Sara Brofferio
email: sara.brofferio@math.u-psud.fr

Dariusz Buraczewski

ON UNBOUNDED INVARIANT MEASURES OF STOCHASTIC DYNAMICAL SYSTEMS

Keywords: 2010 Mathematics Subject Classification. Primary: 37Hxx, 60J05; Secondary: 60K05, 60B15 Stochastic recurrence equation, stochastic dynamical system, invariant measure, affine recursion, reflected random walk, Poisson equation

We consider stochastic dynamical systems on R, i.e. random processes defined by X x n = Ψn(X x n-1 ), X x 0 = x, where Ψn are i.i.d. random continuous transformations of some unbounded closed subset of R. We assume here that Ψn behaves asymptotically like Anx, for some random positive number An (the main example is the affine stochastic recursion Ψn(x) = Anx + Bn). Our aim is to describe invariant Radon measures of the process X x n in the critical case, when E log A 1 = 0. We prove that those measures behave at infinity like dx

x . We study also the problem of uniqueness of the invariant measure. We improve previous results known for the affine recursions and generalize them to a larger class of stochastic dynamical systems which includes, for instance, reflected random walks, stochastic dynamical systems on the unit interval [0, 1], additive Markov processes and a variant of the Galton-Watson process.

1. Introduction 1.1. Stochastic dynamical systems. Let F be the semigroup of continuous transformations of an unbounded closed subset R of the real line R endowed with the topology of uniform convergence on compact sets. In the most interesting examples R is the real line, the half-line [0, +∞) or the set of natural numbers N. Given a regular probability measure µ on F, we define the stochastic dynamical system (SDS) on R by

X x 0 = x; X x n = Ψ n (X x n-1 ), (1.1)
where {Ψ n } is a sequence of i.i.d. random functions, distributed according to µ. The aim of this paper is to study conditions for the existence and uniqueness, as well as behavior at infinity, of an invariant infinite Radon measure of the process X x n , i.e. of a measure ν on R such that

(1.2) µ * F ν(f ) = ν(f ),
for any f ∈ C C (R), where

µ * F ν(f ) = ∫ R E [ f (X x 1 ) ] ν(dx) = ∫ F ∫ R f (Ψ(x))ν(dx)µ(dΨ).
There is quite an extensive literature on the case when the process X n is positive recurrent, that is, it possesses an invariant probability measure. The existence of such a measure can be proved supposing that the process has some contractive property (for example, if Ψ n are Lipschitz mappings with Lipschitz coefficients L n = L(Ψ n ) and E[log L 1 ] < 0), [START_REF] Diaconis | Iterated random functions[END_REF]). This invariant probability measure is well described in several specific cases, such as affine recursions (i.e. Ψ(x) = Ax + B), namely in the seminal paper of H.Kesten [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF]. C.M.Goldie [START_REF] Goldie | Implicit renewal theory and tails of solutions of random equations[END_REF] and recently M. Mirek [START_REF] Mirek | Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps Probab[END_REF] generalized Kesten's theorem to stochastic recursions such that Ψ(x) behaves like Ax for large x. They proved that if EA κ = 1 (and some other hypotheses are satisfied), then lim z→∞ z κ ν{x : |x| > z} = C + > 0.

In other words the measure ν is close at infinity to C+dx x 1+κ . Less is known for the null recurrent case, especially in a general setting. Existence and uniqueness of an invariant Radon measure have been the topic of two recent works: B. Deroin, V. Kleptsyn, A. Navas and K. Parwani [START_REF] Deroin | Symmetric random walks on Homeo + (R). Annals of Probability to appear[END_REF] on symmetric SDS of homeomorphism of R, and M. Peigné and W. Woess [START_REF] Peigné | Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity[END_REF] on the phenomenon of local contraction. We refer to them for a more complete bibliography on the subject. As in the contracting case, affine recursions is one of the first models being systematically approached. A seminal paper in this area is the one of Babillot, Bougerol and Elie [START_REF] Babillot | The random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF]. They proved existence and uniqueness of a Radon measure and gave a first result on its behavior at infinity.

The goal of the present work is twofold. First of all we investigate the behavior at infinity of invariant measures and, for a large class of SDS's, we generalize and improve results known for affine recursions. Secondly, we consider the problem of uniqueness of the invariant measure. We give a relatively simple criterium that can be applied for very concrete examples. 1.2. Behavior at infinity. It turns out that to prove existence and to describe the tail of the measure it is sufficient to control the maps that generate the SDS near infinity. In particular we suppose that they are asymptotically linear, in the sense that there exists 0 ≤ α < 1 such that for all ψ ∈ F (AL α ) |ψ(x) -A α (ψ)x| ≤ B α (ψ)(1 + |x| α ) for all x ∈ R with A α (ψ) and B α (ψ) strictly positive. We study here the critical case, i.e. E[log A α ] = 0.

Existence of an invariant measure supported in R is relatively easy to deduce from the well-known literature, because in this case the SDS is bounded by a recurrent process (we give more details in subsection 2.3). The main result of the paper is the description of the tail of invariant measures at infinity. Theorem 1.3. Suppose that there exists 0 ≤ α < 1 such that the maps Ψ n satisfy (AL α ) µ-a.s. and that

E[log A α ] = 0 and P[A α = 1] < 1, (1.4) E [ (| log A α | + log + |B α |) 2+ε ] < ∞, (1.5)
the law of log A α is aperiodic, i.e. there is no p ∈ R such that log A α ∈ pZ a.s. (1.6) Let ν be an invariant Radon measure ν for the process {X x n } n . Suppose that ν is supported by R and it is positive on any neighborhood of +∞. Then the family of dilated measures δ z -1 * ν(I) := ν(zI) converges vaguely on R * + = (0, ∞) to C + da a as z goes to infinity for some C + > 0, i.e.

lim z→∞ ∫ R * + ϕ(z -1 u)ν(du) = C + ∫ R * + ϕ(a) da a ,
for any ϕ ∈ C C (R * + ). The key example of an asymptotic linear SDS is the affine recursion (called also the random difference equation). Then F is the set of affine mappings of the real line Ψ(x) = Ax + B with A > 0 and the process is given by the following formula (1.7)

X x n = A n X x n-1 + B n , X x 0 = x.
Our results are also valid for Goldie's recursions, e.g. Ψ(x) = max{Ax, B} + C (with A > 0) and Ψ(x) = √ A 2 x 2 + Bx + C (with A, B, C positive). Since the problem can be reduced, without any loss of generality, to the case α = 0 (see Lemma 2.1), our hypotheses essentially coincide, in the one dimensional situation, with the class introduced by M.Mirek [START_REF] Mirek | Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps Probab[END_REF]. Our main theorem should be viewed as an analog of Kesten's and Goldie's results in the critical case.

Other interesting examples can be obtained conjugating asymptotic linear systems with an appropriate homeomorphism. For instance, our result can also be applied to describe invariant measures of SDS on the interval generated by functions that have the same derivative at the two extremities. Theorem 1.3 also says that invariant measures of SDS on [0, +∞) generated by mappings exponentially asymptotic to translations, i.e.

|ψ(x) -

x + u ψ | ≤ v ψ e -x ,
∀x ≥ 0 behave at infinity as the Lebesgue measure dx of R, if E(u ϕ ) = 0. This result can be compared with the Choquet-Deny Theorem saying that the only invariant measure for centered random walks on R is the Lebesgue measure. Another interesting process that is α-asymptotically linear for α > 1/2 is a Galton-Watson evolution process with random reproduction laws. In Section 6, we give more details on the different examples.

Let us mention that in our previous papers [START_REF] Benda | Schwach kontraktive dynamische Systeme[END_REF][START_REF] Brofferio | On the invariant measure of the random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF][START_REF] Brofferio | On solutions of the affine recursion and the smoothing transform in the critical case[END_REF] we have already studied the behavior at infinity of the invariant measure ν for the random difference equation (1.7). However the main results were obtained there under much stronger assumptions, namely we assumed existence of exponential moments, that is

E[A δ + A -δ + |B| δ ] < ∞ for some δ > 0. Theorem 1.
3 improves all our previous results for affine recursions and describes the asymptotic behavior of ν under optimal assumptions, that is the weakest known conditions implying existence of the invariant measure [START_REF] Babillot | The random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF]. To our knowledge, for all the other recursions even partial results are not known.

We would like also to remark that, in the contracting case, Kesten's theorem requires moment of order at least κ and, as far as we know, there exist no results on the behavior of the tail of the invariant probability when the measure is supposed to have only logarithmic moment.

The proof of Theorem 1.3 is given in sections 3 and 4. In order to describe ν at infinity, we give first an upper bound of this measure and prove some regularity properties of its quotient. The techniques we use in the present paper are more powerful than those applied in [START_REF] Brofferio | On the invariant measure of the random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF], and are heavily based on the renewal theory for random walks on the affine group. Among other results we prove directly that ν[-z, z] grows as log z (Proposition 3.1). Next, in Section 4 we consider the Poisson equation for the additive convolution on R

f (x) = µ * f (x) + g(x), where f (x) = ∫ ϕ(e -x u)ν(du) for some ϕ ∈ C C (R * + )
and µ is the law of -log A. Notice that the asymptotic behavior of f and ν is the same, therefore it is sufficient to study f . In the contrast to [START_REF] Brofferio | On the invariant measure of the random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF] we do not solve explicitly this equation. We apply techniques borrowed from the work of Durrett and Liggett [START_REF] Durret | Fixed points of the smoothing transformation[END_REF] (see also Kolesko [18]), reduce the problem to the classical renewal equation with drift and deduce its asymptotic behavior from the renewal theorem.

1.3. Uniqueness of the invariant measure. Another fundamental question is to determine whether the invariant measure is unique or not. The nature of this problem is different from the ones we have considered so far. In fact dniqueness depends on the local behavior of the system and it is no more sufficient to control the random maps only at the infinity.

In the non-contracting case, this problem was studied first by Babillot, Bougerol and Elie [START_REF] Babillot | The random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF] in the context of the affine recursion and they proved uniqueness under the assumptions of Theorem 1.3. Relying on their ideas Benda [START_REF] Benda | Schwach kontraktive dynamische Systeme[END_REF] studied in full generality recurrent and locally contractive SDS's.

The SDS is called recurrent if there exists a closed set L such that every open set intersecting L is visited by X x n infinitely often with probability 1. The SDS is locally contractive if for any x, y ∈ R and every compact set K ⊂ R,

(1.8) lim n→∞ |X x n -X y n | • 1 K (X x n ) = 0 almost surely.
Benda [START_REF] Benda | Schwach kontraktive dynamische Systeme[END_REF] proved that if {X x n } is a recurrent and locally contractive SDS, then it possesses a unique (up to a multiplicative constant) invariant Radon measure. He didn't publish his results, however they have been recently incorporated, with a complete and simplified proof, into two papers of Peigné and Woess [START_REF] Peigné | Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity[END_REF][START_REF] Peigné | Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings[END_REF], where they also investigated ergodicity of SDS generated by Lipschitz maps with centered Lipschitz's coefficient.

Our aim is to consider very concrete families of Lipschitz mappings of R + , as the one presented in Goldie's work [START_REF] Goldie | Implicit renewal theory and tails of solutions of random equations[END_REF]. Although recurrence of the corresponding SDS's is immediate, the main obstacle in applying Benda's theorem is the local contraction hypothesis (1.8). In [START_REF] Peigné | Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings[END_REF] the authors considered the reflected affine stochastic recursion, being a mixture of the reflected random walk (described below) and the affine stochastic recursion (defined in 1.7). Unfortunately the method of hyperbolic extensions they introduce cannot be applied to dynamical systems, whose dependence on the affine recursion cannot be expressed in such a direct way.

A different approach can be found in [START_REF] Deroin | Symmetric random walks on Homeo + (R). Annals of Probability to appear[END_REF], where the authors proved a local contraction property for a symmetric SDS generated by homeomorphisms of R. Their proof is very elegant but is heavily based on the additional assumption that the SDS is generated by invertible mappings distributed according to a symmetric measure. In particular their results cannot be applied to noninvertible SDS, as the one generated by ψ(x) = max{Ax, B} + C, one of the most interesting in applications.

Our contribution to the subject is to give sufficient conditions for uniqueness that can be applied to some concrete mappings of R + = [0, ∞), such as ψ(x) = max{Ax, B} + C and other Goldie's recursions.

Theorem 1.9. Suppose that R = [0, ∞), α = 0 and that the hypotheses of Theorem 1.3 are satisfied. Assume moreover that [START_REF] Aldous | Probability approximations via the Poisson clumping heuristic[END_REF] there exists β > 0 such that P(Ψ[0, +∞) ⊆ [β, +∞)) > 0;

(2) A(Ψ)x ≤ Ψ(x) ≤ A(Ψ)x + B(Ψ) for all x ≥ 0;

(3) the functions Ψ are Lipschitz and their Lipschitz coefficients are equal to A(Ψ).

Then the SDS defined on [0, ∞) by (1.1) is locally contractive. Therefore there exists a unique invariant Radon measure of the process {X x n } on [0, +∞).

The proof of this theorem is contained in Section 5.

1.4. Reflected random walk. The reflected random walk is the SDS defined for

x ∈ R + = [0, ∞), by Y x 0 = x, Y x n = Y x n-1 -u n , (1.10)
where u n is a sequence of i.i.d. real valued random variables with a given law µ.

If u n ≥ 0 a.s., then it was proved by Feller [START_REF] Feller | An introduction to probability theory and its application II[END_REF] that this process possesses a unique invariant probability measure ν, i.e. a measure satisfying

µ * ν(f ) = ∫ R+ ∫ R+ f (|x -y|)ν(dx)µ(dy) = ∫ R+ f (x)ν(dx) = ν(f ).
Moreover the measure ν can be explicitly computed: ν(dx) = (1 -F (x))dx, for F being the distribution function of µ. The process has been also studied in more general settings, when u n admits also negative values (see Peigné, Woess [START_REF] Peigné | Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity[END_REF] for recent results and a comprehensive bibliography).

Here, we are interested in the critical case when Eu n = 0. Peigné and Woess [START_REF] Peigné | Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity[END_REF] proved that if

E(u + 1 ) 3
2 < ∞, for u + 1 = max{u 1 , 0}, then the process {X n } is recurrent on R + . As a consequence of Benda's theorem, the process possesses a unique invariant Radon measure ν on R + (local contractivity is easy to prove). The reflected random walk can be transformed in an asymptotically linear system by conjugating with an invertible function s of [0, +∞) such that s(x) = e x for large x. Then ψ(x) = s(|s -1 (x) -u|) is asymptotically linear with A(ψ) = e -u . Hence Theorem 1.3 can be used to justify that the invariant measure of Y x n behaves at infinity like the Lebesgue measure. Nevertheless in this case one can prove the same result under weaker moment assumptions and a much simpler proof. A short argument based only on the duality lemma and the renewal theorem gives:

Theorem 1.11. Assume Eu 1 = 0, E(u + 1 ) 3 2 < ∞, E(u - 1 ) 2 < ∞ and the law µ of u 1 is aperiodic, then for every ϕ ∈ C C (R + ) lim x→∞ ∫ R+ ϕ(u -x)ν(du) = C + ∫ R+ ϕ(u)du, for some positive constant C + .
The proof of this theorem will be given in Section 6.6.

We are grateful to the referees for their careful reading of the manuscript and many helpful suggestions for for improvement in the presentation.

Notations and preliminary results

Reduction to condition (AL).

Observe first that, conjugating the SDS with an appropriate function, we can suppose without loss of generality that the distance of the random map to a linear function is smaller than some constant. In fact we have the following lemma whose proof is postponed to Appendix A: Lemma 2.1. Let 0 ≤ α < 1. Suppose that ψ satisfies

(AL α ) |ψ(x) -A α x| ≤ B α (1 + |x| α ).
Then the conjugate function

ψ r = r • ψ • r -1 , where r(x) = sign(x)|x| 1-α , satisfies (AL 0 ) with A 0 = A 1-α α .
The appropriate constant B 0 can be chosen such that log

+ B 0 ≤ C α (| log A α | + log + B α + 1
), for the constant C α depending only on α.

If ψ is distributed according to µ, the law ψ r is given by µ r = δ r * µ * δ r -1 , and if ν is a µ-invariant measure then ν r = δ r * ν is µ r -invariant. Thus if Theorem 1.3 holds for ν r then it holds for ν. Indeed

lim z→∞ ∫ R * + ϕ(z -1 u)ν(du) = lim z→∞ ∫ R * + ϕ(z -1 r -1 (u))ν r (du) = lim z→∞ ∫ R * + ϕ(z -1 u 1/(1-α) )ν r (du) = lim z→∞ ∫ R * + ϕ((z -(1-α) u) 1/(1-α) )ν r (du) = C + ∫ R * + ϕ(a 1/(1-α) ) da a = C + (1 -α) ∫ R * + ϕ(a) da a .
In order to simplify our notations, we'll suppose from now on that α = 0, i.e. for all ψ ∈ F (AL)

A(ψ)x -B(ψ) < ψ(x) < A(ψ)x + B(ψ) for all x ∈ R.
Since R is closed, we can extend the property (AL) to all x ∈ R for a suitable continuous extension of ψ to R. With a slight abuse of notation, we will denote with the same letter (e.g. ψ), the map from R to R and its continuous extension that verifies (AL) for all x ∈ R. In the same way, ν will be seen both as a measure on R and as a measure on R whose support is contained in R.

Comparison of X

x n with the affine recursion. We assume that the maps A = A(ψ) and B = B(ψ) from F to R * + = (0, ∞) are measurable and that F is a monoid closed by composition. Assumption (AL) implies lim

x→+∞ x∈R ψ(x)/x = lim x→-∞ x∈R ψ(x)/x = A(ψ), therefore the map A is a homomorphism from F to R * + i.e. A(ψ 1 • ψ 2 ) = A(ψ 1 )A(ψ 2 ).
The choice of B is not unique and it can be chosen as big as needed.

Let {Ψ n } ∞ n=1 be an i.i.d. sequence of random variables with values in F of law µ. We are interested in the study of the iterated stochastic function system

X x n = Ψ n (X x n-1 ) = Ψ n ...Ψ 1 (x) and X x 0 =
x. If hypothesis (AL) is satisfied, the trajectories of the process X x n can be dominated from below and from above by the affine recursions (2.2)

Z x n = A n Z x n-1 -B n and Y x n = A n Y x n-1 + B n
, where, to simplify our notation, we note A n = A(Ψ n ) and B n = B(Ψ n ). We will also assume, according to hypotheses of Theorem 1.3, a logarithmic moment of order 2 + ε and that log A 1 is nontrivially centered. Without any loss of generality, we can also choose B(ψ), such that:

B n ≥ 1 a.s. (2.3) P(A n x + B n = x) < 1 for all x. (2.4)
In such a way the two dimensional process (Z x n , Y x n ) satisfies all the assumptions required by Babillot, Bougerol, Elie [START_REF] Babillot | The random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF]. Thus it is recurrent, locally contractive and possesses a unique invariant measure.

It will be convenient to use in the proof the language of groups. Namely, let G = Aff(R) = R R * + be the group of all affine mappings of R, i.e. the set of pairs (b, a) ∈ R × R * + acting on R: (b, a) : x → ax + b. Then the group product is given by the formula

(b, a) • (b ′ , a ′ ) = (b + ab ′ , aa ′ ),
the identity element is (0, 1) and the inverse element is given by (b, a) -1 = (-b/a, 1/a).

Let µ G be the probability distribution of (B n , A n ) on the group G. Then the random elements g n = (B n , A n ) are i.i.d. random variables in G with law µ G . We define the left and the right random walk on G:

(2.5) L n = g n • ... • g 1 , R n = g 1 • ... • g n .
Then, Y x n = L n (x). A very important role in our proofs will be played by the random walk on R generated by -log A i , i.e.

(2.6)

S n = -(log A 1 + • • • + log A n ),
(we put the sign minus to follow notations of our previous works). Since E log A = 0, the random walk S n is recurrent. Moreover since we assume aperiodicity, the support of S n is just R. We often use the downward and upward sequence of stopping times (2.7)

l n := inf{k > l n-1 : S k < S ln-1 }, t n := inf{k > t n-1 : S k ≥ S tn-1 }
and l 0 = t 0 = 0. Observe that t 1 and l 1 are almost surely finite, but have infinite mean. On the other hand, hypothesis E(| log A| 2+ε ) < ∞ guarantees that S t1 and S l1 are integrable (see [START_REF] Chow | Moments of ladder variables for driftless random walks[END_REF]).

In the sequel we will use, depending on the situation, different convolutions. We define a convolution of a function f on R with a measure η on R as a measure on R given by

(2.8) f * η(K) = ∫ R 1 K (f (u))η(du) = η(f -1 (K)).
Given z ∈ R * + and a measure η on R we define (2.9)

δ z * R * + η(K) = ∫ R 1 K (zu)η(du) = η(z -1 K).

2.3.

Existence of an invariant measure. We conclude this section observing that the existence of the invariant measure on R ⊆ R for a SDS satisfying the hypotheses of Theorem 1.3 follows immediately from recurrence of the process {X x n } and Lin's theorem [START_REF] Lin | Conservative Markov processes on a topological space[END_REF]. More precisely, consider the positive operator

P f (x) = ∫ f (Ψ(x))µ(dΨ) on C b (R). Then, since Z x n ≤ X x n ≤ Y x n and (Z x n , Y x n ) is recurrent, the process {X x n } is recurrent, i.e. there exists a nonnegative function u ∈ C c (R) such that ∑ ∞ n=0 P n u(x) = ∞ for all x.
Therefore, by [START_REF] Lin | Conservative Markov processes on a topological space[END_REF] there exists a non-null invariant Radon measure ν on R of the process {X x n }. Observe that the support of this measure can be bounded (for instance if the functions Ψ fix the point 0, then the Dirac measure at 0 is an invariant measure). In this paper we are interested in measures having unbounded support. A sufficient (but not necessary) condition to ensure that the invariant measure is not bounded is to assume that the random functions Ψ do not fix a compact subset C of R ( that is there is no compact C such that P(Ψ(C) ⊆ C) = 1).

First bounds of the tail of the invariant measure

We start to study the behavior of ν at infinity. In particular we will prove in this section that ν(dx) does not grow faster than dx x , the Haar measure of R * + . The behavior of ν at ∞ is related to the behavior of the family of measures δ z -1 * ν. In this section we prove

Proposition 3.1. Under the hypotheses of Theorem 1.3 we have the following:

(1) There exists

C 0 > 0 such that ν[-z, z] < C 0 (1 + log z) for all z > 1
Moreover, if the support of ν is not bounded on the right, i.e. ν(z, +∞) > 0 for all z ∈ R, then (2) There exist M > 1 and δ > 0 such that ν[z, zM ] > δ for all z ≥ 1.

(3) For all u 2 > u 1 > 0 there exists

C = C(u 1 , u 2 , M ) > 0 such that (3.2) ν[e x+y u 1 , e x+y u 2 ] ν[e x , e x M ] < C(1 + y) for all x > 0, y > 0.
In particular the family of measures

1 ν[z,zM ] δ z -1 * ν on (0, +∞) is vaguely compact when z goes to +∞.
There are two key arguments in the proof of this proposition. One is the following Lemma, that we will use several times in the sequel.

Lemma 3.3. Let ν be a positive µ-invariant measure on R. Then for any pair of intervals

V, U ⊂ R, ν(V ) ≥ P(T W < ∞) • ν(U ) where W = W(V, U ) = {ψ ∈ F | ψ(U ) ⊂ V } and T W is the stopping time defined by T W = inf{n ≥ 0 : Ψ 1 • • • Ψ n ∈ W}.
Proof. Observe that the backward process

M n = Ψ 1 • • • Ψ n * ν(V ) M 0 = ν(V )
is a positive martingale with respect to the filtration generated by the Ψ n . In fact

E(M n |F n-1 ) = Ψ 1 • • • Ψ n-1 * µ * ν(V ) = Ψ 1 • • • Ψ n-1 * ν(V ). Since (Ψ 1 • • • Ψ T W ) -1 (V ) ⊇ U ,
for any fixed n ∈ N, by the optional stopping time theorem,

ν(V ) = E(M T W ∧n ) ≥ E(1 {T W ≤n} Ψ 1 • • • Ψ T W * ν(V )) ≥ P(T W < n)ν(U ).
We let n go to infinity to conclude.

The other crucial observation is that the backward recursion

Ψ 1 • • • Ψ n (x)
is controlled by the right random walk R n on the affine group generated by the product of g i = (B i , A i ) (see (2.5)). More precisely, given g ∈ Aff(R) we denote by a(g) and b(g) its projections on R * + and R respectively, then

a(R n )x -b(R n ) ≤ Ψ 1 • • • Ψ n (x) ≤ a(R n )x + b(R n ).
We use these bounds to estimate the stopping time that appears in Lemma 3.3. In particular as an immediate consequence of the lemma above we obtain

Corollary 3.4. Let W = W (m 1 , m 2 , k 1 , k 2 ) = {(B, A) ∈ Aff(R) | Ak 2 + B ≤ m 2 ; Ak 1 -B ≥ m 1 } and T W = inf{n ≥ 0 : R n ∈ W }. Then we have ν(m 1 , m 2 ) ≥ P[T W < ∞]ν(k 1 , k 2 ).
Proof. The Corollary follows from Lemma 3.

3, taking U = [k 1 , k 2 ], V = [m 1 , m 2 ] and noticing that T W ≥ T W .
Since the potential theory of the affine group is well understood, we have enough tools to estimate P(T W < +∞) in many situations. For a continuous and compactly supported function f on Aff(R) we define the potential

U * δ g (f ) := E [ ∞ ∑ n=0 f (L n g) ] = E [ ∞ ∑ n=0 f (R n g) ] .
A renewal theorem for the potential U , i.e. description of its behavior at infinity, was given in [START_REF] Babillot | The random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF], where the authors proved that for all h ∈ C C (Aff(R)):

(3.5) lim a→0 U * δ (0,a) (h) = ν G ⊗ dx x (h),
for ν G being a suitable non-trivial multiple of the invariant measure of the process Y x n = L n (x) . Now we are ready to prove the following lemma. Lemma 3.6. Suppose (1.4), (1.5), (2.3) and (2.4). There exist a compact subset

V 0 = {(B, A) ∈ Aff(R) | |B| < b 0 , a -1 0 < A < a 0 } and a constant δ > 0 such that: (1) if W z = (0, z) • V 0 = {(B, A) | |B| < z b 0 , z a -1 0 < A < z a 0 }, then P(T Wz < ∞) > δ for all z ≥ 1; (2) if V z = V 0 • (0, z -1 ) = {(B, A) | |B| < b 0 , a -1 0 /z < A < a 0 /z}, then P(T Vz < ∞) > δ 1 + log z for all z ≥ 1. Proof. Step 1. First observe that for every V ⊂ Aff(R) (3.7) U (V -1 V )P(T V < ∞) ≥ U (V ).
In fact

U (V ) = ∞ ∑ n=0 P[R n ∈ V ] = E [ 1 {TV <∞} ∞ ∑ n=T V 1 {RT V R T V n ∈V } ] ≤ P(T V < ∞)U (V -1 V ) where R l n := R -1 l R n = g l+1 • • • g n .
Step 2: Proof of (1). By (3.7) we write (assuming the denominator is nonzero)

(3.8) P(T Wz < ∞) ≥ U (W z ) U (W -1 z W z ) = U ((0, z) • V 0 ) U (V -1 0 V 0 )
.

A simple calculation relates the right random walk on the affine group to the reversed left random walk Ln

= R -1 n = g -1 n • • • g -1 1 . Observe that for any V ⊂ Aff(R) we have U ((0, z)V ) = ∑ n P [ R n ∈ (0, z)V ] = ∑ n P [ R -1 n ∈ V -1 (0, z -1 ) ] = ∑ n P [ Ln (0, z) ∈ V -1 ] = Ȗ ( V -1 (0, z -1 ) ) ,
where Ȗ is the potential of the reversed random walk Ln . Since the law of g -1 n is also centered and verifies the hypotheses of [START_REF] Babillot | The random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF], there exists a unique Radon measure νG on R invariant under μG , the law of g -1 = (B, A) -1 . Then by (3.5)

lim z→+∞ U ((0, z)V ) = lim z→+∞ Ȗ ( V -1 (0, z -1 ) ) = ( νG × dx x ) (V -1 ).
We take sufficiently large V 0 such that

U (W -1 z W z ) = U (V -1 0 V 0 ) > 0 and ( νG × dx x (V -1 0 ) ) > 0
and, in view of (3.7), we conclude.

Step 3: Proof of (2). As in the previous step, by (3.7), we write (3.9)

P(T Vz < ∞) > U (V z ) U (V -1 z V z ) = U ( V 0 (0, z -1 ) ) U ( (0, z)V -1 0 V 0 (0, z -1
)

)
Now we have to estimate U (V z ) from below and U (V -1 z V z ) from above. The later is the most difficult part of the proof.

To deal with this second problem we decompose the centered random walk on the affine group in a contracting part and a dilating part using ladder stopping times. This key idea has been applied in several different ways in important works on the subject, for instance [START_REF] Grincevičjus | A central limit theorem for the group of linear transformations of the line. (Russian)[END_REF][START_REF] Élie | Comportement asymptotique du noyau potentiel sur les groupes de Lie[END_REF][START_REF] Page | A local limit theorem on the semi-direct product of R * + and R d[END_REF][START_REF] Babillot | The random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF]. We use here a potential version. Let {g i } be another sequence of i.i.d. elements of Aff(R) independent and of the same law as {g i }. We define S n , t k , l k as in (2.6) and (2.7). We claim that (3.10)

U (f ) = E [ ∞ ∑ n=0 f (L n ) ] = E [ ∞ ∑ k,i=0 f (R li L t k )
] .

In fact, for n > k define

L k n = g n • • • g k+1 and L k k = e. Observe that E [ ∞ ∑ n=0 f (L n ) ] = E [ ∞ ∑ k=0 t k+1 -1 ∑ i=t k f (L i ) ] = E [ ∞ ∑ k=0 E [ t k+1 -1 ∑ i=t k f (L t k i L t k ) L t k ]] .
Since for fixed k the sequence {L t k t k +i } i≥0 is independent of L t k and has the same law as {L i } i≥0 , by the duality lemma (see lemma 5.4 [START_REF] Brofferio | On the invariant measure of the random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF]) we have

E [ t k+1 -1 ∑ i=t k f (L t k i L t k )|L t k = g ] = E [ t1-1 ∑ i=0 f (L i g) ] = E [ ∞ ∑ i=0 f (R li g) ]
and we obtain (3.10).

Observe that S li (resp. S t k ) is a random walk with finite mean and negative (resp. positive) steps. Take a, b > 2 , then by (3.10) and the classical renewal theorem [START_REF] Feller | An introduction to probability theory and its application II[END_REF], we have

U ([-b, b] × [1/a, a]) = ∞ ∑ k,i=0 P [ b(R li L t k ) ≤ b; -log a ≤ S li + S t k ≤ log a ] = ∞ ∑ k,i=0 P [ e -S l i b(L t k ) + b(R li ) ≤ b; -log a ≤ S li + S t k ≤ log a ] ≤ ∞ ∑ k,i=0 P [ b(R li ) ≤ b; -log a ≤ S li + S t k ≤ log a ] since b(L t k ) ≥ 0 = ∞ ∑ i=0 E [ 1 [b(R l i )≤b] E [ ∞ ∑ k=0 1 {-log a≤S l i +St k ≤log a} g i , i ≥ 0 ]] ≤ C log a ∞ ∑ i=0 P [ b(R li ) ≤ b ]
Since we assume B ≥ 1 a.s., we have for i ≥ 1:

b(R li ) = b(R li-1 R li-1 li ) = e -S l i-1 b(R li-1 li ) + b(R li-1 ) ≥ e -S l i-1
That is

U ([-b, b] × [1/a, a]) ≤ C log a ( 1 + ∞ ∑ i=1 P [ S li-1 ≥ -log b ] ) ≤ C log a (1 + C log b) Therefore, since V -1 z V z ⊆ {(B, A) | |B| ≤ 2b 0 a 0 z, a -2 0 ≤ A ≤ a 2 0 }, we obtain U (V -1 z V z ) ≤ K log a 0 (1 + log z + log(2b 0 a 0 )).
To estimate U (V z ) from below as in the previous case, we just apply the renewal theorem (3.5). Plugging those estimates into (3.9), we conclude.

Proof of Proposition 3.1.

Step 1: Proof of (1). We apply Corollary 3.

4 with [k 1 , k 2 ] = [-z, z] and [m 1 , m 2 ] = [-2b 0 , 2b 0 ]
and consider, according to the notation there, the subset of Aff(R)

W (-2b 0 , 2b 0 , -z, z) = { g ∈ Aff(R) | g([-z, z]) ⊆ [-2b 0 , 2b 0 ] } = { (B, A) | Az + B < 2b 0 } .
This subset contains the set

V z = { (B, A) b -1 0 z < A < b 0 z , |B| < b 0 } .
We can apply Corollary 3.4 and, choosing b 0 large enough, Lemma 3.6.2 to conclude:

ν(-z, z) ≤ ν[-2b 0 , 2b 0 ] P(T Vz < ∞) < C 0 (1 + log z).
Step 2: Proof of (2). Take M > 1 and 0

< k 1 < k 2 . Set [m 1 , m 2 ] = [z, zM ]. Then by Corollary 3.4 ν[z, zM ] ≥ P(T Wz < ∞)ν[k 1 , k 2 ], where W z = W (z, zM, k 1 , k 2 ) = (0, z)W (1, M, k 1 , k 2 ) =: (0, z)W 1 .
Observe that if k 1 , M and M/k 2 tend to infinity, then

W 1 = {(B, A) | Ak 1 -B > 1, Ak 2 + B < M }
grows to Aff(R). Thus, there exists C > 0 such that if k 1 ≥ C,M > C and M/k 2 ≥ C, the set W 1 contains the compact set V 0 defined in Lemma 3.6. Therefore P(T Wz < ∞) is uniformly bounded from below for large values of z. Moreover, since we require the support of ν to be unbounded on the right, one can choose k 2 such that ν[k 1 , k 2 ] > 0 and we conclude.

Step 3: Proof of (3). Let a 0 , b 0 be sufficiently large numbers such that Lemma 3.6 holds. Take M > max{2, 4a 2 0 }. First suppose that u2 u1 < M

V (y) = { (B, A) ∈ Aff(R) | B < b 0 , 2 e y u 1 ≤ A ≤ M e y 2u 2 } .
Since ( M e y 2u2 )/( 2 e y u1 ) = M u1 4u2 > a 2 0 , we can apply Lemma 3.6 and prove that there exists

C > such that ν[e x+y u 1 , e x+y u 2 ] ν[e x , e x M ] ≤ 1 P(T V (y) < ∞) < C(1 + y) for all x > log b 0 , y > 0
By the previous steps, the last inequality is satisfied for 0 < x ≤ log b 0 and all y > 0.

For general

U = [u 1 , u 2 ] with u2 u1 ≥ M 4a 2 0
we can deduce (3.2) covering U with a finite number of small intervals.

Since the law of log A is aperiodic, proceeding as in [START_REF] Babillot | The random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF] and [START_REF] Brofferio | On the invariant measure of the random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF], one can prove that the family of quotient measures is asymptotically invariant under the action of R * + and converges to the Haar measure of R * + . Corollary 3.11. Under the hypotheses of Theorem 1.3

lim inf z→∞ δ z -1 * ν(ϕ) > 0,
where ϕ is an arbitrary nonzero and nonnegative element of C c (0, +∞).

Furthermore for ϕ 1 , ϕ 2 ∈ C c (0, +∞) and ϕ 2 not identically zero:

(3.12) lim z→∞ δ z -1 * ν(ϕ 1 ) δ z -1 * ν(ϕ 2 ) = ∫ R * + ϕ 1 (a) da a ∫ R * + ϕ 2 (a) da a . Therefore (3.13) lim x→+∞ δ e -(x+y) * ν(ϕ) δ e -x * ν(ϕ) = 1 and δ e -(x+y) * ν(ϕ) δ e -x * ν(ϕ) ≤ K ϕ (1 + y) for all x, y > 0.
In particular the function

L(z) = δ z -1 * ν(ϕ) is slowly varying.
Proof. For the reader's convenience we present a sketchy proof (see Proposition 2.2 [START_REF] Brofferio | On the invariant measure of the random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF] for more details). First take a Lipschitz function Φ whose compact support contains (1, M ) and let L(z) = δ z -1 * ν(Φ). Since the family of measures ν z = 1 L(z) δ z -1 * ν is vaguely compact, for every sequence we can extract its subsequence ν zn convergent to a limit measure η.

For every Lipschitz compactly supported function ϕ and Ψ ∈ F there exists a compact set

U = U (ϕ, Ψ) such that ϕ ( Ψ(u) z ) -ϕ ( Au z ) ≤ B z • 1 U ( Au z ) ,
and

lim n→∞ ∫ ϕ ( Ψ(u) zn ) ν(du) - ∫ ϕ ( Au zn ) ν(du) L(z n ) ≤ lim n→∞ C|z -1 n b|ν(a -1 z n U ) L(z n ) ≤ Cη(a -1 U ) • lim n→∞ |z -1 n b| = 0,
Thus the function

h(y) = δ y * η(ϕ) = lim n→∞ δ (0,z -1 n y) * G ν(ϕ) L(z n )
on R * + is superharmonic with respect to the action of µ A , the law A 1 . Since h is positive and continuous, by the Choquet-Deny theorem it must be a constant function, that is δ a * η(ϕ) = η(ϕ) for every a ∈ R * + . Because η(Φ) = 1, then η is a fixed multiple of the Haar measure of R * + and

lim z→+∞ δ z -1 * ν(ϕ) δ z -1 * ν(Φ) = ∫ ϕ(a) da a ∫ Φ(a) da a .
This proves (3.12) and (3.13). In particular, if ϕ is nonzero, by Proposition 3.1, we have lim inf

z→∞ δ z -1 * ν(ϕ) ≥ ∫ ϕ(a) da a • lim inf z→∞ δ z -1 * ν(Φ) > 0. Take k such that the support of ϕ is contained in [1/k, k]. Then e -(x+y) * ν(ϕ) e -x * ν(ϕ) ≤ ν[e x /M, e x M ] e -x * ν(ϕ) ν[e x+y /k, e x+y k] ν[e x /M, e x M ] ≤ K(1 + y),
because the first quotient is bounded.

Homogeneity at infinity

In this section we finish the proof of Theorem 1.3. The main idea of the proof is similar to our previous papers [START_REF] Brofferio | On the invariant measure of the random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF][START_REF] Buraczewski | On invariant measures of stochastic recursions in a critical case[END_REF][START_REF] Brofferio | On solutions of the affine recursion and the smoothing transform in the critical case[END_REF]. Given a nice function ϕ on R * + we define the function

f (x) = ∫ R * + ϕ(e -x u)ν(du).
Behavior at infinity of the measure ν is coded in the asymptotic behavior of f . To describe f we consider it as a solution of the Poisson equation

µ * R f (x) = f (x) + g(x)
where µ is the law of -log A and the function g is defined by the equation above. We cannot use the classical renewal theorem, since the measure µ is centered. In our previous papers we expressed f as a special potential of g. However this approach was technically involved and it was not possible to establish the optimal hypotheses. Here we apply ideas due to Durrett and Liggett [START_REF] Durret | Fixed points of the smoothing transformation[END_REF], who studying similar equation and applying the duality lemma, were able to reduce the problem to the classical renewal theorem. In Proposition 4.1, we determine weak assumptions in the terms of the Poisson equation that enable to control the asymptotic behavior of the solution.

In the second part of the section, we apply this result to our problem. We show that there exist slight perturbations of the functions f and g defined above which satisfy all the required conditions. Finally we deduce our main result proving that the tail of the measure ν converges at infinity. Proposition 4.1. Let µ be a centered probability measure on R with finite moment of order 2 + ϵ for some ϵ > 0 and let f be a continuous function on R such that

(4.2) 0 ≤ f (x) ≤ C(1 + x + ) and ∫ y -∞ f (x)dx ≤ C(1 + y + )
where x + := max{0, x}. Let g be the continuous function on R defined by the Poisson equation:

(4.3) µ * f (x) = f (x) + g(x).
Suppose also that g is directly Riemann integrable, then

(4.4) lim x→+∞ E[f (x + S t )] -f (x) = -1 E[S l ] ∫ R g(x)dx,
where S n is the random walk of law µ and t and l are the stopping times t = inf{n > 0 : S n ≥ 0} and l = inf{n > 0 : S n < 0}.

Moreover, if ∫ R g(x)dx = 0 and ∫ R |xg(x)|dx < ∞, (4.5) lim x→+∞ E [ ∫ x+St x f (z)dz ] = 1 E[S l ] ∫ R x g(x)dx.
The notion of directly Riemann integrable functions is fundamental in the renewal theory and allows to apply the classical renewal theorem to the function g (see for e.g. Feller [START_REF] Feller | An introduction to probability theory and its application II[END_REF]). The proof of this proposition will be given in Appendix A.

Let ν be a µ-invariant Radon measure on R. We would like to apply the previous proposition to the function f (x) = δ e -x * ν(ϕ) for some fixed positive function ϕ ∈ C 1 C (R * + ). Unfortunately we are not able to justify that f satisfies all the required hypotheses. The main reason is that we are not able to control local properties of a general measure ν, namely its behavior near 0. Thus the function f may not be sufficiently integrable at -∞. However it turns out that one can slightly translate the measure ν to overcome the problem.

For this purpose, given ϕ ∈ C 1 C (R * + ) and w 0 > 0 define

f ϕ (x) := ∫ R ϕ(e -x (u -w 0 ))ν(du), g ϕ (x) := µ * R f ϕ (x) -f ϕ (x).
Observe that f ϕ (x) = δ e -x * ν 0 (ϕ) where ν 0 is the measure ν translated by w 0 :

ν 0 (ϕ) = ∫ R ϕ(u -w 0 )ν(du),
i.e. the invariant measure of the SDS obtained by conjugating the original one with the translation by w 0 : ψ 0 (x) = ψ(x + w 0 ) -w 0 . Denote by µ 0 its law. Observe that A(ψ 0 ) = A(ψ) and we can choose B(ψ 0 ) = Aw 0 + w 0 + B, hence µ 0 satisfies our main hypotheses if µ does. Since the translation doesn't change the asymptotic behavior, the measures ν 0 and ν behave in the same way at +∞, namely :

(4.6) lim x→+∞ f ϕ (x) -δ e -x * ν(ϕ) = 0. In fact ∫ +∞ -∞ |ϕ(e -x (u -w 0 )) -ϕ(e -x u)|ν(du) ≤ C ∫ ∞ 0 |e -x w 0 |1 [e x m,e x (M +w0)] ν(du) ≤ C|e -x w 0 | log(e x (M + w 0 )) when supp(ϕ) ⊂ [m, M ].
Summarizing, translation of the invariant measure does not change the problem we study, nor our assumptions. Existence of a corresponding w 0 is provided by the following lemma, whose proof will be given in Appendix A.

Lemma 4.7.

There exists w 0 > 0 such that for all ϕ ∈ C 1 C (R * + ) the functions f ϕ and g ϕ satisfy the hypotheses of Proposition 4.1. Now we are ready to prove our main result.

Proof of Theorem 1.3. We claim that ∫ g ϕ (y)dy = 0. In fact for all y we can apply Corollary 3.11

lim x→+∞ f ϕ (x + y) f ϕ (x) = lim x→+∞ δ e -(x+y) * ν 0 (ϕ) e -x * ν 0 (ϕ) = 1;
thus, since E(S t ) is finite, by dominated convergence E(f ϕ (x + S t )/f ϕ (x)) also converges to 1. Fix ε > 0, then there exists x ε such that for all x ≥ x

E [ f ϕ (x + S t ) ] -f ϕ (x) + 1 ES t ∫ g ϕ (y)dy < ε and E[f ϕ (x + S t )] f ϕ (x) -1 < ε. Therefore f ϕ (x) ≥ | ∫ g ϕ (y)dy|/(εES t ) -1. Since by Lemma 4.7, ∫ x -∞ f ϕ (y)dy < C(1 + x), for all x > x ε > 0 C(1 + x) ≥ ∫ x xε f ϕ (y)dy ≥ ( ∫ g ϕ (y)dy εES t -1 ) (x -x ε ). That is ∫ R g ϕ (y)dy ≤ εES t ( lim inf x→+∞ C(1 + x) x -x ε + 1 ) = εES t (C + 1).
Letting ε ↘ 0, we conclude. In view of Corollary 3.11, the quotient f ϕ (x + y)/f ϕ (x) is uniformly dominated by 1 + S t for x > 0 and 0 < y < S t , thus

lim x→∞ ∫ St 0 f ϕ (x + y) f ϕ (x) dy = ∫ St 0 1 dy = S t . P a.s. By Fatou's lemma (4.8) lim inf x→∞ E [ ∫ St 0 f ϕ (x + y) f ϕ (x) dy ] ≥ E [ lim inf x→∞ ∫ St 0 f ϕ (x + y) f ϕ (x) dy ] = E[S t ]
Therefore by Proposition 4.1

lim sup x→∞ f ϕ (x) = lim sup x→∞ E [ ∫ St 0 f ϕ (x + y)dy ] E [ ∫ St 0 f ϕ (x+y) f ϕ (x) dy ] ≤ 1 E[S l ]E[S t ] ∫ R g ϕ (x)xdx
In particular this proves that f ϕ (x) is bounded above. Since by Corollary 3.11, we already know that f ϕ (x) is bounded below,

∫ St 0 f ϕ i (x+y)
f ϕ (x) dy < CS t . This allow to use the dominated convergence theorem instead of Fatou's Lemma in (4.8) and to replace the inferior limit with the real limit and the inequality with the equality. Thus we have

lim x→∞ f ϕ (x) = lim x→∞ E [ ∫ St 0 f ϕ (x + y)dy ] E [ ∫ St 0 f ϕ (x+y) f ϕ (x) dy ] = 1 E[S l ]E[S t ] ∫ R g ϕ (x)xdx = 1 σ 2 ∫ R g ϕ (x)xdx,
where σ 2 = ∫ x 2 µ(dx) (see [START_REF] Feller | An introduction to probability theory and its application II[END_REF] for the proof that

E[S l ]E[S t ] = σ 2 ).
To conclude take a nonzero nonnegative function Φ ∈ C 1 c (0, +∞). We have proved that the following limit exists lim

z→+∞ δ z -1 * ν(Φ) = lim x→+∞ f Φ (x) = C
and by Corollary 3.11 the constant C is strictly positive. The same corollary also implies that for all ϕ ∈ C c (0, +∞)

lim z→+∞ δ z -1 * ν(ϕ) = lim z→+∞ δ z -1 * ν(ϕ) δ z -1 * ν(Φ) lim z→+∞ δ z -1 * ν(Φ) = C ∫ R Φ(a) da a ∫ R ϕ(a) da a .

Uniqueness of the invariant measure

Proof of Theorem 1.9. Notice first that for any compact set K

lim n→∞ 1 K (X y n )|X y n -X y ′ n | ≤ |y -y ′ | lim sup n→∞ A 1 . . . A n 1 K (X y n ) = |y -y ′ | lim sup n→∞ X y n 1 K (X y n ) X y n A1•••An ≤ lim sup n→∞ C(K) X y n A1•••An . Thus it is sufficient to prove that lim n→∞ X y n A 1 • • • A n = +∞.
Notice that the sequence

X y n A1•••An in nondecreasing. Indeed, since Ψ n (X y n-1 ) ≥ A n X y n-1 , X y n A 1 • • • A n = Ψ n (X y n-1 ) A 1 • • • A n ≥ X y n-1 A 1 • • • A n-1 .
Therefore it is enough to justify that for arbitrary large fixed M > 0 the sequence is a.s. at least once greater than M . Let

U β,γ := {Ψ ∈ F|Ψ[0, +∞) ⊆ [β, +∞) and A(Ψ) < γ} and V α := {Ψ ∈ F|A(Ψ) < α} .
In view of our hypotheses there exist α < 1, β > 0, and γ such that these two sets have positive probability. For a fixed x 0 , take N > 0 such that α N -1 M γx 0 < β and let

ψ 0 = ψ 1 ψ 2 with ψ 1 ∈ U β,γ and ψ 2 ∈ V N -1 α
. We claim that (5.1)

ψ 0 (x) A(ψ 0 )x > M for all 0 ≤ x ≤ x 0 .
In fact

ψ 0 (x) = ψ 1 (ψ 2 (x)) ≥ β > M (γα N -1 x 0 ) > M A(ψ 1 )A(ψ 2 )x > M A(ψ 0 )x.
Observe that since X y n is recurrent, there exists x 0 > 1 such that P[0 ≤ X y n < x 0 i.o.] = 1 for every y ≥ 0. Let us fix y, x 0 and define a sequence T k of hitting times of [0, x 0 ]

T 0 = 0, T k = inf { n > T k-1 + N : X y n < x 0 } .
By recurrence, all T k are almost surely finite. Let Ψ j

i := Ψ j • • • • • Ψ i+1 , then {Ψ T k +N T k } is a sequence of i.i.d. random transformations distributed as µ N . Since µ N (U β,γ V N -1 α ) > 0 there exists almost surely k 0 such that Ψ T k 0 +N T k 0 ∈ U β,γ V N -1 α
. Then, by (5.1), we have:

X y T k 0 +N A 1 • • • A T k 0 +N = Ψ T k 0 +N T k 0 (X y T k 0 ) A 1 • • • A T k 0 +N ≥ Ψ T k 0 +N T k 0 (X y T k 0 )x 0 A T k 0 +1 • • • A T k 0 +N X y T k 0 > M.

Examples

In this section we present some of the more significant classes of stochastic dynamical system to which the results of the previous sections apply. [START_REF] Brofferio | On the invariant measure of the random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF].1. The random difference equation. The first example is naturally the SDS induced by random affinities, that is Ψ n (x) = A n x+B n , for a random pair (B n , A n ) ∈ R×R * + . Then X x n is given by formula (1.7). This process is called the random difference equation or the affine recursion. It is well known that under the assumptions of Theorem 1.3 this process is recurrent and locally contractive, thus it possesses a unique invariant Radon measure ν, see [START_REF] Babillot | The random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF]. Behavior of this measure at infinity was studied previously in [START_REF] Buraczewski | On invariant measures of stochastic recursions in a critical case[END_REF][START_REF] Brofferio | On the invariant measure of the random difference equation Xn = AnX n-1 + Bn in the critical case[END_REF][START_REF] Brofferio | On solutions of the affine recursion and the smoothing transform in the critical case[END_REF] under a number of additional strong hypotheses. Theorem 1.3 provides an optimal result, in the sense that the hypotheses implying existence and uniqueness of the invariant measure, are sufficient also to deduce that this measure must behave at infinity like Cdx x .

Stochastic recursions with unique invariant measure.

Our results can also be applied to a more general class of stochastic recursions that behave at infinity as Ax (i.e. Φ(x) ∼ Ax for large x). In the contracting case (E[log A] < 0) those recursions were studied by Goldie [START_REF] Goldie | Implicit renewal theory and tails of solutions of random equations[END_REF] (see also Mirek [21], who described this class of recursions in general settings, including more examples). Just to give some concrete examples let us mention that our results are valid (under rather obvious and easy to formulate assumptions) for the following examples

• Ψ 1,n (x) = max{A n x, B n } + C n , for A n , B n , C n > 0. • Ψ 2,n (x) = √ A 2 n x 2 + B n x + C n , for A n , B n , C n > 0 and ∆ = B 2 -4A 2 C
≤ 0 In both cases above the mappings Ψ i,n are Lipschitz with the Lipschitz coefficient equal to A. This is obvious for the first example. For the second one, denote

x 0 = -B 2A 2 , D = -∆ 4A 2 . Observe that since Ψ 2,n (x) = √ A 2 (x -x 0 ) 2 + D, its derivative Ψ ′ 2,n (x) = A 2 (x -x 0 ) √ A 2 (x -x 0 ) 2 + D = A 1 √ 1 + D A 2 (x-x0) 2 ↗ A
is an increasing function that tends to A. Hence, under appropriate moment assumptions, the SDS on R + generated by the random functions defined above satisfies assumptions of both Theorem 1.3 and Theorem 1.9. Therefore the corresponding random process possesses a unique invariant measure, which behaves at infinity like Cdx

x . If we do not suppose ∆ = B 2 -4A 2 C ≤ 0, then Ψ 2,n are still asymptotically linear functions to which Theorem 1.3 applies, but we cannot prove uniqueness of an invariant measure. 6.3. Random automorphisms of the interval [0, 1]. SDS's acting on the real line after conjugating by an appropriate function can be seen as random automorphisms of the interval [0, 1] fixing the end points. Our key property (AL) is translated in this setting into requiring that the automorphisms "reflect" at the same way in 0 and in 1, in the sense that the derivative in these two points has to be the same. The B term is then related to the term of order two at these end points (or order 2 -α, if we conjugate a SDS that satisfy (AL α )). More precisely Corollary 6.1. Consider a SDS on [0, 1] defined by random functions ϕ ∈ C([0, 1]) fixing 0 and 1, differentiable at the extremities of the interval and such that

ϕ ′ (0) = ϕ ′ (1) =: a ϕ . Let β 0 1 = inf u∈[0,1/2] (1 -ϕ(u)) > 0, β 0 2 = inf u∈[0,1/2] ϕ(u) u > 0, β 0 3 = sup u∈[0,1/2] ϕ(u) -a ϕ u u 2 < ∞. β 1 1 = inf u∈[1/2,1] ϕ(u) > 0, β 1 2 = inf u∈[1/2,1] 1 -ϕ(u) 1 -u > 0, β 1 3 = sup u∈[1/2,1] ϕ(u) -1 -a ϕ (u -1) (u -1) 2 < ∞. Suppose that E [ | log a ϕ | 2+ε ] < ∞, E [ | log β i k | 2+ε ]
< ∞, for some ε > 0, all i, k, and that E[log a ϕ ] = 0. Then the SDS on [0, 1] is conjugated to an asymptotically linear SDS on R that satisfy the hypotheses of Theorem 1.3. Therefore there exists at least one invariant Radon measure ν on (0, 1) and for every such a measure ν, which charges a neighborhood of 0, there exists a strictly positive constant C such that for all 0 < a < b < 1

lim z→+∞ ν(a/z, b/z) = C log b/a Proof. Let r(u) = - 1 u + 1 1 -u .
be a diffeomorphism of (0, 1) onto R. In the technical Lemma A.4, whose proof is postponed to the Appendix A, we prove that the conjugated function

Ψ ϕ = r • ϕ • r -1 satisfy (AL) for A(Ψ ϕ ) = 1/a ϕ and B(Ψ ϕ ) < C r ( (1 + a ϕ + β 0 3 ) a ϕ β 0 2 + 1 β 0 1 + (1 + a ϕ + β 1 3 ) a ϕ β 1 2 + 1 β 1 1
) ,

where C r depends only on the function r. Thus, under the hypotheses of the corollary, the conjugated SDS satisfies the assumptions of our main theorem. Let µ be the law of ϕ and µ = r * µ * r -1 be the law of the conjugated SDS on R. Then ν is a µ-invariant Radon measure on R if and only if ν = r -1 * ν is a µ-invariant Radon measure on (0, 1). Then by Theorem 1.3 and since |r(u)

+ 1/u| < 2 for 0 < u < 1/2, ν ( a z , b z ) -ν ( - z a , - z b ) = ν ( r ( a z ) , r ( b z )) -ν ( - z a , - z b ) ≤ ν ( - z a -2, - z a + 2 ) + ν ( - z b -2, - z b + 2
) → 0 for z → +∞. Thus lim z→+∞ ν ( a/z, b/z ) = lim z→+∞ ν ( - z a , - z b ) = C log b/a.
6.4. Additive Markov processes and power functions. When an asymptotically linear SDS is conjugated by a homeomorphism of the real line which behaves as the exponential at infinity, it is transformed into a SDS that is asymptotically a translation or, by the reversed conjugation, a power function.

More precisely consider a SDS generated by functions ϕ such that

(6.2) |ϕ(x) -x + sign(x)u ϕ | ≤ v ϕ e -|x|
for some constants u ϕ and v ϕ . This class contains mappings of [0, ∞) that are equal to translations outside a bounded set, that is a Markov additive process as defined in Aldous [1, sections C11, C33].

Let s be a continuous bijection of R such that s(x) = e x for x > 1 and s(x) = -e -x for x < -1.

Then the SDS generated by

ψ ϕ (x) = s • ϕ • s -1 satisfies hypothesis (AL) with A(ψ ϕ ) = e -u ϕ .
Hence, under moment conditions that can be obtained with standard calculations, if E(u ϕ ) = 0 there exists an invariant measure which behaves at infinity as the Lebesgue measure dx, i.e.

lim z→+∞ ν(α + z, β + z) = C(β -α),
for every measure of unbounded support, some constant C > 0 and all β > α.

In a similar way a SDS generated by function ϕ such that

|x| a • sign(x)e -b1 log(|x|+2) α ≤ ϕ(x) ≤ |x| a • sign(x)e +b1 log(|x|+2) α ,
for some α is associated to an α-asymptotically linear system by the reverse conjugation

ψ ϕ (x) = s -1 • ϕ • s and A(ψ ϕ ) = a.
Thus, if E(log a) = 0 and some moments are finite, for any invariant measure ν, whose support in unbounded on the positive half-line, there exists a strictly positive constant C such that for all 1 < α < β

lim z→+∞ ν(α z , β z ) = C log log β log α .
6.5. Population of Galton-Watson tree with random reproduction law. Consider the following model of reproduction of a population. Let {ρ ω |ω ∈ Ω} be the set of probability measures on the set of natural numbers N and λ(dω) be a probability law on Ω. At each generation a law of reproduction ρ ω is chosen according to λ(dω) and each individual j is replaced by r j offsprings, r j chosen according to the law ρ ω and independently from the other individuals. To prevent the extinction of the population a random immigration i ω it added to the population. More formally, if the population consists of x ∈ N individuals, the population of the following generation is

ψ ω,r (x) = i ω + x ∑ j=1 r j ,
where the reproduction law ω ∈ Ω is chosen according to λ(dω), r = {r j } j are i.i.d of law ρ ω and i ω is a random variable. If every generation is independent from the previous one then the evolution of the population is a SDS on R = N of law µ(dψ) = ⊗ρ ω (dr)λ(dω). If Er 2 1 < ∞, the law of iterated logarithm proves that the ψ ω,r are µ-almost surely α-asymptotically linear with an error of order x α for all α > 1/2 and

A(ψ) = A ω = ∫ N
rρ ω (dr) = average number of offspring per individual for ρ ω .

Unlike the classical Galton-Watson process, in our context A ω is not constant, but varies from one generation to another. The key parameter, that decides whether the system is recurrent, is E(log A ω ) = ∫ log A ω λ(dω). To apply Theorem 1.3, we need to control moments of B(ψ). The details are stated in the following lemma. Our estimates are fairly rough and the hypotheses could be probably improved, but this go beyond the purpose of our paper. then there exists a finite constant C α that only depends on α such that

E((log + B(ψ)) 2+ϵ ) ≤ C α ( 1 + E((log + i ω ) 2+ϵ ) + E(r 4 1 )
)

Proof. Observe first

|ψ ω,r (x) -A ω x| x α + 1 ≤ i ω + ∑ x j=1 (r j -A ω ) x α + 1 . Thus (log + B(ψ)) 2+ϵ ≤ C ( (C + log + i ω ) 2+ϵ + sup x∈N ( log + ∑ x j=1 (r j -A ω ) x α + 1 ) 2+ϵ )
Let y j := r j -A ω be centered random variables. For a fixed reproduction law ω denote by P ω the quenched probability. Since under P ω the variables y j are independent, E ω (y j1 y j2 y j3 y j4 ) = 0 if there exists an index j k that is different from all the others. Then standard calculus shows that

E ω [ x ∑ j=1 y j ] 4 = x ∑ j1,j2,j3,j4=1 E ω (y j1 y j2 y j3 y j4 ) = xE ω (y 4 1 ) + 3x(x -1) ( E ω [y 2 1 ] ) 2 ≤ 4x 2 E ω (y 4 1 )
Finally, since α > 3/4, we have

E ( sup x∈N ( log + ∑ x j=1 (r j -A ω ) x α + 1 ) 2+ϵ ) ≤ CE ( ∑ x∈N ( ∑ x j=1 y j x α + 1 ) 4 ) = CE ( ∑ x∈N E ω [ ∑ x j=1 y j 4 ] (x α + 1) 4 ) ≤ CE ( ∑ x∈N 4x 2 E ω (y 4 1 ) (x α + 1) 4 ) = C ∑ x∈N 4x 2 E(y 4 1 ) (x α + 1) 4 < ∞.
6.6. Reflected random walk in critical case. The reflected random walk

Y x n = Y x n-1 -u n
, is an example of an asymptotic translation for which (6.2) holds. Thus we can apply our main Theorem 1.3. However, in this case the same results hold under weaker hypotheses and a much more direct proof. We give here the proof of Theorem 1.11, stated in the introduction.

Proof of Theorem 1.11. Define the upward ladder times of S n = ∑ n i=1 u i : t 0 = 0,

t k+1 = inf{n > t k : S n ≥ S t k },
and put u k = S t k -S t k-1 . Then {u k } is a sequence of i.i.d. random variables and every u k is equal in distribution to S t1 . We define reflected random walk for {u k }:

Y x 0 = x, Y x k+1 = Y x k -u k+1 , then Y x k = Y x t k .
In view of the result of Chow and Lai [START_REF] Chow | Moments of ladder variables for driftless random walks[END_REF], E(u k ) 1 2 < ∞ and this is sufficient for existence of a unique invariant probability measure ν t of the process {Y x k } (see [START_REF] Peigné | Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity[END_REF] for more details). Let us define the measure

ν 0 (f ) = ∫ R+ E [ t-1 ∑ n=0 f (Y x n ) ] ν t (dx).
Notice first that this is a Radon measure. Indeed, define [START_REF] Chow | Moments of ladder variables for driftless random walks[END_REF]). Take any f ∈ C C (R + ), then by the duality Lemma [START_REF] Feller | An introduction to probability theory and its application II[END_REF] (6.4)

l i = inf{n > l i-1 : S n < S li-1 }. Since E(u - 1 ) 2 < ∞, -∞ < ES l < 0 (see
ν 0 (f ) = ∫ R+ E [ t-1 ∑ n=0 f (x -S n ) ] ν t (dx) = ∫ R+ E [ ∞ ∑ n=0 f (x -S ln ) ] ν t (dx).
By the renewal theorem

E [ ∞ ∑ n=0 f (x -S ln ) ] ≤ CE [ #n : α < x -S ln < β ] ≤ C|β -α|, therefore ν 0 (f ) is finite and thus ν 0 is a Radon measure Next, since µ t * ν t = ν t , we have µ * ν 0 (f ) = ∫ R+ ∫ R E [ t-1 ∑ n=0 f (Y |x-y| n ) ] µ(dy)ν t (dx) = ∫ R+ E [ t ∑ n=1 f (Y x n ) ] ν t (dx) = ν 0 (f ).
Therefore ν 0 is a µ invariant Radon measure, so ν 0 = Cν and without any loss of generality we may assume ν = ν 0 . Finally, by (6.4), the Lebesgue theorem and the renewal theorem

lim z→∞ ∫ R+ f (u -z)ν(du) = lim z→∞ ∫ R+ E [ ∞ ∑ n=0 f (x -S ln -z) ] ν t (dx) = 1 -ES l ∫ R+ f (x)dx
and the Theorem is proved.

Appendix A. Proofs of technical results

In this Appendix we give the postponed proofs of the technical results stated in Lemma 2.1, Proposition 4.1, Lemma 4.7. At the end we formulate and prove Lemma A.4, which is used in Section 6.3.

Proof of Lemma 2.1. We will prove the result only for positive x, since for negative values of x the same argument is valid just by conjugating with the map x → -x.

Suppose first x ≥ 1. We have

r ( A α r -1 (x) -B α (1 + |r -1 (x)| α ) ) ≤ ψ r (x) ≤ r ( A α r -1 (x) + B α (1 + |r -1 (x)| α ) ) r ( A α x 1 1-α -B α (1 + x α 1-α ) ) ≤ ψ r (x) ≤ r ( A α x 1 1-α + B α (1 + x α 1-α ) ) r ( A α x 1 1-α -B α c α x α 1-α ) ≤ ψ r (x) ≤ r ( A α x 1 1-α + B α c α x α 1-α )
where c α only depends on α. Suppose further

x > c α B α /A α , then A α x 1 1-α -B α c α x α 1-α > 0 and ( A α x 1 1-α -B α c α x α 1-α ) 1-α ≤ ψ r (x) ≤ ( A α x 1 1-α + B α c α x α 1-α ) 1-α A 1-α α x 1-α 1-α -A -α α x -α 1-α B α c α x α 1-α ≤ ψ r (x) ≤ A 1-α α x 1-α 1-α + (1 -α)A -α α x -α 1-α B α c α x α 1-α A 1-α α x -A -α α B α c α ≤ ψ r (x) ≤ A 1-α α x + (1 -α)A -α α B α c α since for x 0 > 0 and h > 0, by concavity (x 0 + h) 1-α ≤ x 1-α 0 + (1 -α)x -α 0 h and (x 0 -h) 1-α ≥ x 1-α 0 -x -α 0 h. Hence we proved the lemma for x > max{1, c α B α /A α } Now, for x < 1 r ( A α x 1 1-α -B α (1 + x α 1-α ) ) ≤ ψ r (x) ≤ r ( A α x 1 1-α + B α (1 + x α 1-α ) ) -(2B α ) 1-α ≤ ψ r (x) ≤ (A α + 2B α ) 1-α and for x ≤ c α B α /A α . - ( B α ( 1 + ( c α B α A α ) α 1-α )) 1-α ≤ ψ r (x) ≤ ( A α ( c α B α A α ) 1 1-α + B α ( 1 + ( c α B α A α ) α 1-α )) α 1-α
Hence the lemma follows.

Proof of Proposition 4.1.

Step 1. Let t k and l k be the stopping times defined in (2.7). Let U l be the potential of the random walk S l k and let

R(x) := ∞ ∑ k=0 E [ g(x + S l k ) ] = U l (δ x * R g).
Since the function g is directly Riemann integrable and -∞ < ES l < 0, the function R is well defined and finite for every x. Notice also that by the duality lemma [START_REF] Feller | An introduction to probability theory and its application II[END_REF] (A.1)

R(x) = ∞ ∑ k=0 E [ g(x + S l k ) ] = E [ t-1 ∑ k=0 g(x + S k )
] .

Step 2. We claim that

(A.2) E [ f (x + S t ) ] -f (x) = E [ t-1 ∑ k=0 g(x + S k ) ] = R(x).
Indeed, the process f (x + S n ) -∑ n-1 k=0 g(x + S k ) forms a martingale (for this purpose one just has to iterate the Poisson equation (4.3)). Then for any fixed n, T ∧ n is a bounded stopping time, therefore by the optional stopping time theorem we have

f (x) = E [ f (x + S t∧n ) ] -E [ (t∧n)-1 ∑ k=0 g(x + S k )
] .

To justify that we can let n tend to infinity and change the order of the limit and the expected value to obtain (A.2) observe that

E [ f (x + S t∧n ) ] ≤ CE [ 1 + (x + S t∧n ) + ] ≤ CE [ 1 + (x + S t ) + ] < ∞.
The second term is uniformly dominated in n by

E [ t-1 ∑ k=0 |g|(x + S k ) ] = ∞ ∑ k=0 E [ |g|(x + S l k ) ] < ∞,
therefore converges to R(x) when n goes to infinity. This proves that

E [ f (x + S t ) ] -f (x) = R(x) = U l (δ x * R g)
and by the renewal theorem we obtain (4.4).

Step 3. Let

G(x) := ∫ x -∞ g(z)dz. If we suppose ∫ g(x)dx = 0 then G(x) = ∫ +∞ -∞ g(z)dz - ∫ ∞ x g(z)dz = - ∫ +∞ x g(z)dz. Thus |G(x)| ≤ ∫ x -∞ |g(z)|dz 1 (-∞,0] (x) + ∫ ∞ x |g(z)|dz 1 [0,+∞) (x) =: G 1 (x) + G 2 (x),
and G is directly Riemann integrable since functions G i are monotone and integrable on their support:

∫ 0 -∞ G 1 (x)dx = ∫ +∞ -∞ ∫ +∞ -∞ 1 [z<x<0] |g(z)|dx dz = ∫ 0 -∞ |zg(z)| dz < ∞ ∫ +∞ 0 G 2 (x)dx = ∫ +∞ -∞ ∫ +∞ -∞ 1 [z>x>0] |g(z)|dx dz = ∫ +∞ 0 |zg(z)| dz < ∞. Furthermore ∫ ∞ -∞ G(x)dx = ∫ +∞ -∞ ∫ +∞ -∞ 1 [z<x<0] g(z)dx dz - ∫ +∞ -∞ ∫ +∞ -∞ 1 [z>x>0] g(z)dx dz = - ∫ +∞ -∞ zg(z)dz.
Step 4. By the renewal theory, U l (δ x * R G) is well defined and by Fubini's thoerem

∫ x -∞ R(z)dz = ∫ 0 -∞ ∫ x -∞ g(z + u)dzU l (du) = ∫ 0 -∞ ∫ x+u -∞ g(z)dzU l (du) = U l (δ x * R G).
On the other hand

∫ x -∞ R(z)dz = E [ ∫ x -∞ f (z + S t )dz - ∫ x -∞ f (z)dz ] = E [ ∫ x+St x f (z)dz
] .

In fact the two integrals above are finite because by our hypotheses

∫ x -∞ E[f (y + S t )]dy = E [ ∫ x+St -∞ f (y)dy ] ≤ CE[1 + (x + S t ) + ]
and ES t < ∞ since µ has moment of order 2 + ϵ, see [START_REF] Chow | Moments of ladder variables for driftless random walks[END_REF]. Thus we proved

E [ ∫ x+St x f (z)dz ] = δ x * R U l (G)
and we can conclude using again the renewal theorem.

Proof of Lemma 4.7.

Step 1. Let 0 < γ < 1, then the set of v > 0 such that the function u → (u -v) -γ is ν-integrable on (v, +∞) is of full Lebesgue measure. In fact for any interval

[a, b] ⊂ (0, ∞): ∫ b a (∫ ∞ v (u -v) -γ ν(du) ) dv = ∫ 2b a ( ∫ u∧b a (u -v) -γ dv ) ν(du) + ∫ ∞ 2b ( ∫ u∧b a (u -v) -γ dv ) ν(du) ≤ ∫ 2b a ( ∫ 2b-a 0 w -γ dw ) ν(du) + ∫ ∞ 2b (∫ u-a u-b w -γ dw ) ν(du) = C + ∫ ∞ 2b (u -b) -γ (b -a)ν(du) < ∞ Take w 0 such that ∫ ∞ w0 (u -w 0 ) -γ ν(du) < ∞ then f ϕ (x) = ∫ ∞ w0 ϕ(e -x (u -w 0 ))ν(du) ≤ C ∫ ∞ w0 e γx (u -w 0 ) -γ ν(du) ≤ Ce γx ,
this gives good estimates of f ϕ for negative x's.

Step 2. Let supp(ϕ) ⊂ [m, M ]. By Proposition 3.1 the tail of ν is at most logarithmic, therefore for x ≥ 0,

f ϕ (x) ≤ ν([e x m + w 0 , e x M + w 0 ]) ≤ ν([e x m, e x (M + w 0 )]) ≤ C(1 + x). and ∫ x -∞ f ϕ (y)dy ≤ C ∫ R ∫ ∞ -∞ 1 [y<x] 1 [m,M ] (e -y (u -w 0 ))dyν(du) ≤ C ∫ R 1 [w0<u≤e x (M +w0)] log M m ν(du) ≤ C(1 + x + )
This proves (4.2).

Step 3. We need to justify that g ϕ = µ * f ϕ -f ϕ is directly Riemann integrable and moreover ∫ R |xg(x)|dx < ∞. We recall first that, since g is continuous, to prove that it is directly Riemann integrable is sufficient to show that |g| is dominated on (-∞, 0] (resp. on [0, +∞) ) by an integrable nondecreasing (resp. nonincreasing) function. For x < 0 : ) . ) .

µ * f ϕ (x) = ∫ +∞ -∞ f ϕ (x + y)µ(dy) = ∫ -x/2
Before we prove the last inequality, let us check how it implies the lemma. Since log + (z + 1)ρ(z) ≤ C/(1 + z) Lemma A.4. Let ϕ ∈ C([0, 1]) be a function fixing 0 and 1, derivable at 0 and 1 and such that ϕ ′ (0) = ϕ ′ (1) =: a ϕ . Suppose: ϕ(u) -a ϕ u u 2 < ∞.

β 0 1 = inf u∈[0,1/2]
β 1 1 = inf u∈[1/2,1] ϕ(u) > 0, β 1 2 = inf u∈[1/2,1] 1 -ϕ(u) 1 -u > 0, β 1 3 = sup u∈[1/2,1] ϕ(u) -1 -a ϕ (u -1) (u -1) 2 < ∞.
Consider the diffeomorphism of (0, 1) on R

r(u) = - 1 u + 1 1 -u .
Then Ψ ϕ = r • ϕ • r -1 satisfy (AL) for A(Ψ ϕ ) = 1/a ϕ and

B(Ψ ϕ ) < C r ( (1 + a ϕ + β 0 3 ) a ϕ β 0 2 + 1 β 0 1 + (1 + a ϕ + β 1 3 ) a ϕ β 1 2 + 1 β 1 1
) ,

where C r depends only on the function r.

Proof. Since the function r satisfies r(u) = -r(1 -u) and our assumptions on ϕ near 0 and 1 are symmetric, it is sufficient to prove the condition (AL) only for negative x. Since β 0 3 < ∞, by the Taylor expansion we have ) for x → -∞ For x < 0 we write

x a ϕ -Ψ ϕ (x) = x a ϕ -r ( ϕ ( r -1 (x) )) ≤ x a ϕ + 1 ϕ ( r -1 (x) ) + 1 1 -ϕ ( r -1 (x) )
Notice that for x < 0, r -1 (x) ∈ (0, 1/2), therefore the second factor can be bounded by 1

β 0 1
. So, we need just to estimate the first term. We write

I(x) = x a ϕ - 1 ϕ ( r -1 (x) ) = |ϕ(r -1 (x))x -a ϕ | |a ϕ • ϕ(r -1 (x))|
Take M = -r(1/10), then for x ∈ [-M, 0] we have ϕ(r -1 (x)) ≥ β 0 2 r -1 (x) ≥ β 0 2 /10 and hence

I(x) ≤ 10 M + a ϕ a ϕ β 0 2 .
Now we consider x < -M . Since there exists η such that xr -1 (x) > η, by (A.5) and (A.6), we have

I(x) = ϕ(r -1 (x))x + a ϕ • |x| a ϕ • ϕ(r -1 (x)) r -1 (x) • |xr -1 (x)| ≤ 1 a ϕ β 0 2 η • ϕ(r -1 (x))x + a ϕ |x|

4a 2 0.

 2 Take [m 1 , m 2 ] = [e x , e x M ] and [k 1 , k 2 ] = [e x+y u 1 , e x+y u 2 ]. For x > log(b 0 ), the set W (e x , e x M, e x+y u 1 , e x+y u 2 ) = { (B, A) ∈ Aff(R) | Ae x+y u 2 + B ≤ e x M ; Ae x+y u 1 -B ≥ e x } contains the set

Lemma 6 . 3 .

 63 Suppose (dr)λ(dω) < ∞. Let α > 3/4 andB(ψ) = B ω,r = sup x∈N |ψ ω,r (x) -A ω x| x α + 1 ,

2 K( 1 +-x/ 2 K( 1 + 2 K( 1 + 1 +( 1 +Step 4 .

 212121114 (x + y) + )µ(dy) ≤ Ce γ(x/2) + ∫ ∞ |y|)µ(dy) = Ce γ(x/2) |y|)|y| 1+ε µ(dy) ≤ C |x| 1+ε ,since µ has a moment of order 2 + ε. Thus g ϕ 1 (-∞,0] is directly Riemann integrable. Furthermore ∫ 0 |y| + |y| 2 )µ(dy) < ∞ To check that g ϕ is directly Riemann integrable and |xg ϕ (x)| is integrable for positive x we show that:∞ ∑ n=0 sup n≤x<n+1 |xg ϕ (x)| < ∞.Applying µ 0 invariance of ν 0 and since A(ψ 0 ) = A(ψ), we obtain|g(x)| = ∫ ∫ ϕ ( e -x (A(ψ)u) ) -ϕ ( e -x (ψ(u)) ) ν 0 (du)µ 0 (dψ)The function ϕ(x) = ϕ(e x ) is a Lipschitz on R, hence: the convention that log z = -∞ for z ≤ 0 and ρ(y) := min{C 1 y , 2∥ϕ∥ ∞ }. Take now 0 ≤ n ≤ x < n + 1 |x| ϕ ( e -x (Au) ) -ϕ ( e -x ψ(u) ) ≤ log + Au + B m

(

  last expression we use the fact that there exists a constant C such that for all non-increasing functions h : [0, +∞) → [0, +∞) and all M > 0 du) ≤ C(1 + log + M )

( 1 -

 1 ϕ(u)) > 0,

(A. 5 ) 1 x 2

 512 ϕ(u) = au + ϵ ϕ (u) with |ϵ ϕ (u)| ≤ β 0 3 u 2 for u ≤ 1/2. Moreover simple calculus shows that (A.6) r -1 (x) = -1 x + ϵ r -1 (x) with ϵ r -1 (x) = O (

3 (

 3 ϕ r -1 (x) x + ε ϕ (r -1 (x)) x + a ϕ |x| = 1 a ϕ β 0 2 η a ϕ ε r -1 (x)x + ε ϕ ( r -1 (x) ) x ≤ a ϕ ε r -1 (x)x + β 0

  B) 2 + log + B log + A ). last expression is µ 0 -integrable and we conclude. Finally to prove (A.3) we write ∫

	1/2 for z > 0, by (A.3), we have
	∫ ( log + 1 m ≤ C(1 + log + (B/A)) + log + B + log + ( ( (1 + log + B) + Au B + 1) ) ∫ +∞ ρ( Au B 1 ( (1 + log + B)ρ(z) + )ν 0 (du) ≤ C ( 1 + (log + The R 1 (1 + z) 1/2 h ( |u|/M ) ν 0 (du) ≤ ∥h∥ ∞ ν 0 ([-M e, M e]) + ∫ 1 [|u|>eM ] h ( |u|/M ) ν 0 (du) ) (1 + log + (z))	dz z	)
	≤ C(1 + log + M )∥h∥ ∞ +	n=1 ∫ ∞ ∫ e n e n-1 1 ( log + (z) + log + M + 3 ( log + (ze 2 M ) + 1 ) ) h(z) h(z) dz z dz z	.

≤ C(1 + log + M )∥h∥ ∞ + ∞ ∑ n=1 ∫ 1 [e n+1 M ≥|u|>e n M ] ν 0 (du)h(e n ) ≤ C(1 + log + M )∥h∥ ∞ + ∞ ∑ n=1 ( log + (e n+1 M ) + 1 ) h(e n ) ≤ C(1 + log + M )∥h∥ ∞ + ∞ ∑

This research project has been partially supported by Polonium project 7946/2010. D. Buraczewski was partially supported by the NCN grant DEC-2012/05/B/ST1/00692. S. Brofferio was partially supported by the ERC starting grant GA 257110 "RaWG" .