

ξ ′ = (x ′ , z ′ ) ∈ Λ, we also define u ∞ (ξ ′ ) as follows u ∞ (ξ ′ ) = iπ j≥1 θ j-1 1 (g j ) ′ (π X f -j ξ ′ ) e 2iππ X f -j ξ ′ .
Observe that (1, u ∞ (ξ ′ )) corresponds to the direction of the solenoid at ξ ′ .

Lemma 8.9. There exist c > 0, c 0 > 0, c 1 > 0 and C(n 0 ) > 0 such that for any n ≥ n 0 and r < |x -0|, we have (i) The direction u n (ξ) is uniformly bounded away from zero and infinity:

c 0 < |u n (ξ)| < c 1 . (ii) For any ξ ′ ∈ Z 0 -n (ξ) such that |π X ξ -π X ξ ′ | < r
, and any k ≥ 0, the directions satisfy

|u n (ξ) -u n+k (ξ ′ )| ≤ C(n 0 )r + πθ n 1 -θ . Proof. (i) The upper bound c 1 = π/(1-θ) is obvious since |g ′ | > 1.
Isolating the j = 1 term gives the lower bound

|u n (ξ)| ≥ π g ′ (π X (f -1 ξ)) - πθ 1 -θ ≥ π 1 g ′ ∞ - θ 1 -θ =: c 0 .
(ii) The sum in u n+k (ξ ′ ) differs from that of u n (ξ ′ ) only by the last terms j = n + 1, . . . , n + k, and this is bounded by the rest of the convergent series, so it suffices to prove the claim for k = 0. The distortion bound (15) yields, since |π X f -j ξ ′ -π X f -j ξ| < r for all j = 1, . . . , n,

(g j ) ′ (π X f -j ξ)e 2iππ X f -j ξ ′ (g j ) ′ (π X f -j ξ ′ )e 2iππ X f -j ξ -1 ≤ D(n 0 )e D(n 0 )r+2πr .
We now prove Proposition 8. The first term is roughly bounded by the help of the marginal measure μ, whose density is bounded in a neighborhood of x = 0, hence µ(C r,s (x) × D(z, r + s)) ≤ μ(C r,s (x)) ≤ cs, provided r is sufficiently small and s < r. For typical ξ the decay of µ(B(ξ, r)) is ruled by the dimension dim H µ (see [START_REF] Hirata | Statistics of return times: a general framework and new applications[END_REF]), hence choosing δ = Without loss of generality we will suppose, for this part of the proof only, that s = r δ for some fixed δ, 1 < δ < min(2, α dim H µ). Let k be such that θ n+k = s. Let β = s r n ν for some exponent ν that will be fixed later. Note that r 2 ≪ s ≪ r ≪ β. Recall that ℓ n is the minimal integer ℓ ≥ 0 such that a -n+ℓ =0 . We let Z = Z 0 -n (ξ) and set U := g ℓn π X (f -n Z) ∈ Zn-ℓn . Note that the code of U starts with a non zero symbol. To simplify the exposition we introduce the notation

I ⋉ B := {(t, z) : t ∈ I, (t, z) ∈ B} = π -1
X (I) ∩ B, for I ⊂ T 1 and B ⊂ M. We decompose W with the pieces of element Z ′ of the partition Z 0 -n-k which intersect it.

Lemma 8.10.

Each Z ′ ∩ W (with Z ′ ∈ Z 0 -n-k ) is contained in a tube T Z ′ Z ′ ∩ W ⊂ I ⋉ Z ′ ⊂ T Z ′ := {(t, z) ∈ M : t ∈ I, |z -z 0 -(t -t 0 )u Z ′ | < D 1 s} where D 1 = 2 + C(n 0 ) + π/(1 -θ), I ⊂ (x -r, x + r), (t 0 , z 0 ) ∈ Z ′ ∩ W and |u Z ′ -u n (ξ)| < cr.
There are two type of intersections, transversal and non transversal (See Figure 1). (since r| sin β| ≤ rβ and r(1 -cos β) ≤ rβ 2 ≪ s). These two boxes are parallel to the plane defined by the x-axis and the vector u n (ξ), of length r, height of order rβ and thickness s. There are at most Cr/s different Z ′ intersecting W non transversally, since they are separated by a distance of order s and they point in the same direction bounded away from the horizontal axis (the x-axis). Moreover, each Z ′ ∩ W is contained in a set I Z ′ ⋉ Z ′ for some interval I Z ′ of length at most crβ. Lemma 8.11. For any interval I ⊂ T 1 \ {0} and any cylinder

Z ′ ∈ Z 0 -n-k such that Z ′ ⊂ Z we have µ(I ⋉ Z ′ ) ≈ µ(Z ′ )|I|.
Proof. The set f -n-k (I ⋉ Z ′ ) is of the form J × D where the interval J is a subset of the cylinder Z′ = π X (f -n-k Z) ∈ Zn+k and g n+k J = I. By invariance we have µ(I ⋉ Z ′ ) = μ(J). Observe that Z′ and Z have the same code. By distortion [START_REF] Rousseau | Poncaré recurrence for observations[END_REF] we have

|J| | Z′ | ≈ |I| |T 1 | .
This ratio of length transfers to a ratio of measures by [START_REF] Liverani | A probabilistic approach to intermittency[END_REF], proving the lemma since by invariance again μ( Z′ ) = µ(Z ′ ).

Note that except the very particular cylinder

Z ′ = [0 k Z] ∈ Z 0 -n-k which has a measure µ([0 k Z]) ≈ (k + ℓ n ) -1/γ |U | ≈ k -1/γ |U |,
(due ( 18)), all the other Z ′ ∈ Z 0 -n-k contained in Z do have at least a non zero symbol in their code at some position -n -m 0 ∈ {-n -k, . . . , -n -1}. Applying [START_REF] Wasilewska | Limiting distribution and error terms for the number of visits to balls in mixing dynamical systems[END_REF] to them, we obtain

µ(Z ′ ) ≈ (k -m 0 + 1) -1/γ |[a -n-m 0 • • • a 0 ]| (k -m 0 + 1) -1/γ (m 0 + ℓ n ) -1-1/γ |U | (k -1/γ (1 + ℓ n ) -1-1/γ + k -1-1/γ )|U |
(considering separately the cases m 0 ≤ k/2 and m 0 > k/2). Therefore the total measure coming from non transversal intersections is bounded by µ(

Z ′ ∩N T =∅ Z ′ ) [k -1/γ + r s (k -1/γ (1 + ℓ n ) -1-1/γ + k -1-1/γ )]|U |crβ. ( 20
)
For transversal intersections, the angle is at least β. Thus Z ′ ∩ W ⊂ I Z ′ ⋉ Z ′ for some interval I Z ′ of length less than s/β. Hence the measure of the intersection µ(Z ′ ∩ W )

s β µ(Z ′ ). Therefore µ( Z ′ ∈N T Z ′ ) s β µ(Z) s β c(1 + ℓ n ) -1/γ |U |. (21)
Finally, the ball B(ξ, r) contains the set I ⋉Z * for some cylinder Z * ∈ Z 0 -n-2 , Z * ⊂ Z, and an interval I of length say r/4. Thus

µ(B(ξ, r)) ≥ µ(I ⋉ Z * ) µ(Z * )r (1 + ℓ n ) -1-1/γ |U |r. (22)
Putting ( 20), ( 21) and ( 22) together, we obtain

µ(W ) µ(B(ξ, r)) (I) + (II) + (III) + (IV )
where

(I) = k -1/γ β(1 + ℓ n ) 1+1/γ = s r O(n -1/γ+ν+(1+1/γ)/p ) (II) = k -1/γ rβ s = O(n -1/γ+ν ) (III) = k -1-1/γ rβ s (1 + ℓ n ) 1/γ+1 = O(n -1-1/γ+ν+(1/γ+1)/p ) (IV ) = s rβ (1 + ℓ n ) = O(n -ν+1/p ).
The first term (I) goes to zero exponentially fast in n. The second term (II) goes to zero provided ν < 1/γ, the third one provided ν < 1/γ + 1 -(1/γ + 1)/p and the last one provided ν > 1/p. If γ > 1/2, all these conditions are satisfied whenever the interval (γ, 1/γ -γ) is nonempty (i.e. whenever γ < 1/ √ 2), taking ν in the interval and p sufficiently close to 1/γ. If γ ≤ 1/2, (I), (II), (III) and (IV ) go to zero provided p is close enough to 2.

Billiard in a stadium

Let ℓ > 0. We consider the convex domain Q of the plane corresponding to the union of the rectangle [-ℓ/2, ℓ/2]×[-1, 1] with the two discs of radius 1 centered at (-ℓ/2, 0) and (ℓ/2, 0) respectively. This domain Q is called stadium.

Figure 2. The stadium billard

The billiard system (M, f, µ) will describe (at reflection times) the evolution of a point particle moving in Q with elastic reflection off ∂Q and going straight on between two reflection times. The phase space M corresponds to the set of unit vectors based on ∂Q and pointing inwards. We define it as follows:

M := R 2(π + ℓ)Z × - π 2 , π 2 .
A particle has configuration x = (r, ϕ) ∈ M if its position p ∈ ∂Q has curvilinear abscissa r (∂Q being oriented counterclockwise and r = 0 corresponding to the point (ℓ/2, -1)) and if its reflected vector makes the angle ϕ with n(p), where n(p) is the unit normal vector to ∂Q at p oriented inwards. The transformation f : M → M maps a configuration at a reflection time to the configuration at the next reflection time.

This transformation preserves the probability measure µ on M given by dµ = cos ϕ 4(π + ℓ) drdϕ.

Here again we consider the metric d((r, ϕ), (r ′ , ϕ ′ )) := max(|r -r ′ |, |ϕ -ϕ ′ |) (but our result also holds for the other usual metrics).

f(x)

x Figure 3. The billard map f . Theorem 9.1. For the billiard system (M, f, µ) in the stadium described above, the conclusion of Theorem 3.1 holds true for µ-almost every x ∈ M, i.e. for µ-almost every x ∈ M, the number of visits to B(x, r) up to time ⌊t/µ(B(x, r))⌋ converges in distribution (with respect to µ) to a Poisson random variable of mean t.

Proof. The fact that (M, f, µ) can be modeled by a Young tower satisfying (P1)-(P5) can be found in [START_REF] Markarian | Billiards with polynomial decay of correlations[END_REF][START_REF] Chernov | Billiards with polynomial mixing rates[END_REF][START_REF] Bálint | Limit theorems in the stadium billiard[END_REF]. Namely, the fact that (P2)-(P3) hold with α = 1 (and so (2)) comes from Propositions 2.1 and 2.3 of [START_REF] Bálint | Limit theorems in the stadium billiard[END_REF] and the fact that (1) holds with ζ = 2 is proved in Section 9 of [START_REF] Chernov | Billiards with polynomial mixing rates[END_REF]. Finally, because of the continuity and positivity of the density function of µ with respect to the Lebesgue measure, (3) holds for every x ∈ M (See the Appendix for details).

In particular for Billiard systems one can take a = b = 1. ≤ 4s a r -a θ ρ(1 -θ) a µ(B(x, 2r)), since s a ≥ s a k+1 = 2 -k-1 (ρθ n (1 -θ)) a and r -a ≥ (θ n ) -a θ -a . Now, for s ∈ [s 1 , r], we have s/r ≥ ρλ(1 -θ) and so µ(B(x, r + s) \ B(x, r)) ≤ µ(B(x, 2r)) ≤ s a r -a (ρλ(1 -θ)) -a µ(B(x, 2r)).

Proposition 7 . 1 .

 71 Let σ < dim H µ. For µ-a.e. x ∈ M we have R 2 (B(x, r), ⌊r -σ ⌋) = o(µ(B(x, r))).

Proposition 8 . 2 . 2 / 2 ,

 8222 If γ < √ for µ-almost every x ∈ M, the assumption (3) on the coronas is satisfied.

2 .

 2 Proof. Let ξ = (x, z) ∈ M with x = 0. Since B(ξ, r) = (x -r, x + r) × D(z, r),setting C r,s (x) and C r,s (z) the coronas in T 1 and C respectively, we get C r,s (ξ) = (C r,s (x) × D(z, r + s)) ∪ ((x -r, x + r) × C r,s (z)).

1+α 2 dim

 2 H µ and s = r δ the last bound satisfies cs = o(µ(B(ξ, r))). The second term deserves more attention. It suffices to show that µ(W ) = o(µ(B(ξ, r))), where W := (x -r, x + r) × C r,s (z).

Figure 1 .

 1 Figure 1. The green rods represent the pieces of cylinders Z ′ intersecting W . The non transversal intersections happen in the red boxes. Let v n = u n (ξ)/|u n (ξ)|. We say that Z ′ intersects W non transversally if there exists ξ ′ = (x ′ , z ′ ) ∈ Z ′ ∩ Λ such that the angle between u ∞ (ξ ′ ) and the circle of center z and radius |z -z ′ | is smaller than β. But since ∠(v n , u ∞ (ξ ′ )) r ≪ β, such a transversal intersection can only occur in (x -r, x + r) × {z ′ ∈ C : z ′ = z + (±r + q)v n e ip , |q| < s, |p| β}. Now, for any ξ ′′ ∈ ((x -r, x + r) ⋉ Z ′ ) ∩ Λ, we have ∠(u ∞ (ξ ′′ ), u ∞ (ξ ′ ))| r ≪ β. So for such a Z ′ , the intersection Z ′ ∩ Λ ∩ W is contained in one of the boxes N T N T = (x -r, x + r) × {z + pv n + (±r + q)iv n : |p| < 2rβ, |q| < s}

Proposition A. 2 .( 1 - 2 2 Proposition A. 3 .

 21223 If µ is non-atomic and satisfies (23) at some x ∈ supp(µ), then b ≥ a.Proof. Assume that µ satisfies (23) with b < a. Let η ∈ (0, a -b). Then there exists n 0 such that, for every integer n ≥ n 0 , we haveµ(B(x, 2 -n ) \ B(x, 2 -n-1 )) ≤ 2 -ηn µ(B(x, 2 -n )), i.e. (1 -2 -ηn )µ(B(x, 2 -n )) ≤ µ(B(x, 2 -n-1)). Therefore, for any n > n 0 , we haveµ(B(x, 2 -n 0 )) -ηn ) ≤ µ(B(x, 2 -n )) and so µ({x}) = lim n→+∞ µ(B(x, 2 -n )) > 0 if µ(B(x, 2 -n 0 )) > 0.Now let us prove Proposition 3.Let θ < 1. For any a ∈ (0, 1), for every x ∈ M, there exists a constant C 0 such that there exists a sequence r n (x) ∈ (θ n+1 , θ n ) such that, for all n, (23) holds with b = a for the sequence r = r n (x).Proof. Up to a change of θ in one of its integer roots, we assume without loss of generality that 1/2 < θ < 1. Fix x ∈ M and n ∈ N. Let λ < 1/2 and set ρ = (1 -2λ)/3. We define a measure on [0, 1] by setting m([0, t]) = µ(B(x, θ n+1 + t(θ n -θ n+1 ))) -µ(B(x, θ n+1 )) for t ∈ [0, 1]. We construct a sequence of nested intervals I k by dichotomy as follows. Let I 0 = (0, 1). Given I k (for some k ≥ 0), we consider two intervals of length λ|I k | included in I k , at distance ρ|I k | from the left and right endpoints of I k respectively and we define I k+1 as the one with smallest measure. We have |I k | = λ k and m(I k ) ≤ 2 -k m(I 0 ). Let {t * } be the intersection of the I k 's. By construction, since t * ∈ I k the ρλ k -neighborhood of t * is contained in I k-1 . Let us prove that the radius r = θ n+1 + t * (θ n -θ n+1 ) satisfies (23) with a = -ln 2/ ln λ, with a constant C 0 not depending on n. Let s k := ρλ k θ n (1 -θ). For every k ≥ 1 and every s ∈ [s k+1 , s k ], we observe that µ(B(x, r + s) \ B(x, r)) ≤ µ(B(x, r + s k ) \ B(x, r)) ≤ m(I k-1 ) = 2 -k+1 m(I 0 ) ≤ 2 -k+1 µ(B(x, θ n ) \ B(x, θ n+1 )) ≤ 2 -k+1 µ(B(x, 2r)), since θ n < r θ < 2r

Appendix A. Measure of coronas

In this section we discuss various conditions under which Assumption (3) holds and more precisely on the following condition: ∃C 0 , r 0 > 0 such that

Indeed, we observe that, if α dim H µ > b/a, (23) implies (3) for any δ ∈ (b/a, α dim H µ).

Proposition A.1. Suppose that µ is absolutely continuous with respect to the Lebesgue measure.

(i) For any continuity point x of the density, we have ( 23) with a = b = 1 and so (3) since α dim H µ > 1. (ii) If the density is in L p loc for some p > 1 then Assumption (23) holds a.e. with a = 1/q and b = (1 + d(q -1))/q, where 1/p + 1/q = 1.

Proof. (ii) Suppose that the density h is in L p on some open set V and take x ∈ V with B(x, 2r) ⊂ V . By the Hölder inequality for any s < r µ(B(x, r + s) \ B(x, r))

The upper bound is of order r (d-1)/q s 1/q while the measure of the ball is lower bounded by Const × r d for a.e. x.


Edmundo Kronmüller

Ira A Noveck

Mohammed Ikram Krarzia

Rawan Charafeddine

A perspective-free interpretation of negation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A PERSPECTIVE-FREE PRAGMATIC INTERPRETATION OF NEGATION

Edmundo Kronmüller (Escuela de Psicologia Universidad Católica, Santiago, Chile) Ira Noveck, Mohamed Krarzia, & Rawan Charafeddine (Laboratoire sur le langage, le cerveau et la cognition - CNRS, Lyon, France) This study investigates the time-course of perspective taking during reference resolution (Keysar et al. 2000;Hanna et al., 2003;Barr, 2008), but here, instead of an utterance containing a direct reference to an object (e.g. "click on the upside-down funnel"), speakers refer to objects using an utterance containing a negation (e.g. "click on the one that is not the shiny cylinder"). Because listeners hear a negation, the process of recognizing the intended referent is free from the influence of bottom-up processes that can bias the interpretation toward one referent in the domain, like priming or some other automatic activation of speaker-specific word representations. These bottom-up processes have been proposed as the main mechanisms for early speaker-specific effects in interpreting referential expressions (Brennan & Hanna, 2009;Brown-Schmidt, 2009).

In order to carry out this investigation, participants played the role of listeners in a referential communication game and followed directions from two speakers. One of the speakers was a confederate who interacted live with the participant; the participant-listener and the live speaker sat facing one other, each with her own computer screen. The other speaker was a virtual partner from whom the participants heard pre-recorded directions. The task consisted of selecting pictures depicting unfamiliar objects with no conventional names. While the listener's screen showed three pictures, participants believed that the live speaker's screen had just two of the three objects they themselves were seeing and that the speaker had come up with the names or expressions for the objects at that exact moment. In fact, the confederate's screen showed only the "target" picture plus the exact expression the confederate speaker was required to utter. Participants did not know which object was missing from the speaker's view.

In the test trial, three objects appeared on the listener's screen. One object was the negated object that had been referred to twice before by the live speaker in earlier filler trials, allowing for earlier entrainment (Garrod & Anderson, 1987). Another object was the unmentioned object, that hadn't been referred to directly before, leaving it nameless. The third object was the competitor object, through which we introduced the experimental manipulation. In the Same Speaker condition, the live speaker entrained the competitor object by referring to it twice before (as with the negated object). In the Different Speaker condition, it was the pre-recorded speaker who entrained the competitor object. Critically, only the listener knew about the name the virtual speaker gave to this object because the instructions from this speaker were provided through headphones, leaving the other speaker -the live speaker-unaware of the existence of an entrained expression for the competitor object, and even unaware of the identity of the objects the virtual speaker asked the participant to select. Finally, in the None condition, the competitor object was never referred to, leaving it without a name (just like the unmentioned object) and thus leaving the participant with a genuinely ambiguous referential situation.

In these test trials, the live speaker asks the participant to select an object by contrasting it with another object in the screen (the negated object): "click on the one that is not the shiny cylinder". In the None condition, either the competitor or the unmentioned object could be the potential referents, given that both didn't have a name. In the Same condition, only the unmentioned object was a potential referent given that the competitor was referred to twice by the live speaker. Finally, in the Different condition, and assuming that participants consider their partner's perspective, both objects --the unmentioned and the competitor objects --were possible referents (remember, the competitor object didn't have a name for the live speaker either). Note that the speaker's expressions are thoroughly cooperative given that listeners believed that the live speaker saw only two objects.

Twenty-four French speakers participated in the experiment. Participants selected the unmentioned object in the Same Speaker condition in 90% of the trials and in 52% of trials in the None condition. This indicates that participants interpreted the object referred to by the negation of one object as the object without a name. Interestingly, in the Different Speaker condition, the selection rate for the unmentioned object was 61%, revealing that perspective information was considered for final referent assignment (Different vs None: p = .064; Different vs Same: p < .001). Indeed, from the perspective of the live speaker, neither the unmentioned object nor the competitor object had a name, so both are potential referents.

We looked at time course data on the interpretation process. Eye-tracking reveals that 1 and 2 seconds after negation on-set, interference produced by the competitor objectmeasured as the proportion of time looking at the unmentioned object minus the proportion of time looking at the competitor-is equal for the Same and Different conditions (p = .438), but stronger for the None condition (Different vs None: p = .028)(see figure). This shows that participants have not yet realized that from the point of view of the live speaker, the competitor object is also a potential referent for the utterance, as they looked directly at the unmentioned object in the Same and Different conditions. In the None condition, however, because the competitor object has not been named, it is equally good as a referent for the utterance as the unmentioned object. In between 2 and 3 seconds after the on-set of negation, the interference of the competitor object is stronger in the Different than in the Same condition (p = .037), showing that only at this moment and not earlier, do participants begin to integrate speaker's perspective information and realize that in the Different speaker condition, the competitor object was unnamed for the live speaker.

These results suggest that listeners make a fast pragmatic interpretation of negation that does not necessarily rely on perspective information, not even partially, but that relies on pragmatic expectations. Specifically, listeners assume the speaker would refer to an already named object with that same name and not with a negation. Following this expectation of cooperativeness, if the speaker used a negation, then the referent should be a nameless object. Only after this perspective-free interpretation, participants restrict their domain of reference by considering the partner's knowledge. These results are relevant because when referring to something by using negation, listeners con not fast-map a referent to an expression, having to always use information that is not in the linguistic code. This study shows that early on in a first interpretation, pragmatic expectations are already deployed but perspective taking has not yet taking place.