Stefano Galatolo

J Ér

Ôme Rousseau

Benoit Saussol

SKEW PRODUCTS, QUANTITATIVE RECURRENCE, SHRINKING TARGETS AND DECAY OF CORRELATIONS

We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recurrence (also with respect to given observables) and hitting time scale behavior depend on the arithmetical properties of the extension.

By this we show that those systems have a polynomial decay of correlations with respect to C r observables, and give estimations for its exponent, which depend on r and on the arithmetical properties of the system.

We also show examples of systems of this kind having not the shrinking target property, and having a trivial limit distribution of return time statistics.

 [START_REF] Abadi | Inequalities for the occurence time of rare events in mixing processes. The state of the art, Markov Process[END_REF][START_REF] Saussol | An introduction to quantitative Poincaré recurrence in dynamical systems[END_REF][START_REF] Barreira | Dimension and recurrence in hyperbolic dynamics[END_REF]and references therein, just to mention a few of these aspects:

• information theory in the study of repetitions of words in sequences of symbols [START_REF] Ornstein | Entropy and data compression schemes[END_REF],

• probability in dynamical Borel-Cantelli lemma [START_REF] Galatolo | The dynamical Borel-Cantelli lemma and the waiting time problems[END_REF], shrinking targets problems [START_REF] Hill | The ergodic theory of shrinking targets[END_REF], exponential law, Poisson statistics and extreme value theory [START_REF] Freitas | Hitting Time Statistics and Extreme Value Theory[END_REF],

• geometric measure theory in the relation with Hausdorff dimension [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF][START_REF] Galatolo | Dimension via waiting time and recurrence[END_REF][START_REF] Galatolo | Dimension and hitting time in rapidly mixing systems[END_REF] • geometry, in logarithm laws for the geodesic flow and similar(see e.g. [START_REF] Sullivan | Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics[END_REF][START_REF] Kleinbock | Logarithm laws for flows on homogeneous spaces[END_REF][START_REF] Galatolo | Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds[END_REF][START_REF] Masur | Logarithmic law for geodesics in moduli space, Mapping class groups and moduli spaces of Riemann surfaces[END_REF]) • nonlinear analysis in recurrence plots (see e.g. [START_REF] Marwan | Recurrence Plots for the Analysis of Complex Systems[END_REF]). Quite recent results showed relations between quantitative indicators representing the scaling behavior of return times and hitting times in small targets, decay of correlations and arithmetical properties. More precisely, let us consider a discrete time dynamical system (X, T, µ), where (X, d) is a metric space and T : X → X is a measurable map preserving a finite measure µ. Let us consider two points in X and the time which is necessary for the orbit of x to approach y at a distance less than r τ r (x, y) = min{n ∈ N + : d(T n (x), y) < r}. We consider the behavior of τ r (x, y) as r → 0. In many interesting cases this is a power law τ r (x, y) ∼ r R . When x = y this exponent R gives a quantitative measure of the speed of recurrence of an orbit near to its starting point, and this will be a quantitative recurrence indicator. When x = y the exponent is a quantitative measure of how fast the orbit starting from x approaches a point y. It is indeed a measure for the scaling behavior of the hitting time of an orbit starting from x to a sequence of small targets: the balls B r (y) centered in y of radius r. To extract the exponent the recurrence rate (R(x, x) ) and hitting time scaling exponent (R(x, y) ) are defined by R(x, y) = lim sup r→0 log τ r (x, y) log r , R(x, y) = lim inf r→0 log τ r (x, y) log r .

A general philosophy is that in "chaotic" systems these scaling exponents are equal to the local dimension. Indeed, several results relate those exponents to the scaling behavior of the measure of the target set B r (y), the local dimension d µ (y) of the measure µ at y (see e.g. [START_REF] Boshernitzan | Quantitative recurrence results[END_REF], [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF], [START_REF] Saussol | Recurrence rate in rapidly mixing dynamical systems[END_REF], [START_REF] Rousseau | Poincaré recurrence for observations[END_REF], [START_REF] Saussol | Recurrence rate in rapidly mixing dynamical systems[END_REF], [START_REF] Su | Convergence of random walks on the circle generated by an irrational rotation[END_REF], [START_REF] Galatolo | Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems[END_REF], [START_REF] Galatolo | Dimension via waiting time and recurrence[END_REF], [START_REF] Galatolo | Dimension and hitting time in rapidly mixing systems[END_REF], [START_REF] Galatolo | Lorenz like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence[END_REF], [START_REF] Galatolo | Hitting time and dimension in axiom A systems, generic interval exchanges and an application to Birkoff sums[END_REF], [START_REF] Philipp | Some metrical theorems in number theory[END_REF]). More precisely: in [START_REF] Saussol | Recurrence rate in rapidly mixing dynamical systems[END_REF] (see also [START_REF] Rousseau | Poincaré recurrence for observations[END_REF]) and [START_REF] Galatolo | Dimension and hitting time in rapidly mixing systems[END_REF] it is proved that if a system has superpolynomial decay of correlations with respect to Lipschitz observables (equivalently for C k observables, see Appendix), then the recurrence and hitting time exponents are equal to the local dimension of the invariant measure (see Theorem 4 for the precise result).

On the other hand the definition of the above scaling exponents also recalls diophantine approximation. Indeed in "non chaotic systems" like Rotations, Interval exchanges or reparametrizations of rotations, their behavior is related to arithmetical properties of the system and not to the dimension (see [START_REF] Boshernitzan | Diophantine properties of IETs and general systems: Quantitative proximality and connectivity[END_REF], [START_REF] Kim | The recurrence time for interval exchange maps[END_REF], [START_REF] Kim | The waiting time for irrational rotations[END_REF], [START_REF] Galatolo | Long hitting time, slow decay of correlations and arithmetical properties[END_REF], [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF] and Section 2.3 below).

It is worth to remark that there exist mixing systems (with subpolynomial rate, see [START_REF] Galatolo | Long hitting time, slow decay of correlations and arithmetical properties[END_REF]) where the exponents are not related to dimension but to arithmetical properties.

Outline of the results.

The question remained open, if there are polynomially mixing systems (with respect to Lipschitz observables) such that the above defined recurrence-hitting time exponents are different from dimension, and the question whether this can happen for superpolynomially mixing system with respect to smoother observables (C ∞ observables e.g.).

We consider in this paper a skew product of a chaotic map with a rotation with some finite Diophantine type (see below and Section 3 for more precise definitions) and we will show that such a system posses these properties. To be more precise, we are interested in systems of the type (ω, t) → (T ω, t + ϕ(ω)) on a phase space (ω, t) ∈ Ω × R d /Z d , where T is a piecewise expanding map, and on the second coordinate we have a family of isometries controlled by a certain function ϕ. The most basic, yet nontrivial example is the case where T is the doubling map on S 1 , and on the second coordinate we have circle translations.

Ergodic properties of a skew products as above have been quite well studied in the literature as they are one of the most simple example of systems which is in some sense weakly chaotic. The qualitative ergodic theory of these systems has been studied by Brin. A quantitative result about its speed of mixing was given in [8], where it is proved that provided the rotation's angle has some diophantine properties, such systems have at least polynomial decay of correlations with respect to Lipschitz and C k observables, moreover they have superpolynomial decay of correlations with respect to C ∞ ones. More recently, in [START_REF] Gouezel | Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences[END_REF] it was shown that under suitable assumptions, including the above cited most basic example, if ϕ is C 1 and not cohomologous to a piecewise constant function, then the decay of correlations is exponential (for Hölder observables).

In this paper, we investigate the quantitative recurrence and hitting time indicators of such kind of systems and its relations with decay of correlations, in the complementary case where ϕ is piecewise constant. In this case we show the following facts:

• The decay of correlations under Lipschitz or C k observables is polynomial and we give a concrete estimation for its exponent depending on the arithmetical properties of the system (in the case of multidimensional rotations the linear diophantine type of the angle is involved, see Sections 3.2, 5, 6.1). • The recurrence and hitting time exponents are related both to the dimension and the arithmetical properties of the rotation and we give estimations for these exponents (see Sections 4,6.1). • We show that if the Diophantine type is large, the statistics of return time and hitting time has a trivial limiting distribution along some subsequence.

In particular, there is no convergence to the exponential law for these statistics. • Using a multidimensional rotation with a suitable angle (with intertwined partial quotients) the resulting skew product is an example of a system with polynomial decay of correlations, but not satisfying the monotone shrinking target property and having a trivial distribution of return and hitting times. We remark that a mixing system without the monotone shrinking target property was shown in [START_REF] Fayad | Mixing in the absence of the shrinking target property[END_REF]. However the speed of decay of correlations of that example is less than polynomial ( [START_REF] Galatolo | Long hitting time, slow decay of correlations and arithmetical properties[END_REF]), see Section 6.1.

We can hence consider these skew products as a borderline case for the relations between recurrence, hitting time and decay of correlations.

An ingredient of our proofs is an estimate of a discrepancy for random walks generated by multidimensional irrational rotations. This result (which is a generalization of the one given in [START_REF] Su | Convergence of random walks on the circle generated by an irrational rotation[END_REF]) is proven in the appendix. There we also prove some other technical results which are more or less known or at least expected but which are, as far as we know, not present in this form in the literature.

2. Background. . It is relatively easy to obtain that for a general systems ([3], [START_REF] Galatolo | Dimension via waiting time and recurrence[END_REF]):

Proposition 1. Let (X, T, ν) be a dynamical system over a separable metric space and ν be an invariant Borel measure. For each y

(2) R(x, y) ≥ d ν (y) , R(x, y) ≥ d ν (y) holds for ν-almost each x ∈ X. Moreover, if X is a measurable subset of R d for some d, then (3) R(x, x) ≤ d ν (x) , R(x, x) ≤ d ν (x)
for ν-a.e. x ∈ X.

Rapid mixing and consequences.

A sufficient condition for these inequalities to become equalities is that the system is mixing sufficiently rapidly. This is measured by the decay of correlations.

Definition 2 (Decay of correlations). Let φ, ψ : X → R be observables on X belonging to the Banach spaces B, B ′ . Let Φ :

N → R such that Φ(n) → n→∞ 0. A system (X, T, ν
) is said to have decay of correlations with speed Φ with respect to observables in B and B ′ if

(4) φ • T n ψdν -φdν ψdν ≤ ||φ|| B ||ψ|| B ′ Φ(n)
where || || B ,|| || B ′ are the norms 1 in B and B ′ . We say that the decay is polynomial with exponent between a and b if

a ≤ lim inf n→∞ -log Φ(n) log n ≤ lim sup n→∞ -log Φ(n) log n ≤ b.
We say that the decay of correlation is superpolynomial if lim n α Φ(n) = 0, ∀α > 0.

The decay of correlations has been studied for many systems, mostly those with some form of hyperbolicity. It depends on the class of observables, and it is obvious that if a system has decay of correlations with some given speed with respect to Lipschitz or C p observables, then it has also with respect to C k ones when k ≥ p (with at least the same speed). There is also a converse which will be used later. Lemma 3. If the space where the dynamics act is a manifold X ⊂ R d (possibly with boundary) and the system has decay of correlations with respect to C p , C q observables with speed Φ p,q (n) then it has also with respect to C k , C ℓ observables with speed Φ k,ℓ (n) ≤ Φ p,q (n)

1 p k + q ℓ -1 .
In the case k = 1 or ℓ = 1 the same is true with Lipschitz (observables and norm) instead of C 1 .

The proof of the above lemma is postponed to the appendix. Now we can state a result linking decay of correlations and local dimension with recurrence and hitting time ( [START_REF] Galatolo | Dimension and hitting time in rapidly mixing systems[END_REF], [START_REF] Rousseau | Poincaré recurrence for observations[END_REF]). We see that in rapidly mixing systems the recurrence and hitting time exponents are necessarily equal to the local dimension of the invariant measure.

R(x) = R(x) = d ν (x)
for ν-almost each x.

2.3.

Basic results on circle rotations. We will see that in the class of system we are interested in, arithmetical properties are important for quantitative recurrence and hitting time scaling behavior. The simplest case of system where this happen is the case of rotations on the circle. We state some basic results, linking quantitative recurrence, shrinking targets and arithmetical properties in circle rotations. Definition 5. Given an irrational number α we define the type of α as the following (possibly infinite) number:

γ(α) = inf{β : lim inf q→∞ q β qα > 0}.
1 It is worth to remark that under reasonable assumptions, a non uniform statement: φ • T n ψdµφdµ ψdµ ≤ C φ,ψ Φ(n) can be converted in an uniform one, like (4). See [START_REF] Chazottes | Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems[END_REF]. 2 The limit dν (y) = lim r→∞ log ν(B(y,r)) log r exists.

Let q n being the sequence of convergent denominators of α (see the appendix for some recalls on continued fractions). It is worth to remark that the above definition is equivalent to the following [START_REF] Cassels | An introduction to Diophantine approximation[END_REF] γ(α) = lim sup n→∞ log q n+1 log q n .

Every number has type ≥ 1. The set of number of type 1 is of full measure; the set of numbers of type γ has Hausdorff dimension 2 γ+1 . There exist numbers of infinite type, called Liouville numbers; their set is dense and uncountable and has zero Hausdorff dimension.

The behavior of quantitative recurrence and hitting time in small targets for circle rotations with angle α depend on the type of α, as it is described in the following statement Theorem 6. ( [START_REF] Kim | The waiting time for irrational rotations[END_REF], [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF]) If T α is a rotation of the circle, y a point on the circle then for almost every x R(x, y) = γ(α), R(x, y) = 1.

and for each x

(7) R(x, x) = 1 γ(α) , R(x, x) = 1.
It is interesting to see that the statement implies the impossibility to construct a system with (8) R(x, y) > d ν (y)

for typical x with a circle rotation, and it is worth to remark that similarly such an example cannot be constructed by an interval exchange [START_REF] Boshernitzan | Diophantine properties of IETs and general systems: Quantitative proximality and connectivity[END_REF]. The situation is different if we can consider two dimensional rotations (see Section 6.1). We will exploit this and construct a system having even polynomial decay and still satisfying (8).

2.4. Statistics of hitting/return time. Let us consider a family of centered balls B r = B r (y) such that ν(B 0 ) = 0 and ν(B r ) = 0 for r > 0. We will consider the statistical distribution of return times in these sets. We say that the return time statistics of the system converges to g for the balls B r , if

(9) lim r→0 ν({x ∈ B r , τ r (x, y) ≥ t ν(Br ) }) ν(B r ) = g(t).
If g(t) = e -t we say that the system has an exponential return time limit statistics. Such statistics can be found in several systems with some hyperbolic behavior in some class of decreasing sets, however other limit distributions are possible.

Starting from all the space instead of the ball B r defines the hitting time statistics: [START_REF] Fayad | Mixing in the absence of the shrinking target property[END_REF]

lim r→0 ν({x ∈ X, τ r (x, y) ≥ t ν(B r ) }) = g(t).
We will show that in our system there is not exponential statistics, even more, the limiting distribution can be trivial.

3. The class of skew products under consideration 3.1. Definition. We will consider a class of systems constructed as follows. The base is a measure preserving system (Ω, T, µ). We assume that T is a piecewise expanding Markov map on a finite-dimensional Riemannian manifold Ω, that is:

• there exists some constant β > 1 such that D x T -1 ≤ β -1 for every x ∈ Ω. • There exists a collection J = {J 1 , . . . , J p } such that each J i is a closed proper set and (M1) T is a C 1+η diffeomorphism from int J i onto its image;

(M2) Ω = ∪ i J i and int J i ∩ int J j = ∅ unless i = j; (M3) T (J i ) ⊃ J j whenever T (int J i ) ∩ int J j = ∅.
J is called a Markov partition. It is well known that such a Markov map is semiconjugated to a subshift of finite type. Without loss of generality we assume that T is topologically mixing, or equivalently that for each i there exists n i such that T ni J i = Ω. We assume that µ is the equilibrium state of some potential ψ : Ω → R, Hölder continuous in each interior of the J i 's. The sets of the form J i0,...,iq-1 := q-1 n=0 T -n J in are called cylinders of size q and we denote their collection by J q . In this setting it is well known that the pointwise dimension d µ (x) exists and d µ (x) = d µ , µ-almost everywhere.

The system is extended by a skew product to a system (M, S) where M = Ω× T d , T d is the d-torus and S : M → M defined by [START_REF] Freitas | Hitting Time Statistics and Extreme Value Theory[END_REF] S(ω, t) = (T ω, t + αϕ(ω))

where α = (α 1 , ..., α d ) ∈ T d and ϕ = 1 I is the characteristic function of a set I ⊂ Ω which is an union of cylinders. In this system the second coordinate is translated by α if the first coordinate belongs to I. We endow (M, S) with the invariant measure ν = µ × m (m is the Haar measure on the torus).

We make the standing assumption that (NA) for any u ∈ [-π, π], the equation f e iuϕ = λf • T , where f is Hölder (on the subshift) and λ ∈ S 1 , has only the trivial solutions λ = 1 and f constant. This is equivalent to the fact that the map (ω, s) → (T ω, s + uϕ(ω)) is weaklymixing on Ω × T. The simple case where I is a nonempty union of size 1 cylinders such that both I and I c contain a fixed point fulfills this assumption. 3 We will indicate by π 1 : Ω × T d → Ω and π 2 : Ω × T d → T d the two canonical projections, moreover, on Ω × T d we will consider the sup distance.

We will suppose that α is of finite Diophantine type. We introduce two definitions of Diophantine type, which generalize Definition 5 to the higher dimensional cases. The notation ||.|| will indicate the distance to the nearest integer vector (or number) in R d (or R) and |k| = sup 0≤i≤d |k i | the supremum norm.

Definition 7. The Diophantine type of α = (α 1 , ..., α d ) for the linear approxi- mation is γ l (α) = inf{γ , s.t.∃c 0 > 0 s.t. k • α ≥ c 0 |k| -γ ∀0 = k ∈ Z d }
3 Any solution has a modulus |f | constant. The first fixed point gives then e iu = λ while the second gives λ = 1. The existence of such f would then contradicts the ergodicity of T .

Definition 8. The Diophantine type of α = (α 1 , ..., α d ) for the simultaneous approximation is

γ s (α) = inf{γ , s.t.∃c 0 > 0 s.t. kα ≥ c 0 |k| -γ ∀0 = k ∈ N}.

3.2.

Upper bound on decay of correlations. We have an explicit upper bound for the rate of decay of correlations. Proposition 9. For Lipschitz observables the rate of decay is O(n -1 2γ ) for any γ > γ l (α).

For C p , C q observables, the rate of decay is O(n -1 2γ max(p,q,p+q-d) ) for any γ > γ l (α).

Remark 10. We remark that the rate of decay of correlations is superpolynomial for C ∞ observables.

The proof of the proposition is based on the following statement Lemma 11. If the observables A, B are respectively of class C p and C q on M , p + q > d and M Adν = 0, the correlations satisfy

C n (A, B) := M AB • S n dν ≤ C A C p B C q n -ℓ+ǫ for each ǫ > 0, where ℓ = p+q-d 2γ l (α) . Proof of Proposition 9. Let x, y ∈ N * . By Lemma 3 we have Φ p,q (n) = O(Φ px,qy (n) 1 x+y-1 ). By Lemma 11 we know that Φ px,py (n) = O(n -(px+qy-d) 2γ l (α) +ǫ ) for any ǫ > 0. Hence Φ p,q (n) = O(n -ℓ+ǫ ) with ℓ = 1 2γ px + qy -d x + y -1 .
Taking the supremum over x, y ∈ N * gives the conclusion.

The idea of the proof of Lemma 11 is to expand in Fourier series the observables with respect to the second variable t and see that C n (A, B) corresponds to an infinite sum of terms which are similar to correlation integrals of the Fourier coefficients (which are still functions of ω ) with respect to a suitable operator.

Let L be the Perron-Frobenius operator for (Ω, T, ψ) defined for, say a bounded function a, by

L(a)(ω) = T (ω ′ )=ω e ψ(ω ′ ) a(ω ′ ), a : Ω → C.
Without loss of generality we assume that the potential ψ is normalized in the sense that L1 = 1. In particular this implies that for any measurable bounded functions a and b, and any integer n,

Ω ab • T n dµ = Ω L n (a)bdµ. Given u ∈ C 0 (T d , C) we denote its Fourier coefficients by û(k) = T d e -2iπ<k,t> u(t)dm(t), k ∈ Z d
and recall that u is equal to its Fourier series

u(t) = k∈Z d û(k)e 2iπ<k,t> , t ∈ T d . Given u ∈ R we define the complexified Perron-Frobenius operator L u L u (a) = L(e iuϕ a)
where ϕ is the functions involved in the definition of the skew product, as above.

Note that L u is indeed the Perron Frobenius operator of the potential

ψ + iuϕ.
For a given ω ∈ Ω we denote the k-th Fourier coefficient of A(ω, •) by Â(ω, k).

Lemma 12. The correlations can be expressed by the Fourier expansion of the observables and the complexified Perron-Frobenius operator:

C n (A, B) = k∈Z d Ω L n 2π k,α ( Â(•, -k)) B(•, k)dµ.
Proof. We have, using the Fourier expansion of B that (denoting

S n ϕ(ω) = n-1 i=0 ϕ(T i ω)) C n (A, B) = k∈Z d Ω×T d A(ω, t) B(T n ω, k)e 2iπ k,t+αSnϕ(ω) dµ(ω)dm(t) = k∈Z d Ω B(T n ω, k)e 2iπ k,αSnϕ(ω) T d A(ω, t)e 2iπ k,t dm(t)dµ(ω) = k∈Z d Ω B(T n ω, k)e 2iπ k,αSnϕ(ω) Â(ω, -k)dµ(ω) = k∈Z d Ω L n 2π k,α ( Â(•, -k)) B(•, k)dµ.
This proves the lemma.

In the above sum, the Fourier coefficients corresponding to large k will be estimated by the regularity of the observables. To estimate the other coefficients we use the next proposition.

Strictly speaking, to finish the proof we have to work on the subshift of finite type generated by the Markov partition. There is nothing essential, but at some point we will need that the Perron-Frobenius operator preserves the space of Hölder functions of exponent η. For convenience we will keep the same notation (Ω, T, µ) for the symbolic dynamical system, and leave the details to the reader.

Proposition 13 ([21]

). There exists two constants c 0 > 0 and c 1 > 0 such that, as an operator acting on Hölder functions of exponent η, for any n and u ∈

[-π, π] the operator norm | • | η,η satisfies |L n u | η,η ≤ c 0 e -c1nu 2 .
Proof. This property follows classically from quasi-compactness of the Perron-Frobenius operators [START_REF] Hennion | Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness[END_REF]:

First, by a perturbative argument and since ϕ is not cohomologous to a constant, there exists β > 0 and some constants c 0 and c

1 such that if u ∈ (-β, β) then |L n u | η,η ≤ c 0 e -c1nu 2 .
Next whenever u = 0 mod 2π, the operator L u has no eigenvalues of modulus (larger than or) equal to one by the assumption (NA), hence the spectral radius is smaller than one, which gives

lim n→∞ | L n u | 1/n η,η < 1.
An uniform exponential contraction follows then by compacity of [-π, π] \ (β, β). Changing if necessary the values of the constants, the advertised upper bound also holds in the case u ∈ (-β, β).

Proposition 14. For any γ > γ l (α) there exists c 2 > 0 such that for any non-zero

k ∈ Z d |L n 2π k,α | η,η ≤ c 0 exp(-c 2 |k| -2γ n). Proof. Let γ > γ l (α)
. By definition of the Diophantine type of α, we have k, α ≥ c|k| -γ for some constant c > 0 and any k ∈ Z d . The result follows by Proposition 13.

We are now able to prove the main Lemma.

Proof of Lemma 11. We split the estimate into two parts, depending whether k is large or not. Let N be an integer which will be chosen later.

When k is large the estimate is simple:

Since A ∈ C p and B ∈ C q we have Â(ω, k) ∞ ≤ A p |k| -p and B(ω, k) ∞ ≤ B q |k| -q . Hence Ω L n 2π k,α ( Â(•, -k)) B(•, k)dµ ≤ A p B q |k| -p-q .
Therefore, since the operator L n 2π k,α does not expand the sup-norm, for any integer N we get [START_REF] Galatolo | Dimension and hitting time in rapidly mixing systems[END_REF] |k|>N Ω

L n 2π k,α ( Â(•, -k)) B(•, k)dµ ≤ C 1 A p B q N d-p-q .
The case k = 0 is done independently: Indeed, we have

Ω Â(•, 0)dµ = Ω×T d
A dµ ⊗ m = 0 by hypothesis. Then the usual decorrelation estimate gives that for some C, λ > 0 we have ( 13)

Ω L n ( Â(•, 0)) B(•, 0)dµ ≤ C 2 A 1 B ∞ λ n
(recall that ||.|| 1 here is the C 1 norm). On the other hand, for small k = 0 we make use of Proposition 14:

Observe that Â(•, k) ≤ A p |k| 1-p (here ||.|| is the Holder norm).

Ω L n 2π k,α ( Â(•, -k)) B(•, k)dµ ≤ A p |k| 1-p c exp(-c|k| -2γ n) B q |k| -q . Therefore (14) 1≤|k|≤N Ω L n 2π k,α ( Â(•, -k)) B(•, k)dµ ≤ c A p B q 1≤|k|≤N |k| 1-p-q exp(-cn|k| -2γ ).
which is easily seen to be superpolynomially decaying in n, if we choose N = n 1/2γ-ǫ , since

1≤|k|≤N |k| 1-p-q exp(-cn|k| -2γ ) ≤ N d+1-p-q exp(-cnN -2γ ).
Putting together the three estimates ( 12), ( 13) and ( 14) gives

C n (A, B) ≤ const A p B q n -ℓ ,
for any ℓ < p+q-d 2γ .

4. Quantitative recurrence, shrinking targets and arithmetical properties

In this section we give estimations on the quantitative recurrence and hitting time exponent of skew products, as defined before. These results show that both the quantitative recurrence and the hitting time exponent depend on the arithmetical properties of α.

4.1. Deterministic and random multidimensional rotations. The quantitative recurrence and hitting time exponents of our system is influenced by the underlying rotation. Multidimensional rotations have not been completely investigated from this point of view. We shall start with some general considerations on recurrence, hitting times and discrepancy on multidimensional rotations and random walks generated by such rotations. These results will be used in the following.

Let us consider a rotation x → x + α on T d , α = (α 1 , ...α d ). It is easy to see that for this map, the recurrence rates R(x, x) and R(x, x) do not depend on x, hence there are constants that we denote by

R(x, x) = rec(α), R(x, x) = rec(α).
Reproducing the proof of [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF] for one dimensional rotations, it is easy to see that

Proposition 15. Let us consider a rotation x → x + α on T d , α = (α 1 , ...α d ). We have rec(α) = 1 γ s (α)
.

Concerning the hitting time exponents R(x, y) and R(x, y) a similar thing happens (using that R(x, y) = R(xy, 0) for any x, y, together with the invariance R(x + α, 0) = R(x, 0)): there exists also two constants hit(α) and hit(α) such that for any y and a.e. x R(x, y) = hit(α), R(x, y) = hit(α).

The following proposition is quite similar to the transference principle between homogeneous and inhomogeneous diophantine exponents (see [START_REF] Cassels | An introduction to Diophantine approximation[END_REF]).

Proposition 16. Let us consider a rotation

x → x + α on T d , α = (α 1 , ...α d ). We have hit(α) ≥ γ l (α).
Proof. Let γ < γ l (α). Then there is a sequence of vectors k p = (k

p ) = (A t ) -1 [π(C ′ p )],
and then Leb(π(C p )) = 2 p 2 . This allows us to apply Borel Cantelli lemma and deduce that the set of points belonging to C p for infinitely many values of p is a zero measure set.

We recall the classical notion of discrepancy for a rotation of angle α ∈ R d . Let P d be the set of rectangles

R = i [a i , b i ] ⊂ [0, 1] d . Let |R| = i (b i -a i ). D n (α) = sup R∈P d 1 n card 0 ≤ k ≤ n -1 : kα ∈ R + Z d -|R| .
It is governed by the linear Diophantine type of α Proposition 17 ([26]). The discrepancy satisfies, for any γ > γ l (α),

D n (α) = O n -1/γ .
In addition, the second coordinate of the skew product evolves as a random rotation, thus we will need a notion of discrepancy of the random walk on the torus αS n ϕ driven by the Gibbs measure µ, defined by

D µ n (α) = sup R∈P d µ ω | αS n ϕ(ω) ∈ R + Z d -|R| .
In this random setting the upper bound become Proposition 18 (After [START_REF] Su | Convergence of random walks on the circle generated by an irrational rotation[END_REF]). The discrepancy satisfies, for any γ > γ l (α),

D µ n (α) = O ( √ n) -1/γ .
The proof is postponed to the appendix.

4.2.

Hitting time. We start with an easy consequence of the proof of Proposition 16.

Theorem 19. Suppose that γ l (α) > d µ (π 1 (y)) + d. Then, there exists a sequence r n → 0 such that the hitting time statistics to balls B(y, r n ) has a trivial limiting distribution.

In particular these systems are polynomially mixing but the distribution of hitting time to balls does not converge to the exponential law.

Proof. We assume without loss of generality that π 2 (y) = 0. Let γ < γ l (α) and ǫ > 0 such that γ-2ǫ > d µ (π 1 (y))+d. In the proof of Proposition 16 it is shown that there exists a sequence r p → 0 such that the hitting time to the r p -neighborhood of Z d by the sequence t + nα is at least r -γ+ǫ p , for all t ∈ T d outside a set of measure 2/p 2 . Since S n ϕ ≤ n, this implies that τ rp (x, y) ≥ r -γ+ǫ p . Let s > 0. For any r sufficiently small s ν(B(y,r)) < r -γ+ǫ p . Therefore, for any s > 0,

ν x : τ r (x, y) ≤ s ν(B(y, r)) ≤ 2 p 2
provided p is sufficiently large.

The preceding result allows us to estimate the hitting time exponents in our skew product.

Theorem 20. In the skew product (M, S, ν) described above, at each target point y ∈ M it hold (15) R(x, y) ≥ max(d µ (π 1 (y)) + d, hit(α)) and R(x, y) ≥ max(d µ (π 1 (y)) + d, hit(α))

for ν-a.e. x ∈ M (recall that π i are the two canonical projections, as defined above).

Proof. The inequality R(x, y) ≥ d µ (π 1 (y)) + d follows for ν-a.e. x from the general inequality (2), since the invariant measure is the product measure of µ and the Haar measure on the torus, then d µ×m (y) = d µ (π 1 (y)) + d. In particular τ r (x, y) → ∞ as r → 0.

Furthermore, for ν-a.e. x = (ω, t), the upper hitting time exponent R(t, 0) for the rotation by angle α is equal to hit(α). Let h > hit(α). There exists a sequence r p → 0 such that t + nα > r p for n = 1, . . . , ⌊r -h p ⌋. Since 0 < S n ϕ(ω) ≤ n for any n sufficiently large and τ rp (x, y) → ∞, we get that τ rp (x, y) ≥ r -h p for any p sufficiently large. hence R(x, y) ≥ h. The first inequality follows since h is arbitrary.

The corresponding statement for R can be done with a simplification of the same proof (since now the estimates will be valid for any r and not only for a sequence r p ).

The question if the above general inequality in Theorem 20 is sharp arises naturally. We show that if µ is a Bernoulli measure and d = 1 this is the case. Proposition 21. We assume that all the branches of the Markov map T are full, i.e. T (J i ) = Ω for all i, that µ is a Bernoulli measure i.e. µ([a 1 . . .

a n ]) = µ([a i ]) • • • µ([a n ])
, and I depends only on the first symbol, i.e. I is an union of 1-cylinders (recall that ϕ = 1 I ).

Then for any y, for ν-a.e. whenever r is sufficiently small, hence

card{k ≤ N r : t + kα < r} ≥ N r (2r) d -D Nr (α) ≥ r -ξ-ǫ (2 d r d -r ξ+ǫ γ ) ≥ r d-ξ-ǫ . (16) Denote by K r = {k 1 r < k 2 r < • • • < k p r } the set of integers k < N r such that t + kα < r. Notice that k 1
r is the first hitting time of [-r, r] by the orbit of t under the rotation of angle α, therefore by Theorem 6, for a.e. t we have lim inf r→0 log k 1 r / log(1/r) ≥ 1. In particular for any r sufficiently small, (17) k 1 r ≥log r. We now fix some typical t and consider r so small that ( 16) and ( 17) hold. Let b ∈ (0, µ(A)).

Let G = {ω ∈ Ω : ∀n = k 1 r , ..., N r , S n ϕ(ω) > bn}. Let q be the minimal integer such that any q-cylinder has a diameter less than r/2. Note that q = O(-log r). Let B be the union of all q-cylinders that intersect the ball B(ω

′ , r/2). When ω ∈ G, if S n ϕ(ω) ∈ K and T n (ω) ∈ B then τ r (ω) := τ r ((ω, t), (ω ′ , 0)) ≤ n ≤ N r /b.
Let t j (ω) be the smallest integer n such that S n ϕ(ω) = k j . This means S n ϕ(ω) = k j > S n-1 ϕ(ω). Since the increments of S n ϕ are 0 or 1 the sets Γ (nj

) := {t 1 = n 1 , . . . , t p = n p }, n 1 < • • • < n p , form a partition of Ω. Therefore µ(τ r > 1 b N r ) ≤ µ(Ω \ G) + n1<•••<np µ(Γ (nj ) ∩ p j=1 T -nj B c ) (⋆)
.

A large deviations estimates show that there exists c ∈ (0, 1) such that µ(S n ϕ > bn) ≤ c n for any n sufficiently large. With [START_REF] Galatolo | Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds[END_REF] this gives the estimate for the first term

µ(Ω \ G) ≤ Nr n=k 1 r µ(S n ≤ bn) ≤ c k 1 r /(1 -c) ≤ r -log c /(1 -c).
We estimate the second term (⋆). Set n ′ 1 = n 1 , n ′ 2 = n 2n 1 , . . . , n ′ p = n pn p-1 and define (k ′ j ) in the same way. The set Γ (nj ) ∩ j T -nj B c is equal to the intersection of the sets

{S n ′ 1 ϕ = k ′ 1 > S n ′ 1 -1 ϕ}, T -n1 (B c ∩ {S n ′ 2 ϕ = k ′ 2 > S n ′ 2 -1 ϕ}), . . . T -np-1 (B c ∩ {S n ′ p ϕ = k ′ p > S n ′ p -1 ϕ}), T -np B c .
Since ϕ depends only on the first symbol, these sets depend on different coordinates. Thus by the Bernoulli property the measure of the intersection is the product of the measures:

(⋆) = µ({S n ′ 1 ϕ = k ′ 1 > S n ′ 1 -1 ϕ}) p-1 j=2 µ(B c ∩ {S n ′ p ϕ = k ′ p > S n ′ p -1 ϕ, })µ(B c ).
Since for any j = 1, . . . , p we have

∞ n ′ j =1 µ(B c ∩ {S n ′ j ϕ = k ′ j > S n ′ j -1 ϕ}) = µ(B c ) = 1 -µ(B). A summation over n 1 < • • • < n p then gives (⋆) = (1 -µ(B)) p .
By [START_REF] Galatolo | The dynamical Borel-Cantelli lemma and the waiting time problems[END_REF] we have p ≥ r d-ξ-ǫ . Whenever r is sufficiently small we have

µ(B) ≥ µ(B(ω ′dµ(ω ′ )+ǫ/2 . Therefore (1 -µ(B)) p ≤ 1 -r dµ(ω ′ )+ ǫ 2 r -(dµ (ω ′ )+ ǫ 2 ) r -ǫ 2 ≤ exp(-r -ǫ 2 ).
Setting r n = e -n we get that n µ(τ rn > 1 b N rn ) < ∞. By Borel-Cantelli we conclude that for µ-a.e. ω, if n sufficiently large then

τ rn ((ω, t), (ω ′ , 0)) ≤ 1 b r -ξ-ǫ n .
This implies that R((ω, t), (ω ′ , 0)) ≤ ξ + ǫ. The conclusion follows since ε is arbitrary.

A bound for R(x, y) similar to the one given in Proposition 20 also hold. Since this is related with the dynamical Borel-Cantelli lemma and similar topics we postopone it to Section 6.1. Proof. Since the recurrence does not depend on its initial value, we fix arbitrarily some t ∈ T d . Let ǫ > 0, set γ = γ s (α)ǫ and take δ = d µ + 2ǫ.

We consider a set K with µ(K) > 1ǫ arbitrarily close to 1, such that for any r there exists a cover of K by balls of diameter r with cardinality less than cr -dµ-ǫ for some constant c > 0 (see e.g. [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF] for a detailed construction).

Fix some q ∈ N * such that qα < q -γ . At this time q the rotation makes by definition a close return to the origin. The idea is that many of its multiples kq will inherit this property. More precisely, let r = r(q) = q -γ 1+δ . For any integer k < r -δ we still have

kqα ≤ k qα ≤ r -δ q -γ = r -δ+δ+1 = r.
Since S n (ω, t) = (T n (ω), t + αS n ϕ(ω)), if an integer n satisfies (i) S n ϕ(ω) ≤ qr -δ , (ii) S n ϕ(ω) is a multiple of q and (iii) n is a r-return for T then n is an r-return for S.

Let F be the finite extension of T defined on Ω × Z q by F (ω, z) = (T (ω), z + ϕ(ω)).

(we put the discrete metric on Z q ). This map preserves the probability measure µ × 1 q H 0 where H 0 is the counting measure on Z q . Note that if n ≤ qr -δ is an r-return for F then S n ϕ(ω) = 0 mod q, therefore n is also an r-return for S. Therefore if τ F r (ω, z) denotes the r-return for F and τ S r (ω, t) is the r-return for S then τ S r (ω, t) ≤ τ F r (ω, z) whenever the latter is less than qr -δ .

We now take a cover of K by balls {B i } of diameter r which has the properties mentioned before.

We have, repeating the computation in [START_REF] Barreira | Hausdorff dimension of measures via Poincaré recurrence[END_REF] (in particular using Kǎc's lemma for F )

µ(ω ∈ K : τ S r (ω, t) > qr -δ ) ≤ µ × 1 q H 0 ((ω, z) : τ F r (ω, z) > qr -δ ) ≤ i j∈Zq µ × 1 q H 0 ((ω, z) ∈ B i × {j} : τ F Bi×{j} (ω, z) > qr -δ ) ≤ i j∈Zq r δ q Bi×{j} τ F Bi×{j} dµ × 1 q H 0 ≤ cr ǫ .
We now take a sequence q n ∈ N * going to infinity such that q n α ≤ q -γ n and n r(q n ) ǫ < ∞. By Borel-Cantelli we get that for µ-a.e. ω ∈ K we have [START_REF] Galatolo | Lorenz like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence[END_REF] τ r(qn) (ω, t) ≤ r(q n ) -δ-1+δ γ for any n sufficiently large. Thus, writing x = (ω, t) we get

R(x, x) ≤ δ + 1 + δ γ ,
for ν-a.e. x ∈ K × T d . The conclusion follows letting ǫ → 0.

We remark in particular that when γ s is large, the lower quantitative recurrence exponent becomes smaller than the dimension of the measure ν, which is d µ + d.

The fact that this rapid recurrence occurs at the same scale for µ-a.e. points enables us to deduce the following result.

Theorem 24. Suppose that γ s (α) > 1+dµ d . For ν-a.e. x, there exists a subsequence r n → 0 such that the return time statistics in balls B(x, r n ) has a trivial limiting distribution.

In particular, these systems are polynomially mixing but the return time distribution to balls does not converge to the exponential law.

Proof. We use the same notation of the proof of Theorem 23. Consider ǫ > 0 so small that δ + 1+δ γ + 3ǫ < d µ + d. Take n 0 so large that the set H of points ω ∈ Ω such that (18) holds for any n > n 0 has a measure µ(H) > 1ǫ.

Let ω ∈ Ω be such that for any r sufficiently small r dµ+ǫ ≤ µ(B(ω, r)) < r dµ-ǫ and µ(B(ω, r)∩H)/µ(B(ω, r)) → 1 as r → 0. The fact that the pointwise dimension is µ-a.e. equal to d µ and Lebesgue density theorem shows that this concerns µ-a.e. points of H.

Given r > 0 we set L r = ⌈log 2 r⌉. For any r sufficiently small we have

µ(B(ω, 2L r r)) µ(B(ω, L r r)) ≤ r -3ǫ .
Hence there exists an integer

k r ∈ [L r , 2L r ] such that (19) µ(B(x, k r r)) -µ(B(x, (k r -1)r)) ≤ 1 -log r µ(B(x, k r r)), otherwise µ(B(ω, 2L r r))(1 - 1 -log r ) Lr ≥ µ(B(ω, L r r))
which would contradict the previous inequality provided r is sufficiently small. Let r n = k r(qn) r(q n ). If ω ′ ∈ H ∩ B(x, r n ) \ B(x, r nr(q n )) we have by ( 18) that τ rn (ω ′ , t) ≤ r(q n ) -dµ+d-3ǫ ≤ µ(B(x, r n ))r ǫ n for any n sufficiently large. For any s > 0, once r n ≤ s we get, setting x = (ω, t), ν B(x,rn) (y :

τ rn (y) ≥ s µ(B(x, r n )) ) ≤ µ B(ω,rn) (H c ∩ B(ω, r n -r(q n ))).
Using [START_REF] Galatolo | Long hitting time, slow decay of correlations and arithmetical properties[END_REF] and the fact that ω is a Lebesgue density point of H we conclude that the upper bound goes to zero as n → ∞.

Theorem 25. In a skew product as above the lower recurrence rate is bounded from below by

R(x, x) ≥ min d µ 1 -1 2γ l (α) , d µ + d ν -a.e.x. Proof. Let γ > γ l (α). Let ǫ > 0. Set ∆ = min ((d µ -3ǫ)/(1 -1/2γ), d µ + d).
We take a set K of measure µ(K) > 1ǫ and r 0 > 0 such that for any ω ∈ K and r ∈ (0, r 0 ) we have µ(B(ω, 2r)) ≤ r dµ-ǫ .

Given r > 0 we fix k as the smallest integer such that any cylinder Z ∈ J k has a diameter less than r. Let Z ∈ J k . Let B(Z, r) be the union of balls ω ′ ∈Z B(ω ′ , r).

Hence if Z ∩ K = ∅ then µ(B(Z, r)) ≤ r dµ-ǫ .
Given an integer n, let

W (r, k, n) = {ω ∈ Ω : T n (ω) ∈ B(ω, r
) and αS n ϕ(ω) < r}.

Recall that ϕ = 1 I is constant on m-cylinders for some integer m ≥ 1. Assume that n ≥ m + k and let us decompose its Birkhoff sum as

S n ϕ = S k-m ϕ + S m ϕ • T k-m + S n-k-m ϕ • T k + S m ϕ • T n-m .
Denote by E the range of S m ϕ. We have card E ≤ p 2m < ∞. The sum S k-m ϕ is constant on Z and we denote its common value by q Z . Then for any ω in Z we have

S n ϕ(ω) = q Z + u + S n-k-m ϕ(T k ω) + v for some u, v ∈ E. Notice that S n-k-m ϕ(ω) is σ(T -k J n-k
) measurable, thus the ψ-mixing property 4 of the measure µ implies that

µ(K ∩ W (r, k, n)) = Z∈J k µ(Z ∩ K ∩ W (r, k, n)) ≤ Ψ(0) 2 Z∈J k ,Z∩K =∅ u,v∈E µ(Z ∩ T -k { α(q Z + u + v + S n-k-m ϕ) < r} ∩ T -n B(Z, r)) ≤ Ψ(0) 2 Z∈J k ,Z∩K =∅ u,v∈E µ(Z)µ( α(q Z + u + v + S n-k-m ϕ) < r)µ(B(Z, r)) ≤ Ψ(0) 2 p 4m r dµ-ǫ D µ n-k-m (α) + (2r) d . Take δ ∈ (0, min(d µ , ∆)). When r -δ ≤ n ≤ r -∆ we have, for some constant c, µ(K ∩ W (r, k, n)) ≤ cr dµ-ǫ n -1 2γ + r d . Therefore r -δ ≤n≤r -∆ µ(K ∩ W (r, k, n)) = O cr dµ-ǫ (r -∆ ) 1-1/2γ + r d = O(r ǫ ),
by our choice of ∆. This shows that the probability that for some ω ∈ K there is a r-return of x = (ω, t) between the times r -δ and r -∆ is O(r ǫ ). By a Borel Cantelli argument, for µ-a.e. ω ∈ K, there are no returns in this time interval. On the other hand, by Theorem 4 the recurrence rate for the base map T only is equal to d µ > δ, therefore there are no returns in the time interval 1, . . . , r -δ also. The result follows since ǫ is arbitrary. 4.4. Observed systems. In the previous sections, we studied the return time and hitting time of the skew-product and we showed that their exponents depend on the arithmetical properties of α. A natural question is to study only the return of the second coordinate instead of the whole system. We are going to study the Poincaré recurrence for a specific observation [START_REF] Rousseau | Poincaré recurrence for observations[END_REF], the canonical projection π 2 : Ω × T d → T d and prove that the recurrences rates for the observation do not depend on T . Definition 26. Let x ∈ Ω × T d , r > 0 and p ∈ N. We define the p-noninstantaneous return time for the observation r) . 4 The ψ-mixing property is the following: there exists some sequence Ψ(n) ց 0 such that for any k, any A ∈ σ(J k ) and any measurable set B we have

τ obs r,p (x, x) = inf k > p, π 2 S k (x) ∈ B (π 2 (x),
|µ(A ∩ T -k-n B) -µ(A)µ(B)| ≤ Ψ(n)µ(A)µ(B).
As previously, we define the lower and upper non-instantaneous recurrence rates for the observation In the following proposition, we will prove that the recurrence rates for this observation only depends on the underlying rotation.

R obs (x, x) = lim
Proposition 27. For ν-almost every x ∈ M R obs (x, x) = rec(α)(= 1 γ s (α) ) and R obs (x, x) = rec(α).
Proof. First of all, we can remark that for every x = (ω, t) ∈ Ω × T d , every r > 0 and every p ∈ N * τ obs r,p (x, x) ≥ τ r (t, t) where τ r (t, t) is the return time of t with respect to the rotation with angle α. This gives us that

R obs (x, x) ≥ rec(α) and R obs (x, x) ≥ rec(α).
On the other hand, we have

τ obs r,p (x, x) = inf k > p, S k (ω, t) ∈ Ω × B (t, r) = inf {k > p, t + αS k ϕ(ω) ∈ B (t, r)} = inf {k > p, αS k ϕ(ω) ≤ r} . (20)
Birkhoff ergodic theorem gives for µ-almost every ω ∈ Ω lim

k→+∞ 1 k S k ϕ(ω) = µ(I) =: a.
Then, for µ-almost every ω ∈ Ω there exists N ∈ N such that ∀k > N , S k ϕ(ω) > a 2 k. By definition of rec(α), there exists a sequence r n → 0 and a sequence of integers q n ≥ p such that τ rn (t, t) ≤ q n and lim qn log rn = rec(α). For all n > N a/2 we have S ⌈ 2 a qn⌉ ϕ(ω) > q n and thus

τ obs rn,p (x, x) ≤ 2 a q n .
Letting n goes to infinity and then p to infinity proves that

R obs (x, x) ≤ rec(α).
We proceed similarly with R obs .

The hitting time of the observed system may be defined similarly, Definition 28. Let x, y ∈ Ω × T d , r > 0. We define the hitting time for the observation

τ obs r (x, y) = inf k > 0, π 2 S k (x) ∈ B (π 2 (y), r) .
As previously, we define the lower and upper hitting time exponents for the observation Also for the hitting time, with an obvious modification of the proof for return times, one can show that the exponents do not depend on T : Proposition 29. For every y ∈ M and for ν-almost every x ∈ M R obs (x, y) = hit(α) and R obs (x, y) = hit(α).

R obs (x, y) = lim inf

Decay of correlations, lower bounds

In [START_REF] Galatolo | Long hitting time, slow decay of correlations and arithmetical properties[END_REF]Corollary 3] the following relation between decay of correlations and hitting time exponent is proved : Proposition 30. If a map on a finite dimensional Riemannian manifold M , has an absolutely continuous invariant measure with strictly positive density at some point y 0 5 and s.t. R(x, y 0 ) = R 0 , x-a.e., then the speed of decay of correlations (over Lipschitz observables) of the system is at most a power law with lower exponent (see Definition 2)

p = lim inf n→∞ -log Φ(n) log n ≤ 2 dim M + 2 R 0 -dim M .
We remark that the assumptions on the absolute continuity of the invariant measure can be largely relaxed (see [START_REF] Galatolo | Long hitting time, slow decay of correlations and arithmetical properties[END_REF]). By this proposition and Proposition 20 we have easily:

Proposition 31. The decay of correlations with respect to Lipschitz observables of a skew product on M = Ω × T d , such that µ is absolutely continuous with positive density is a power law with exponent

(21) p ≤ 2 dim M + 2 max(dim M, γ l (α)) -dim M .
These bounds are probably not sharp but the advantage is that they clearly show the dependence on the Diophantine type of α. A comparison with the upper bound given in Proposition 9 gives that for the skew product of the doubling map and a circle rotation endowed with the Lebesgue measure, the exponent p satisfies

1 2γ(α) ≤ p ≤ 6 max(2, γ(α)) -2
. 5 The density is greater than some positive number in a neighborhood of y 0 .

6. Systems with R(x, y) > d and consequences 6.1. A skew product with lower hitting time exponent larger than the dimension. With a suitable torus translation it is possible to obtain a system [START_REF] Galatolo | Long hitting time, slow decay of correlations and arithmetical properties[END_REF] where the lower hitting time exponent is bigger than the local dimension (R(x, y) > d ν for typical x, y) this leads to some other consequence as the lack of a dynamical Borel-Cantelli property, or the triviality of the limit return time distribution. Using a rotation of this kind in a skew product we obtain Theorem 32. There exists a system which has polynomial decay of correlation over C r observables, superpolynomial decay w.r.t. C ∞ ones and • its lower exponent of decay of correlations with respect to Lipschitz observables satisfies 1 16 ≤ lim inf n→∞

-log Φ(n) log n ≤ 8 13 ;
• the lower hitting time indicator R(x, y) is bigger than the local dimension;

• it has no Monotone Shrinking Target property;

• it has trivial limit return time statistics. Let us describe this example. Let us consider a rotation T α , α=(α 1 , α 2 ) on the torus T 2 ∼ = R 2 /Z 2 by the angle α. Suppose γ 1 , γ 2 are respectively the types of α 1 and α 2 and denote by q n and q ′ n the partial convergent denominators of α 1 and α 2 . To obtain a rotation with large lower hitting time indicator let us take ξ > 2 and let Y ξ ⊂ R 2 be the class of couples of irrationals (α 1 , α 2 ) given by the following conditions on their convergents to be satisfied eventually:

q ′ n ≥ q ξ n ; q n+1 ≥ q ′ n ξ .
We note that each Y ξ is uncountable and dense in [0, 1] × [0, 1] and each irrational of the couple is of type at least ξ 2 . Indeed if α 1 is such a number: q n+1 ≥ q ξ 2 n (then q n ≥ q ξ 2n 1 ) and then γ(α 1 ) ≥ lim sup n→∞ log q ξ 2 n log qn = ξ 2 . We also remark that

γ l (α) ≥ max(γ(α i )) ≥ ξ 2 .
With some more work (see Propositions 43 and 44 in Subsection 6.4) it is possible to obtain the following: Proposition 33. There is an angle α ∈ Y 4 which is of finite Diophantine type for the linear approximation. More precisely γ l (α) = 16. This implies that the following example exists.

Example 34. Let us consider the skew product (M, S) where M = [0, 1]× T 2 with α ∈ Y 4 , such that γ l (α) = 16 and T preserves an absolutely continuous invariant measure µ with strictly positive density.

We will see below that the example satisfies all items of Theorem 32. The reason we take a rotation with angle in Y ξ is that the lower hitting time indicator is bounded from below by ξ. Theorem 35 ([19]). If T (α1,α2) is a rotation of the two torus by a vector (α 1 , α 2 ) ∈ Y ξ and y ∈ T 2 , then for almost every x ∈ T 2 R(x, y) ≥ ξ.

In particular we remark that if ξ > 2 then in this example the lower hitting time indicator is bigger than the local dimension. Using Theorem 20 we get: Proposition 36. In a skew product with a rotation by an angle included in Y ξ , at each target point y it hold [START_REF] Hill | The ergodic theory of shrinking targets[END_REF] R(x, y) ≥ max(d µ (π 1 (y)) + 2, ξ)

for ν-a.e. x.

Now we can obtain the first item in Theorem 32.

Proposition 37. In a skew product on M = Ω × T 2 with a rotation by an angle α ∈ Y 4 with γ l (α) = 16 as above, the decay of correlations with respect to Lipschitz observables satisfies

1 16 ≤ lim inf n→∞ -log Φ(n) log n ≤ 8 13 .
Proof. The second inequality is obtained putting d = 2, R ≥ 4 2 ,(see Theorem 20, recalling that if α = (α 1 , α 2 ) then γ l (α) ≥ max i (γ(α i ))) in Proposition 30. The first inequality is obtained by Proposition 9.

Obviously the requirement on the dimension of µ is independent from the others, and such system exists. Hence we have the second item of Theorem 32.

Proposition 38. In the above example R(x, y) > d ν (y) almost everywhere.

Proof. Recall that we consider β ∈ Y 4 R(x, y) ≥ 4 for each y and almost each x. Here d ν (y) = 3 almost everywhere.

In the next subsections we clarify and discuss the other items of the proposition and finally prove the existence of the angles mentioned in Proposition 33.

6.2.

No dynamical Borel-Cantelli property and monotone shrinking targets. Let us consider a family of balls B i = B ri (y) with i ∈ N centered in y and such that r i → 0. In several systems, mostly having some sort of fast decay of correlations or generical arithmetic properties, the following generalization of the second Borel-Cantelli lemma can be proved: [START_REF] Kim | The recurrence time for interval exchange maps[END_REF] ν

(B i ) = ∞ ⇒ ν(lim sup i T -i (B i )) = 1
or equivalently T i (x) ∈ B i for infinitely many i, when x is typical with respect to ν. We recall that there are mixing systems where the above statement does not hold (an example was given in [START_REF] Fayad | Mixing in the absence of the shrinking target property[END_REF], nevertheless in this example the speed of decay of correlations is less than polynomial, see [START_REF] Galatolo | Long hitting time, slow decay of correlations and arithmetical properties[END_REF]). From what it is said above it easily follows that in our examples (which are polynomially mixing with respect to C r observables and superpolynomial mixing with respect to C ∞ ones) the above property is also violated hence proving the second item of Theorem 32.

Definition 39. We say that the system has the monotone shrinking target property if [START_REF] Kim | The recurrence time for interval exchange maps[END_REF] holds for every decreasing sequence of balls in X with the same center.

In [START_REF] Galatolo | The dynamical Borel-Cantelli lemma and the waiting time problems[END_REF] the following fact is proved:

Theorem 40. Assume that there is no atom in X, if the system has the monotone shrinking target property, then for each y we have lim inf log µ(Br (y)) > 1 and then by Theorem 40 the system has not the monotone shrinking target property.

6.3. Trivial return and hitting time distribution. The following statement shows that if the logarithm law (hitting time exponent=dimension ) does not hold then the return time statistic has a trivial limit (compare with Theorem 24 where the result holds for a subsequence).

Theorem 42. ( [START_REF] Galatolo | Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems[END_REF]) If (X, T, ν) is a finite measure preserving system over a metric space X and

(25) R(x, y) > d ν (y)
a.e., then the system has trivial limit return time statistic in sequence B r (y). That is, the limit in (9) exists for each t > 0 and g(t) = 0.

The mixing system without logarithm law given in [START_REF] Galatolo | Long hitting time, slow decay of correlations and arithmetical properties[END_REF] hence has trivial return limit statistic in each centered sequence of balls, this gives an example of a smooth mixing systems with trivial limit return time statistics. As said before, this example, as the Fayad example has slower than polynomial decay of correlations. Since in Example 34 we found a system with polynomial decay of correlations with respect to Lipschitz observables but lower hitting time indicator is bigger than the local dimension hence the third item of Theorem 32 is established. 6.4. Construction of a diophantine angle with intertwined partial quotients. We briefly recall the basic definitions and properties of continued fractions ( for a general reference see e.g. [START_REF] Cassels | An introduction to Diophantine approximation[END_REF]) that will be needed in the sequel. Let α be an irrational number, denote by [a 0 ; a 1 , a 2 , . . .] its continued fraction expansion:

α = a 0 + 1 a 1 + 1 a 2 + . . . =: [a 0 ; a 1 , a 2 , . . .].
The integers a 0 , a 1 , a 2 . . . are called partial quotients of α and are all positive except for a 0 . As usual, we define inductively the sequences p n and q n by:

p -1 = 1, p 0 = 0, p k+1 = a k+1 p k + p k-1 for k ≥ 0; q -1 = 0, q 0 = 1, q k+1 = a k+1 q k + q k-1 for k ≥ 0.
The fractions p n /q n are called the convergents of α, as they do in fact converge to it. Moreover they can be seen as best approximations of α in the following sense.

As usual denote by x := min n∈Z |x -n| the distance of a real number form the integers. Then q = q n for some n if and only if qα < q ′ α for every positive q ′ < q and p n is the integer such that q n α = |q n αp n |.

We will see that there are 2 dimensional angles which are of finite type for the linear approximation, but have intertwined partial quotients, in a way that they belong to some nontrivial class Y ξ , with ξ = 4, which is big enough to have an example which can be used in Section 6.1. The proof is quite similar to the ones given in [9, Section 7]. Since we need a slightly different result and we need to change some step we write it explicitly here.

Proposition 43. There are α and α ′ such that their partial quotients q n , q ′ n satisfy for any n:

(1)

q ′4 n-1 ≤ q n ≤ 4q ′4 n-1 (2) q 4 n ≤ q ′ n ≤ 4q 4 n (3) q n ∧ q ′ n-1 = 1 (4) q ′
n ∧ q n = 1 where a ∧ b = 1 means that a and b are relatively prime.

Proof. We will construct α, α ′ and their partial quotients q n , q ′ n by constructing the appropriate coefficients a n , a ′ n . Now let us proceed by induction, assuming that a 0 , ..., a n-1 , a ′ 0 , ..., a ′ n-1 have been constructed satisfying items 1-4, now let us construct a n .

Since q ′ n-1 ∧ q n-1 = 1 then there is an integer τ n < q ′ n-1 such that τ n q n-1 ≡ -q n-2 (mod q ′ n-1 ), so that q ′ n-1 divides τ n q n-1 + q n-2 . Now, choose ρ n is such that ρ n ∧ q ′ n-1 = 1 and

q ′4 n-1 ≤ ρ n q n-1 ≤ 2q ′4 n-1
and define a n = τ n + ρ n . With this choice of a n q n = a n q n-1 + q n-2 = ρ n q n-1 + τ n q n-1 + q n-2 .

Since τ n ≤ q ′ n-1 we have q ′4 n-1 ≤ q n ≤ 4q ′4 n-1 and item 1) is satisfied at step n. On the other hand, from the inductive hypotesis q n-1 ∧ q ′ n-1 = 1 and from our choice of ρ n it follows ρ n q n-1 ∧ q ′ n-1 = 1, on the other hand τ n q n-1 + q n-2 is a multiple of q ′ n-1 . Consequently q n ∧ q ′ n-1 = 1, and item 3) of the inductive step is proved at step n. Now we construct a ′ n in the same way: andρ ′ n is such that ρ ′ n ∧ q n = 1 and q 4 n ≤ ρ ′ n q ′ n-1 ≤ 2q 4 n (recall that q n was already constructed just above) then

a ′ n = τ ′ n + ρ ′ n with τ ′ n q ′ n-1 ≡ -q ′ n-2 (mod q n ),
q ′ n = a ′ n q ′ n-1 + q ′ n-2 = ρ ′ n q ′ n-1 + τ ′ n q ′ n-1 + q ′ n-2
, and hence q 4 n ≤ q ′ n ≤ 4q 4 n , proving Item 2). Finally, we use the already proved relation: q n ∧ q ′ n-1 = 1 to obtain q ′ n ∧ q n = 1 (Item 4) at step n as before.

Proposition 44. The vector (α, α ′ ) described above is such that γ l ((α, α ′ )) = 16.

Proof. As remarked before, it is easy to see that γ(α) = γ(α ′ ) = ξ 2 = 16, hence γ l ((α, α ′ )) ≥ 16. For the opposite inequality, we have to show that for any (k, l) ∈ Z 2 , with max(|k|, |l|) sufficiently large 16 .

||kα + lα ′ || ≥ 1 (max(|k|, |l|))
If k or l = 0 then the problem is reduced to a one dimensional one, and the exponent is the diophantine exponent of α and α ′ which was already remarked to be 16. Now, let us suppose l, k = 0 and q ′ n-1 ≤ max(|k|, |l|) ≤ q ′ n for some n. Let us suppose q ′ n-1 ≤ max(|k|, |l|) ≤ q n (the case q n ≤ max(|k|, |l|) ≤ q ′ n will be treated below).

We recall some general properties of continued fractions:

|α - p n q n | ≤ 1 q n q n+1 , |α ′ - p ′ n-1 q ′ n-1 | ≤ 1 q ′ n-1 q ′ n . As |k| ≤ q n , |l| ≤ q n |kα + lα ′ -k p n q n -l p ′ n-1 q ′ n-1 | ≤ 1 q n+1 + q n q ′ n q ′ n-1 ≤ 1 q 16 n + q n q 4 n q ′ n-1 = o( 1 q n q ′ n-1
).

for n (and consequently max(|k|, |l|) ) sufficiently large. On the other hand, since q n ∧ q ′ n-1 = 1 and

q n ∧ p n = 1, k ≤ q n implies ||k p n q n -l p ′ n-1 q ′ n-1 || ≥ 1 q ′ n-1 q n .
With the above estimation we have that for large n ||kαlα ′ || ≥ 1 2q ′ n-1 q n thus using again the inequalities between the various q n and q ′ n we have

||kα -lα ′ || ≥ 1 8q ′5 n-1 . Since q ′ n-1 ≤ max(|k|, |l|) we obtain (26) ||kα -lα ′ || ≥ 1 8(max(|k|, |l|)) 5 .

Now let us consider the case q

n ≤ max(|k|, |l|) ≤ q ′ n : again |α - p n q n | ≤ 1 q n q n+1 , |α ′ - p ′ n q ′ n | ≤ 1 q ′ n q ′ n+1 . As |k| ≤ q ′ n , |l| ≤ q ′ n |kα + lα ′ -k p n q n -l p ′ n q ′ n | ≤ q ′ n q n q n+1 + 1 q ′ n+1 ≤ 1 q ′ 8 n + q ′ n q ′4 n q n = o( 1 q n q ′ n ).
On the other hand, since q n ∧ q ′ n-1 = 1 and

q n ∧ p n = 1, k ≤ q ′ n implies ||k p n q n -l p ′ n q ′ n || ≥ 1 q ′ n q n .
With the above estimation we have that for large n ||kαlα ′ || ≥ 1 2q ′ n q n thus using again the inequalities between the various q n and q ′ n we have

||kα -lα ′ || ≥ 1 8q 5 n .
Since q n ≤ max(|k|, |l|) we obtain [START_REF] Marwan | Recurrence Plots for the Analysis of Complex Systems[END_REF] ||kαlα ′ || ≥ 1 8(max(|k|, |l|)) 5 .

Finally, [START_REF] Kuipers | Uniform distribution of sequences[END_REF] together with [START_REF] Marwan | Recurrence Plots for the Analysis of Complex Systems[END_REF] This implies that

f ǫ C p ≤ f C k sup |α|≤p-k (D α ρ ǫ ) L 1 ≤ f C k sup |α|≤k ǫ -(p-k) |D α (ρ)|dx ≤ Cǫ -(p-k) f C k
where C is a constant depending on the function ρ. Moreover, for any ǫ and δ, it holds that ||ff ǫ || ∞ ≤ ǫ k ||f || C p , ||gg δ || ∞ ≤ δ ℓ ||g|| C ℓ . Now let us estimate the decay of correlations of f and g by their regularized functions:

f • T n gdν ≤ (f • T n + f ǫ • T n -f ǫ • T n ) (g + g δ -g δ )dν ≤ |(f • T n -f ǫ • T n ) (g -g δ )| dν + |(f • T n -f ǫ • T n ) (g δ )| dν + |(f ǫ • T n ) (g -g δ )|dν + |(f ǫ • T n ) (g δ )|dν ≤ ǫ k δ ℓ ||f || C k ||g|| C ℓ + ǫ k ||f || C k ||g|| 1 + δ ℓ ||f || 1 ||g|| C ℓ + f ǫ C p g δ C q Φ k,ℓ (n) ≤ ǫ k δ ℓ ||f || C k ||g|| C ℓ + ǫ k ||f || C k ||g|| 1 + δ ℓ ||f || 1 ||g|| C ℓ + Cǫ k-p δ q-ℓ f C k g C ℓ Φ p,q (n)
≤ ||f || C p ||g|| C q (ǫ k δ ℓ + (ǫ k + δ ℓ ) (1 + diam(X)) + Cǫ k-p δ ℓ-q Φ p,q (n)).

since ||f || 1 ≤ f C p (1 + diam(X)). This will be essentially minimized when ǫ k = δ ℓ = ǫ k-p δ ℓ-q Φ p,q (n). This gives

f • T n gdν ≤ ||f || C k ||g|| C ℓ Φ(n) kℓ pℓ+qk-kℓ .
In the case k = 1 or ℓ = 1 the same estimate is valid for Lipschitz observables 6with the C 1 norm replaced by the Lipschitz one. 7.2. Discrepancy estimates. We prove here Proposition 18.

Proof. We follow the line of [START_REF] Su | Convergence of random walks on the circle generated by an irrational rotation[END_REF] (Theorem 5.5) who established this result in the case of the simple symmetric random ±α on the circle (that is d = 1 there and the increments where ±α with independent equal probability).

The discrepancy of a probability distribution Q on the torus is defined by the maximal difference, among all rectangles R of P d , between Q(R) and its Lebesgue measure. Recall Erdös-Túran-Koksma inequality [START_REF] Kuipers | Uniform distribution of sequences[END_REF] who estimates the discrepancy in terms of the Fourier transform Φ Q : Let γ > γ l (α). There exists some constant c 2 such that for any h = 0, h, α > c 2 |h| -γ .

For K ⊂ D and H ∈ N let

H K = {h ∈ N d : 0 ≤ h i ≤ H if i ∈ K, h i = 0 otherwise} and Γ N (H K ) = 0 =h∈H K 1 r(h)
e -c1N h,α 2 .

We will show that (29)

0<|h|∞≤H 1 r(h) |Φ Q (h)| = O( H γ-1 √ N ).
According to [START_REF] Masur | Logarithmic law for geodesics in moduli space, Mapping class groups and moduli spaces of Riemann surfaces[END_REF], the proposition will follow from [START_REF] Ornstein | Entropy and data compression schemes[END_REF] with the choice H = N We prove it by induction on the cardinality of K ⊂ D. For K = ∅ there is nothing to prove. Let ℓ ≤ d -1 and assume that it is true for any K ⊂ D of cardinality ℓ.

Let K ⊂ D with cardinality ℓ + 1. The sum for h ∈ H K restricted a face h i = 0, for some i ∈ K, corresponds to the ℓ-dimensional situation, where the estimate holds. Therefore, it suffices to prove the estimate for the sum restricted to h i ≥ 1 for any i ∈ K. In this case r(h) is simply the product i∈K h i . Benoit Saussol, Université Européenne de Bretagne, Université de Brest, Laboratoire de Mathématiques CNRS UMR 6205, 6 avenue Victor le Gorgeu, CS93837, F-29238 Brest Cedex 3, France E-mail address: benoit.saussol@univ-brest.fr URL: http://www.math.univ-brest.fr/perso/benoit.saussol/

2. 1 .

 1 General inequalities for hitting and recurrence time exponents. Recall that the upper and lower local dimension of ν at y are defined as d ν (y) = lim sup r→∞ log ν(B(y,r)) log r and d ν (y) = lim inf r→∞ log ν(B(y,r)) log r

Theorem 4 .

 4 If (X, T, ν) has superpolynomial decay of correlations with respect to Lipschitz observables, as above and the local dimension d ν (y) exists 2 then (5) R(x, y) = R(x, y) = d ν (y) for ν-almost each x. If moreover X ⊆ R d for some d, then

Remark 22 .

 22 x we have R(x, y) ≤ max(d µ (π 1 (y)) + d, dγ l (α)). Under the assumptions of the proposition: • If γ l (α) > d µ +d then the hitting time exponent is larger than the dimension of the invariant measure a.e. • In dimension d = 1 we get the equality for all y R(x, y) = max(d µ (π 1 (y)) + 1, γ(α)), for ν-a.e. x. Proof. Without loss of generality we assume that y = (ω ′ , 0). Let ξ = max(d µ (ω ′ ) + d, dγ l (α)) and let ǫ > 0. Let γ = γ l (α) + ǫ/d. For r > 0 set N r = r -ξ-ǫ . For all t we have by Proposition 17 that D Nr (α) ≤ N -1 γ r

4. 3 .

 3 Quantitative recurrence. Theorem 23. We have R(x, x) ≤ d µ + 1+dµ γ s (α) for ν-a.e. x.

r→0 log τ

  obs r (x, y) log r and R obs (x, y) = lim sup r→0 log τ obs r (x, y) log r .

r→0

  log τ r (x, y) log ν(B r (y))= 1 for ν-a.e. x.We hence easily have:Proposition 41. In a skew product as above over M = Ω × T 2 , if α ∈ Y ξ and ξ > d µ + d then the system has not the monotone shrinking target property. Proof. Let us consider y such that the local dimension d µ (π 2 (y)) = d µ (this is a full measure set). The statement follows from the definition of local dimension: it holds log µ(Br (y)) log r → d µ (π 1 (y)) + d. Br (y) (x) log µ(B r (y)) log µ(B r (y)) log r ) = R(x, y) by definition. Since in our examples R(x, y) > d µ + d. Then lim log τ Br (y) (x)

Appendix 7 . 1 .

 71 imply that for any (k, l) ∈ Z 2 , with max(|k|, |l|) sufficiently large||kαlα ′ || ≥ 1 8(max(|k|, |l|)) 5 ≥ 1 (max(|k|, |l|))16 .7.Changing regularity of the observables. We prove the lemma which allows to give a better bound on the decay of correlations, by passing through higher regularity observables.Proof of Lemma 3. Let k ≤ p and ℓ ≤ q. Let f ∈ C k and g ∈ C ℓ be Lipschitz observables such that f dµ = gdµ = 0. Let us consider a regularization by convolution. As usual, let us consider a function ρ ∈ C ∞ (R d ), ρ ≥ 0 having support in B 1 (0) and such that ρdx = 1. Let ǫ > 0. Then consider ρ ǫ (x) = ǫ -d ρ( x ǫ ), this function has support in B ǫ (0) and still ρ ǫ dx = 1. Let us consider a multiindex α = (α 1 , ..., α d ) ∈ N d , then remark thatD α ρ ǫ = ǫ -d D α ρ x ǫ = ǫ -d ǫ -|α| (D α ρ) x ǫ hence D α ρ ǫ L 1 = |D α ρ ǫ |dx = ǫ -|α| |D α ρ|dx.Now consider f ǫ defined byf ǫ (x) = f * ρ ǫ (x) = f (y)ρ(xy)dyand g ǫ = g * ρ ǫ (the convolution of g and ρ ǫ with respect to Lebesgue measure on R d ). For any β ≤ α such that |β| ≤ k we getD α f ǫ (x) = D β f (y) D α-β ρ ǫ (xy)dy ≤ f C k D α-β ρ ǫ L 1 .

1 ,

 1 |h i |). Let D = {1, . . . , d}. We apply it with Q the distribution of αS N ϕ under µ which satisfies|Φ Q (h)| = Ω e -2iπ h,αSN ϕ dµ = Ω L N 2π h,α (1)dµ ≤ c 0 e -c1N h,α 2 ,by Proposition 13.

  it suffices to show that for K = D(30) Γ N (H K ) = O( H γ-1 √ N ).

  Given a : N d → R, n ∈ N d and i ∈ D let a i (n) = a(n)a(n + e i ), a i (n) = ni ki=1 a(n + (k in i )e i ).We extend this notation to multiindices I ⊂ {1, . . . , d}, defining a I (n) and a I (n) in the obvious way. Note n -I = n -i∈I e i . We use the following multidimensional Abel's summation formula, for any a, b : N d → R: (31) (ab) K = I⊂K (b I a K ) I (n -I).

  A short overview of hitting and recurrence time. The study of hitting time or recurrence time in dynamical systems has several facets. Many of them are described by the surveys

	7.2. Discrepancy estimates	28
	References	29
	1. Introduction
	1.1.
	1. Introduction
	1.1. A short overview of hitting and recurrence time
	1.2. Outline of the results
	2. Background.
	2.1. General inequalities for hitting and recurrence time exponents
	2.2. Rapid mixing and consequences
	2.3. Basic results on circle rotations
	2.4. Statistics of hitting/return time
	3. The class of skew products under consideration
	3.1. Definition
	3.2. Upper bound on decay of correlations
	4. Quantitative recurrence, shrinking targets and arithmetical properties
	4.1. Deterministic and random multidimensional rotations
	4.2. Hitting time
	4.3. Quantitative recurrence
	4.4. Observed systems
	5. Decay of correlations, lower bounds
	6. Systems with R(x, y) > d and consequences
	6.1. A skew product with lower hitting time exponent larger than the
	dimension
	6.2. No dynamical Borel-Cantelli property and monotone shrinking targets
	6.3. Trivial return and hitting time distribution
	6.4. Construction of a diophantine angle with intertwined partial quotients
	7. Appendix
	7.1. Changing regularity of the observables

  2p 2 . For our purpose it is sufficient to prove that for a full measure set of t ∈ T d , eventually (as p → ∞)||nα 1 + t 1 || < r p , ..., ||nα d + t d || < r p =⇒ n > d|k p | γ log |k p | . Let us fix p and choose t = (t 1 , ..., t d ) such that ||t 1 k 1,p + t 2 k 2,p + ... + t d k d,p || > d p 2 . Now suppose n is such that ||nα 1 + t 1 || < r p , ..., ||nα d + t d || < r p and remark that d p 2 < ||k p • t|| = ||k p • (-nα) + k p • (t + nα)|| ≤ n||k p • α|| + d|k p |||t + nα|| ≤ n|k p | -γ + d|k p |r p Now, it remains to show that the set of t such that ||t • k p || > d p 2 eventually has full measure. Indeed, let us consider the Z d -periodic set C Now the measure of these sets must be evaluated on the torus T d = R d /Z d . Let us denote by π the canonical projection R d → T d . Let us now consider the set C ′ p = {x ∈ R d s.t. ||x • e 1 || ≤ d p 2 }. First let us remark that Leb(π(C ′ p )) = 2 p 2 . Now remark that being a nonsingular integer matrix, A t induces a measure preserving map A

				≤ n|k p | -γ +	d 2p 2
	and thus n ≥	d|kp| γ 2p 2 ≥	d|kp| γ log |kp| .
	Let r p =	|kp| -1

1,p , ..., k d,p ) with |k p | → ∞ such that ||k p • α|| < |k p | -γ . By choosing a subsequence we can also suppose without loss of generality that p ≤ 1 2 log(|k p |). p = {t ∈ R d s.t. ||t • k p || ≤ d p 2 }. Now, choose an integer nonsingular matrix such that Ae 1 = k p (e 1 is the first vector of the canonical basis of R d ) then C p = {t ∈ R d s.t. ||A t t • e 1 || ≤ d p 2 }. t on the torus and π(C

We recall that, by Rademacher Theorem, a Lipschitz function has derivatives defined almosteverywhere and the derivatives are a.e. bounded by the Lipschitz constant of the function. This clearly suffices to make the proof works.

This work was partially supported by the ANR Perturbations (ANR-10-BLAN 0106), Indam, DynEurBraz.

We apply this formula to b(h) = i∈K h -1 i and a(h) = e -c1N h,α . We first estimate a K (h):

Therefore, each of the interval [0, η), [η, 2η), ..., contains at most one k, α with 1 ≤ k ≤ h, and the first interval does not contain any of them since 0, α ∈ [0, η).

for some constant C. On the other hand, we have b

This implies that, for some constant C ′ and any I ⊂ K,

where n I,K,H i = H -1, H, 0 according to i ∈ I, K, D \ K. The inducing step (30) follows by [START_REF] Rousseau | Poincaré recurrence for observations[END_REF].


Alex Boulton

Language awareness and medium-term benefits of corpus consultation

Data-driven learning (DDL) involves learners exploring corpora to discover language rather than 'being taught'. This is claimed to have a number of advantages, and while empirical evidence to date is encouraging, it is less conclusive than might be hoped -partly, it is argued, as it tends to focus on short-term goals by testing knowledge and retention of selected language items covered. It may be, however, that the real advantages of DDL lie in longer-term benefits, cognitive/constructivist as well as purely linguistic; in addition to 'incidental' learning and greater learner autonomy, these include language awareness and noticing ability, all of which are more difficult to assess. This paper reports on a medium-term experiment comparing noticing skills between experimental and control groups of lower-intermediate learners of English in an architecture school in France. The control groups were taught in the usual way, while the last 15 minutes of the experimental classes were given over to exploring the British National Corpus on line, with learners working in pairs on specific language points. Both experimental and control groups were tested after 12 weeks -not on the language points covered, but on their sensitivity to other language items in a previously unseen text.

The time-frame of the present experiment is not enough to allow the full effects of DDL to be revealed, but does provide some indication that its main advantages go well beyond the shortterm learning outcomes generally examined in current research. If further long-term work can confirm these findings, it might go some way towards helping DDL to reach a wider audience.

Introduction

For over two decades, researchers (e.g. [START_REF] Johns | From printout to handout: grammar and vocabulary teaching in the context of data-driven learning[END_REF]) have been promoting the use of electronic corpora in language learning. In its most common form, learners use concordance lines such as in Figure 1 to detect the common, probable patterns of usage in context, concentrating on collocation, colligation, frequency, distribution, and so on. A number of studies have attempted to evaluate some aspect of this 'data-driven learning' (DDL) approach: in a survey of 20 such research projects, Boulton (forthcoming) finds that the results are generally encouraging, but rarely overwhelmingly positive. It is notable that the research is often designed around short-term returns on predefined language items; in other words, many of them Trends in Computer-Assisted Language Learning: Working Together. Madrid: Macmillan ELT, attempt to show that practical and logistical barriers (cf. Boulton 2009a) can be overcome under the right circumstances -especially given enthusiastic teachers with sufficient resources. However, it might be that the main advantages of DDL lie more in its long-term effects (Chan & Liou 2005: 241;[START_REF] Yoon | More than a linguistic reference: the influence of corpus technology on L2 academic writing[END_REF]): developing a range of essential cognitive processes, increasing language sensitivity and ability to cope with authentic language, enhancing motivation due to greater learnercenteredness, and adopting a more 'naturalistic' approach to patterns (as opposed to rules) coupled with an interactive, discovery approach fosters better language learning ability on the whole, leading to greater autonomisation for lifelong learning. If aspects such as these do represent the main strengths of DDL, then studies that do not address them directly clearly have their limitations. This paper focuses on one potential long-term advantage, namely in promoting noticing skills, a feature frequently mentioned in the DDL literature1 (e.g. [START_REF] Allan | Can a graded reader corpus provide 'authentic' input?[END_REF][START_REF] Aston | Learning with corpora: an overview[END_REF][START_REF] Bernardini | A 'trainee' translator's perspective on corpora[END_REF][START_REF] Bondi | Small corpora and language variation: reflexitivity across genres[END_REF][START_REF] Gabrielatos | Corpora and language teaching: just a fling or wedding bells? Teaching English as a Second Language[END_REF]Johns 1997;[START_REF] Johns | Integrating corpus-based CALL programs and teaching English through children's literature[END_REF][START_REF] Mauranen | Spoken corpus for an ordinary learner[END_REF][START_REF] Meunier | New types of corpora for new educational challenges: collecting, annotating and exploiting a corpus of textbook material[END_REF][START_REF] O'sullivan | Enhancing a process-oriented approach to literacy and language learning: the role of corpus consultation literacy[END_REF][START_REF] Pérez-Paredes | Some lessons students learn: selfdiscovery and corpora[END_REF][START_REF] Thompson | Modal verbs in academic writing[END_REF][START_REF] Yoon | More than a linguistic reference: the influence of corpus technology on L2 academic writing[END_REF]. Noticing is also present in many DDL materials (e.g. [START_REF] Mckay | Teaching the syntactic, semantic and pragmatic dimensions of verbs[END_REF][START_REF] Johns | From printout to handout: grammar and vocabulary teaching in the context of data-driven learning[END_REF][START_REF] Bowker | Working with specialized language: a practical guide to using corpora[END_REF]Thurstun & Candlin 1997;[START_REF] Tribble | Concordances in the classroom[END_REF], whereas Meunier and Gouverneur (2009: 195) find it is often absent in more traditional materials. Indeed, Carter (1998: 51) claims that communicative teaching on the whole "has not encouraged in students habits of observation, noticing, or conscious exploration." While traditional deductive approaches may allow the teacher to 'do the noticing for the learner', inductive approaches are entirely dependent upon noticing (see [START_REF] Schaffer | A comparison of inductive and deductive approaches to teaching foreign languages[END_REF] for a comparison), and DDL is largely inductive in nature. Of course, DDL has no monopoly on noticing: on the whole, it is implicit at some level in virtually all approaches, especially where the focus is on learning (featuring problem-solving, discovery learning, task-based or process-oriented approaches, for example) rather than teaching. [START_REF] Cook | The philosopher pulled the lower jaw of the hen.' Ludicrous invented sentences in language teaching[END_REF] even claims that it is an advantage of invented sentences that they do promote noticing. Noticing is closely related to a number of other features, such as focus on form, consciousness, language awareness, sensitisation, and so on. It is particularly associated with [START_REF] Schmidt | The role of consciousness in second language learning[END_REF], who claimed that it preceded understanding; but unlike understanding, "noticing is the necessary and sufficient condition for converting input to intake" (p. 129). While we may learn, know and use language effectively without conscious understanding, noticing is essential -or at least, "more noticing leads to more learning" (Schmidt 1994: 18). Also [START_REF] Schmidt | Attention[END_REF]; [START_REF] Gass | Input, interaction, and the second language learner[END_REF]; though see [START_REF] Robinson | Individual differences and the fundamental similarity of implicit and explicit adult second language learning[END_REF] for a contrary view.

Method

This study involved the participation of 59 students (average age just over 20; 48% women) in their second year of architecture studies in France -34 in experimental groups (EG), 25 in control groups (CG). Their main motivations are for architecture and not languages: despite eight years of English, their levels of proficiency are fairly low (averaging about 450 points on an in-house TOEIC, i.e. about A2-B1 on the CEFR) -hence the relevance of a new approach, a claim made by [START_REF] Boulton | Data-driven learning: taking the computer out of the equation[END_REF]. The primary objective is to score at least 650 points on the TOEIC by the end of the following year. To this end, the common language programme focuses on listening skills (in the computer room) as well as reading (mainly grammar and vocabulary). Conditions for both were essentially the same: a total of 30 hours of English in 90-minute weekly classes on the same days and times, with the same materials, facilities and evaluations, but different teachers. For the EG only, the last 10-20 minutes of class time was devoted to corpus-based activities in the computer room; due to various constraints, this was on an irregular basis for only 12 of the classes, a total time of about 3 hours. Students worked in pairs of their own choice, using [START_REF] Davies | BYU-BNC: The British National Corpus[END_REF] interface to the British National Corpus. In an attempt to integrate the DDL activities and make them relevant to the learners' immediate concerns as recommended by [START_REF] Somogyi | Using the concordancer in vocabulary development for the Cambridge Advanced English (CAE) course[END_REF], all DDL language points covered arose during class, either in the current week, or as revision / reinforcement from the previous week, or preparation for the following week. As time was at a premium and the same core syllabus had to be completed as for the other classes, it was not possible to offer extensive training. Rather, after a brief 5-minute introduction the first time, detailed Trends in Computer-Assisted Language Learning: Working Together. Madrid: Macmillan ELT, p. 39-46. instructions for each DDL activity were printed out; examples are given in Figure 2. This of course limits the students' autonomy, but means they can start corpus consultation on relevant points immediately, and by the end of the course should be sufficiently familiar with the procedures to undertake further corpus work on their own should they so wish.

HUNDRED(S) (OF) a)

Is it possible to have a number with a plural (e.g. hundreds)? Try the NEWS section for hundred and hundreds. Is either followed by of? What do you conclude? b)

Compare with thousand, million, billion; then ten and dozen -any differences? How would you translate each in French?

PASSIVES a)

Search only in NEWS for is said; how would you translate this in French? What structure(s) typically follow it? b)

What would you call is said: present continuous, present perfect, present passive, present simple? c)

Try for the following verbs (be + past participle); do they all exhibit the same usage? Which are the most frequent? believe, expect, know, report, suppose, think. At the end of the course, the EG learners completed a questionnaire, the results of which are reported in [START_REF] Boulton | Corpora for all? Learning styles and data-driven learning[END_REF] where the focus was on learning styles. Additionally, learners in both CG and EG were given a short test to assess their spontaneous noticing skills. As there is no standard procedure for assessing noticing (Williams 2005: 681), the activities were designed around a familiar support, a short (345-word) newspaper article with a business focus on a topic previously covered by all students. 2 The participants were told they would have 5 minutes to read the text, which would subsequently be retrieved before they answered questions on it. The first focus-on-form (FOF) question type simply required them to remember which of two words had been used in the text, each pair having similar form (e.g. achievable / available) or similar meaning (e.g. March / January). The second focus-on-meaning (FOM) question type asked them to choose the best translation of a word from the text (e.g. SQUEEZE: satisfaire, contraindre, commencer, accroître). Crucial for the purpose of noticing here is the fact that none of the language points selected had been overtly covered during the course, and the learners did not know what type of questions they would be asked (cf. Hulstijn 1997).

Alex Boulton. 2011. Language awareness and medium-term benefits of corpus consultation. In A. Gimeno Sanz (ed.), New Trends in Computer-Assisted Language Learning: Working Together. Madrid: Macmillan ELT, p. 39-46.

The results of the experiment are given in Table 1, with the mean scores, standard deviation and results of a two-tailed p-test for the experimental and control groups for each question type. The EG performed better than the CG on both questions types, especially for FOM. However, the differences in neither case were statistically significant. Combining both sets of data increases the statistical difference, but this is still not significant at the usual levels on a two-tailed paired t-test (p>0.05).

Conclusions

This paper reports on a simple experiment designed to test whether the processes involved in basic corpus consultation lead to medium-term benefits in noticing skills. The experimental group does indeed seem to improve more than the control group, but the difference does not meet the standard levels of significance. The constraints of the course limited the corpus work possible, in terms of overall time spent (about 3 hours), the length of each session (10-20 minutes including time lost for various reasons) and the irregular rhythm (roughly once every two weeks on average). Although [START_REF] Flowerdew | Corpus linguistics for academic literacies mediated through discussion activities[END_REF] claims short sessions may be preferable, the learners in the study by [START_REF] Gan | Modeling teaching with a computer-based concordancer in a TESL preservice teacher education program[END_REF] would have preferred full two-hour classes. The time constraints also ruled out significant training, and it may be that "noticing things in corpus data is an acquired skill even for linguistically relatively sophisticated learners" (Mauranen 2004: 99). The lack of training also inhibited more autonomous work (though see Boulton 2009c), and the activities conducted thus had to be tightly controlled; this further limits student input, with (one might suppose) a concomitant reduction in motivation during the process and reduced impact on noticing skills. Although these limitations are likely to be typical of many language classrooms where limited time can be devoted to alternative approaches, especially in a computer laboratory, the modest improvement does suggest that greater benefit would be recorded from longer and more in-depth corpus work. In short, if a little DDL leads to a modest improvement, then more DDL might lead to more substantial differences. Trends in Computer-Assisted Language Learning: Working Together. Madrid: Macmillan ELT, p. 39-46.

The overall evidence to date shows that DDL can bring immediate benefits, but these are relatively small, and there is a case to be made that its real advantages lie in longer-term outcomes. The present study suggests that for DDL to fulfil its potential in promoting substantial long-term benefits, occasional short sessions are not enough. While it seems reasonable to suppose that more time spent in corpus work would lead to more significant improvement, DDL is not an entire method, and in most contexts will vie for class time with other materials and techniques. The question then turns from whether it is effective, to whether it is an efficient use of learners' time over long periods. An alternative but equally important step forward will be to see how DDL can be used outside the classroom, a possibility largely untested to date (Chambers 2007: 13).

Figure 1 .

 1 Figure 1. Concordance of depend. http://corpus.byu.edu/bnc/

Figure 2 .

 2 Sample activities.

In this context, it refers mainly to spontaneous noticing by the learners, rather than teacherdirected focus on form.

AlexBoulton. 2011. Language awareness and medium-term benefits of corpus consultation. In A. Gimeno Sanz (ed.), New Trends in Computer-Assisted Language Learning: Working Together. Madrid: MacmillanELT,

"Mortgage lending increases but remains historically low". Sandra Haurant. 14/04/09. http://www.guardian.co.uk