Éric Olivier
email: eric.olivier@univ-amu.fr

Alain Thomas
email: alain-yves.thomas@laposte.net

PROJECTIVE CONVERGENCE OF COLUMNS FOR INHOMOGENEOUS PRODUCTS OF MATRICES WITH NONNEGATIVE ENTRIES

Keywords: 1991 Mathematics Subject Classification. 15B48, 28A12. Key words and phrases. Infinite products of nonnegative matrices, Multifractal analysis, Bernoulli convolutions

Let Pn be the n-step right product A1 • • • An, where A1, A2, . . . is a given infinite sequence of d×d matrices with nonnegative entries. In a wide range of situations, the normalized matrix product Pn/ Pn does not converge and we shall be rather interested in the asymptotic behavior of the normalized columns PnUi/ PnUi , where U1, . . . , U d are the canonical d × 1 vectors. Our main result in Theorem A gives a sufficient condition (C) over the sequence A1, A2, . . . ensuring the existence of dominant columns of Pn, having the same projective limit V : more precisely, for any rank n, there exists a partition of {1, . . . , d} made of two subsets Jn = ∅ and J c n such that each one of the sequences of normalized columns, say PnUj n / PnUj n with jn ∈ Jn tends to V as n tends to +∞ and are dominant in the sense that the ratio PnU j n / PnUj n tends to 0, as soon as j n ∈ J c n . The existence of sequences of such dominant columns implies that for any probability vector X with positive entries, the probability vector PnX/ PnX , converges as n tends to +∞. Our main application of Theorem A (and our initial motivation) is related to an Erdős problem concerned with a family of probability measures µ β (for 1 < β < 2 a real parameter) fully supported by a subinterval of the real line, known as Bernoulli convolutions. For some parameters β (actually the so-called PV-numbers) such measures are known to be linearly representable: the µ β -measure of a suitable family of nested generating intervals may be computed by means of matrix products of the form PnX, where An takes only finitely many values, say A(0), . . . , A(a), and X is a probability vector with positive entries. Because, An = A(ξn), where ξ = ξ1ξ2 • • • is a sequence (one-sided infinite word) with ξn ∈ {0, . . . , a}, we shall write Pn = Pn(ξ) the dependence of the n-step product with ξ: when the convergence of Pn(ξ)X/ Pn(ξ)X is uniform w.r.t. ξ, a sharp analysis of the measure µ β (Gibbs structures and multifractal decomposition) becomes possible. However, most of the matrices involved in the decomposition of these Bernoulli convolutions are large, sparse and it is usually not easy to prove the condition (C) of Theorem A. To illustrate the technics, we consider one parameter β for which the matrices are neither too small nor too large and we verify condition (C): this leads to the Gibbs properties of µ β .

1. Introduction 1.1. Generalities. Given A = (A 1 , A 2 , . . . ) an infinite sequence of d × d matrices we consider the right products P n = A 1 . . . A n . Among the numerous ways for studying such products, the problem of the (entrywise) convergence of the sequence P 1 , P 2 , . . . itself (notion of Right Convergent Product) is solved by Daubechies and Lagarias [START_REF] Daubechies | Sets of matrices all infinite products of which converge[END_REF][START_REF] Daubechies | Corrigendum/addendumto: Sets of matrices all infinite products of which converge[END_REF]. The probabilistic approach is exposed in the book of Bougerol and Lacroix [START_REF] Bougerol | Products of Random Matrices with Applications to Schrődinger Operators[END_REF] with a large range of results about the products of random matrices. Given a probability measure µ on the set of complex-valued d × d matrices, [4, Part A III Theorem 4.3] gives two sufficient conditions for the convergence in probability of the normalized columns of P n to a random vector, as well as for the almost sure convergence to 0 of the angle between any couple of rows of P n . The first condition (strong irreducibility in [4, Part A III Definition 2.1]) means the non existence of a reunion of proper subspaces of C d being stable by left multiplication by each element in the support of µ. The second condition the n-step product with ξ: when the convergence of P n (ξ)X/ P n (ξ)X is uniform w.r.t. ξ, a sharp analysis of the Bernoulli convolution µ β (Gibbs structures and multifractal decomposition) becomes possible.

The paper is organized as follows. Section 2 is devoted to the presentation of the background necessary to establish Theorem A, the proof of the theorem itself being detailed in Section 3. We illustrate in Section 4 many aspects of Theorem A through several elementary examples. In Section 6, we show to what extent Theorem A may be used to analyse the Gibbs properties of the linearly representable measures : we give two examples, the first one (Section 7) being the Kamae measure, related with a self-affine graph studied by Kamae [START_REF] Kamae | A characterization of self-affine functions[END_REF], the second one (Section 8) being concerned with the Bernoulli convolutions µ β already mentioned. In the latter case, most of the matrices involved in the linear decomposition of µ β are large, sparse and it is usually not easy to verify condition (C) of Theorem A. To illustrate the technics involved, Section 8 is devoted to the PV-number β ≈ 1.755 s.t. β 3 = 2β 2 -β + 1, leading to (1)

A(0) :=          
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0

          A(1) :=          
0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

          A(2) :=          
1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

         
In Section 9, we verify (in details) that condition (C) holds for (most of) the products P n (ξ) = A(ξ 1 ) • • • A(ξ n ), where A(0), A(1) and A(2) are the matrices in (1) related to µ β , so that we are able to establish the Gibbs properties of this measure.

A complete bibliography about infinite products of matrices can be found in the paper of Victor Kozyakin [START_REF] Kozyakin | An annotated bibliography on convergence of matrix products and the theory of joint/generalized spectral radius[END_REF].

Acknowledgement. -Both authors are grateful to Ludwig Elsner and collaborators for their comments on a preliminary version of the present work; in particular this has incitated us to clarify the relations between Theorem A and the rank 1 asymptotic approximation of the normalized matrix products under condition (C) (See Section 5).

1.2. Statement of Theorem A. Given a r × 1 vector V , we note [START_REF] Bezhaeva | Oseledets: Erdős measures, sofic measures, and Markov chains[END_REF] I(V ) := 1 ≤ i ≤ r ; V (i) = 0 and ∆(V ) := i∈I(V )

U i ;

the inclusion I(V ) ⊂ I(V ) is thus equivalent to the inequality ∆(V ) ≤ ∆(V ). Major ingredients for Theorem A are the sets H 1 , H 2 (Λ) and H 3 (λ) (for λ ≥ 0 and Λ ≥ 1), whose elements are d 1 × d 2 matrices and respectively defined by setting

A ∈ H 1 ⇐⇒ ∀j 0 , j 1 , ∆(AU j 0 ) ≥ ∆(AU j 1 ) or ∆(AU j 0 ) ≤ ∆(AU j 1 ) ;

A ∈ H 2 (Λ) ⇐⇒ A(i 0 , j 0 ) = 0 =⇒ AU j 0 ≤ ΛA(i 0 , j 0 ) ; (4)

A ∈ H 3 (λ) ⇐⇒
A(i 0 , j 0 ) = 0, A(i 0 , j 1 ) = 0 =⇒ AU j 1 ≤ λA(i 0 , j 0 ) . [START_REF] Bowen | Some systems with unique equilibrium states[END_REF] For any nonnegative matrix A = 0, the following two constants are well defined: [START_REF] Boyle | Hidden Markov processes in the context of symbolic dynamics[END_REF] Λ A := min Λ ≥ 1 ; A ∈ H 2 (Λ) and λ A := min λ ≥ 0 ; A ∈ H 3 (λ) .

Definition 1.1. Let A = (A 1 , A 2 , . . . ) be a sequence of d×d matrices and 0 = s 0 = s 1 < s 2 < . . . a sequence of integers; given n ≥ 0 there exists k = k(n) ≥ 0 s.t. s k+1 ≤ n < s k+2 and we note

P n := A 1 • • • A n and Q n := A s k +1 • • • A n ,
where by convention P 0 = Q 0 is the d × d identity matrix; hence, for any n ≥ 0, ( 7)

P n = P s k Q n and k ≥ 1 =⇒ P s k = Q s 1 • • • Q s k .
The sequence A satisfies condition (C) w.r.t. 0 ≤ λ < 1 ≤ Λ < +∞ and (s 0 , s 1 , . . . ) if Theorem A. Let A = (A 1 , A 2 , . . . ) be a sequence of d × d matrices satisfying condition (C); then, there exist H probability vectors V 1 , . . . , V H (1 ≤ H ≤ d) with

(8) n ≥ s 2 =⇒ Q n ∈ H 1 ∩ H 2 (Λ) ∩ H 3 (λ).
(9) ∆(V 1 ) ≥ • • • ≥ ∆(V H ) while V h-1 = V h (for any 1 < h ≤ H)
and there exists ∅ = J h (n) ⊂ {1, . . . , d} (1 ≤ h ≤ H and n ∈ N large enough), giving the partition [START_REF] Denker | Ergodic Theory on Compact Spaces[END_REF] (i, j) ; P n (i, j) = 0 = H h=1 I h × J h (n), (where

I h := I(V h ))
for which the following assertions hold: (i) : for 1 ≤ h ≤ H and j 1 , j 2 , . . . a sequence in {1, . . . , d} s.t. j n ∈ J h (n), then lim n→+∞ P n U jn / P n U jn = V h ;

(ii) : for 1 < h ≤ H and j 1 , j 2 , . . . and j 1 , j 2 , . . . two sequences in {1, . . . , d},

(j n , j n ) ∈ J h-1 (n) × J h (n) =⇒ lim n→+∞ P n U j n / P n U jn = 0 ;
(iii) : there exists real numbers ε 1 , ε 2 , • • • with ε n → 0 as n → +∞ such that, for any X ∈ S d for which each P n X = 0, one has for any n ≥ 1: [START_REF] Dumont | Number of representations related to a linear recurrent basis[END_REF] P

n X P n X -V h X (n) ≤ Λ X • ε n where h X (n) = min h ; I(X) ∩ J h (n) = ∅ .
We define the upper/lower top Lyapunov exponent of A = (A 1 , A 2 , . . . ) to be the quantities [START_REF] Elsner | Norm conditions for convergence of infinite products[END_REF] χ top (A) := lim inf

n→+∞ 1 n log A 1 • • • A n and χ top (A) := lim sup n→+∞ 1 n log A 1 • • • A n
and we speak of the top Lyapunov exponent denoted χ top (A) whenever both lower and upper exponents χ top (A) and χ top (A) coincide. A straightforward consequence of part (iii) of Theorem A, is the following corollary.

Corollary A. Let A = (A 1 , A 2 , . . . ) be a sequence of d × d matrices satisfying condition (C); then, here exists a unique V ∈ S d (the top Lyapunov direction) such that for any vector X with positive entries,

lim n→+∞ P n X/ P n X = V .
We notice that, if n) is the ordered list of the singular values of P n (n ≥ 1), e nχ 1 (n) is also the euclidean norm of P n , hence from [START_REF] Elsner | Norm conditions for convergence of infinite products[END_REF] lim inf n→+∞ χ 1 (n) = χ top (A) and lim sup

e nχ 1 (n) ≥ • • • ≥ e nχ d (
n→+∞ χ 1 (n) = χ top (A).
The exponents of the form χ top (A) are found in a probabilistic framework which roots in the seminal work by Furstenberg & Kesten [START_REF] Furstenberg | Products of Random matrices[END_REF]. To fix ideas, let T : Ω → Ω be a continuous transformation, where (for simplicity) Ω is a compact metric space and suppose that µ is a T -ergodic borelian probability measure. We also consider that A : Ω → M d (C) is a given (borelian) map from Ω to the space M d (C) of the complex d × d matrices. We note P n (ω) = A(ω)A(T (ω)) • • • A(T n-1 (ω)), with the convention that P 0 (ω) is the d × d identity matrix; under these conditions (n, ω) → P n (ω) is a submultiplicative process since P n+m (ω) ≤ P n (ω) • P m (T n (ω)) and Kingman's Subadditive Ergodic Theorem ensures the existence of a constant χ 1 (µ) ≥ -∞ s.t. [START_REF] Erdös | On a family of symmetric Bernoulli convolutions[END_REF] lim n→+∞ 1 n log P n (ω) = inf 1 n log P n (ω) ; n ≥ 1 = χ 1 (µ) µ-a.s.

The quantity χ 1 (µ) is the top-Lyapunov exponent of the random process (n, ω) → P n (ω) , for Ω weighted by µ. In view of the definition in [START_REF] Elsner | Norm conditions for convergence of infinite products[END_REF], notice that for ω being a µ-generic point in Ω (hence for µ-a.e. ω) one has χ top A(ω) = χ 1 (µ), where A(ω) := (A(ω), A(T (ω)), A(T 2 (ω)), . . . ).

A more general framework for characteristic exponents associated with matrix products is given by Oseledets Theorem [START_REF] Oseledets | A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems[END_REF]: indeed, if

e nχ 1 (n,ω) ≥ • • • ≥ e nχ d (n,ω)
form the ordered sequence of the singular values of P n (ω), then (for µ being T -ergodic) each exponent χ k (n, ω) tends µ-a.s. toward a limit χ k (µ) which is the k-th Lyapunov exponent of µ.

(The larger Lyapunov exponent χ 1 (µ) coincides with the top Lyapunov exponent as defined in [START_REF] Erdös | On a family of symmetric Bernoulli convolutions[END_REF], hence his name.) The theory of Lyapunov exponents related to matrix products is a wide domain of research: we mention relationships with Hausdorff dimension of stationary probabilities and multifractal analysis of positive measures (see for instance [START_REF] Ledrappier | Quelques propriétés des exposants caractéristiques[END_REF][20] [START_REF] Feng | Lyapunov exponent for products of matrices and Multifractal analysis. Part I: Positive matrices[END_REF][START_REF] Feng | The variational principle for products of non-negative matrices[END_REF][START_REF] Feng | Lyapunov exponents for products of matrices and Multifractal analysis. Part II: General matrices[END_REF][19] [START_REF] Dai | Extremal ergodic measures and the finiteness property of matrix semigroups[END_REF]).

Notations and background

2.1. Projective distance and contraction coefficient. The definitions and the properties of the Hilbert metric δ H (•, •) and of the Birkhoff (or ergodic) contraction coefficient τ B (•) may be found in the very beginning of Subsection 3.4 of Seneta's book [START_REF] Seneta | Non-negative matrices and Markov chains[END_REF]; we shall need (in particular in the proof of Theorem A) an adaptation/generalization of some concepts. Indeed, the Birkhoff contraction coefficient τ B (A) of a square matrix A (with nonnegative entries) belongs to the unit interval [0 ; 1] with the crucial property that τ B (A) < 1 if and only if A has positive entries: however, the usual framework of Theorem A is concerned with sparse matrices. Hence, we shall consider a contraction coefficient map A → τ (A) whose domain is made of the (non necessarily square) matrices having nonnegative entries, and such that τ (A) < 1 if and only if the positive entries are positioned on a rectangular submatrix.

Definition 2.1. (i) : The d 1 × d 2 nonnegative matrix A = (A(i, j)), distinct from the null matrix, is said to satisfy hypothesis (H) if there exist two nonempty sets I A ⊂ {1, . . . , d 1 } and J A ⊂ {1, . . . , d 2 } such that A(i, j) = 0 ⇐⇒ (i, j) ∈ I A × J A ;

(ii) : the δ-coefficient of A is either δ(A) := +∞ if A does not satisfy (H), or otherwise, [START_REF] Feng | Lyapunov exponent for products of matrices and Multifractal analysis. Part I: Positive matrices[END_REF] δ(A) := max log A(i, j) • A(k, ) A(k, j) • A(i, ) ; (i, k) ∈ I A × I A , (j, ) ∈ J A × J A ;

(iii) : the generalized contraction coefficient of A is:

(15) τ (A) := tanh δ(A) 4 .
Proposition 2.2. τ (A) < 1 if A satisfies (H) and τ (A) = 1 otherwise.

If A = (X Y ) is a d × 2 nonnegative matrix (i.e. X = AU 1 and Y = AU 2 ), then we shall make an abuse of notations writing δ(X, Y ) instead of δ(A): we call δ(X, Y ) the projective distance between X and Y (likewise we shall note δ(X , Y ) := δ(A ) = δ(A) for row vectors). So the restriction of δ to the simplex S d is an extended metric, and defines a topology on S d . Let ∅ = I ⊂ {1, . . . , d}; if one assumes that ∆(X) = ∆(Y ) with I(X) = I(Y ) = I then, the projective distance δ(X, Y ) (= δ(X , Y )) is finite and given by the two following equivalent expressions: [START_REF] Feng | The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers[END_REF] δ(X, Y

) = max i,j∈I log X(i)Y (j) Y (i)X(j) = max i∈I log X(i) Y (i) + max i∈I log Y (i) X(i) . Moreover, δ(X, Y ) coincides with δ X/ X , Y / Y : hence δ(•, •) is entirely determined by its values on the simplex S d . The map δ(•, •) : S d × S d → [0 ; +∞] is closed to a metric but recall that δ(X, Y ) = +∞ means ∆(X) = ∆(Y ).
• X -Y ≤ δ(X, Y ) ≤ X -Y min i∈I {X(i), Y (i)} . (17) 1 d
Proof. Part (i) is straightforward from the definition of δ(•, •) while part (ii) is a key property of Hilbert projective metric (see [51][26]). To prove the double inequality in [START_REF] Feng | Lyapunov exponents for products of matrices and Multifractal analysis. Part II: General matrices[END_REF] of part (iii), we shall use the following inequalities,

-Y ∞ = X(i 0 ) -Y (i 0 ) for i 0 ∈ I, so that 1 ≥ X(i 0 ) ≥ Y (i 0 ) > 0; however, since X, Y ∈ S d , it is necessary that i (X(i) -Y (i)) =
0 and thus, there exits i 1 s.t. X(i 1 ) -Y (i 1 ) ≤ 0; in particular max i∈I log(Y (i)/X(i)) ≥ 0: then, we use the lower bound in [START_REF] Feng | Equilibrium states for factor maps between subshifts[END_REF] together with the second expression of δ(X, Y ) in [START_REF] Feng | The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers[END_REF] to write

δ(X, Y ) ≥ log X(i 0 )/Y (i 0 ) = X(i 0 ) -Y (i 0 ) ≥ X -Y ∞ ≥ 1 d • X -Y ,
proving the lower bound in [START_REF] Feng | Lyapunov exponents for products of matrices and Multifractal analysis. Part II: General matrices[END_REF]. For the corresponding upper bound, let i 0 (resp. i 1 ) s.t.

X(i 0 )/Y (i 0 ) = max i (X(i)/Y (i)) (resp. Y (i 1 )/X(i 1 ) = max i (Y (i)/X(i)) ). If i 0 = i 1 then X = Y , so that δ(X, Y ) = X -Y = 0 and the desired upper bound holds. Now suppose i 0 = i 1 : because i (X(i) -Y (i)) = 0, we know that 1 ≥ X(i 0 ) ≥ Y (i 0 ) ≥ ε and 1 ≥ Y (i 1 ) ≥ X(i 1 )
≥ ε, where ε := min i∈I {X(i), Y (i)}: using the upper bound in [START_REF] Feng | Equilibrium states for factor maps between subshifts[END_REF] together with the second expression of δ(X, Y ) in [START_REF] Feng | The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers[END_REF] gives

δ(X, Y ) = log X(i 0 )/Y (i 0 ) + log Y (i 1 )/X(i 1 ) ≤ 1 ε • X(i 0 ) -Y (i 0 ) + 1 ε • Y (i 1 ) -X(i 1 ) ≤ 1 ε • X -Y . Remark 2.5. Let X * , X 1 , X 2 , • • • ∈ S d ; if δ(X n , X * ) → 0 as n → +∞ then ∆(X n ) = ∆(X * ) as soon as δ(X n , X * ) < +∞. So one has the equivalence (19) lim n→+∞ δ(X n , X * ) = 0 ⇐⇒ lim n→+∞ X n -X * = 0 and ∆(X n ) = ∆(X * ) for n large enough .
We make the simple remark that if each one of the nonzero entries X n (i) are bounded from bellow by ε > 0, then the normed convergence X n → X * is equivalent to δ(X n , X * ) → 0, with more precisely a convergence X n → X * w.r.t. the metric δ(•, •) over S I(X * ) . This is the key point of Lemma 3.6 in the proof of Theorem A in Section 3.

Given X and Y two arbitrary d × 1 vectors with positive entries and B a square d × d allowable (without null row nor null column) nonnegative matrix, one recovers that (see [51, § 3.4])

δ(X , Y ) = δ H (X , Y ) and τ (B) = τ B (B) := sup δ H (X B, Y B) δ H (X , Y ) ; X, Y non colinear , which gives the contraction property of δ H (•, •) and τ B (•) that is (20) δ H (X B, Y B) ≤ δ H (X , Y )τ B (B
(AB) = 0 ≤ δ(A)τ (B)), or there exists i = j both in {1, . . . , d 1 } and k = both in {1, . . . , d 3 } so that δ(AB) = δ(A B ),
where A B has positive entries and

A := A(i, 1) • • • A(i, d 2 ) A(j, 1) • • • A(j, d 2 ) B :=    B(k, 1) B( , 1) . . . . . . B(k, d 2 ) B( , d 2 )    .
Since A is supposed to satisfy (H), the set J A ⊂ {1, . . . , d 2 } is non empty and because A B has positive entries, it is licit to consider the maximal (necessarily non empty) set I ⊂ J A for which the A B has rank 1 and δ(A B ) = 0). However, for A (resp. B ) being a submatrix of A (resp. B), one has δ(A ) ≤ δ(A) (resp. τ (B ) ≤ τ (B)).

I × {1, 2} submatrix B of B is allowable. Now, if A is the {1, 2} × I submatrix of A ,
2.2. Some properties of H 1 , H 2 (•) and H 3 (•). To begin with, suppose that A is a nonnegative If A ∈ H 2 (Λ) satisfies condition (H) (i.e. its nonzero entries are positioned on a rectangular submatrix), then the δ-coefficient of A -which is finite -may actually be bounded by means of Λ. Indeed, if A(i 0 , j 0 )A(i 1 , j 1 )A(i 1 , j 0 )A(i 0 , j 1 ) = 0 then, the condition A ∈ H 2 (Λ) implies

d 1 × d 2 matrix in H 2 (Λ) ∩ H 3 (λ) with A(i 0 , j 0 ) = 0 while A(i 0 , j 1 ) = 0: if A(i, j 0 ) = 0, then (23) AU j 1 ≤ λ • A(i 0 , j 0 ) ≤ (λΛ) • A(i, j 0 ). Lemma 2.7. Given λ ≥ 0, Λ ≥ 1 and A ∈ H 2 (Λ) ∩ H 3 (λ), the following proposition holds: if ∆(AU j 0 ) > ∆(AU j 1 ) then max i A(i, j 1 ) ≤ AU j 1 ≤ (λΛ) min i A(i, j 0 ) = 0 .
A(i 0 , j 0 ) ≤ AU j 0 ≤ Λ • A(i 1 , j 0 ) and A(i 1 , j 1 ) ≤ AU j 1 ≤ Λ • A(i 0 , j 1 ), so that log A(i 0 , j 0 ) A(i 1 , j 0 ) • A(i 1 , j 1 ) A(i 0 , j 1 ) ≤ log Λ • A(i 1 , j 0 ) A(i 1 , j 0 ) • Λ • A(i 0 , j 1 ) A(i 0 , j 1 ) ≤ log(Λ 2 ). Lemma 2.8. If a d 1 × d 2 matrix 0 = A ∈ H 2 (Λ) satisfies condition (H) then, δ(A) ≤ log(Λ 2 ).
We shall also need some stability properties of H 2 (•) and H 3 (•) w.r.t. matrix multiplication.

Lemma 2.9. Let A and B be two matrices with non negative entries, of size

d 1 × d 2 and d 2 × d 3 respectively; then, (i) : if A ∈ H 3 (λ a ) and B ∈ H 3 (λ b ) then AB ∈ H 3 (λ a λ b ); (ii) : if A ∈ H 2 (Λ a ) ∩ H 3 (λ a ) and B ∈ H 2 (Λ b ) then AB ∈ H 2 Λ a + λ a Λ b .
Proof. (i) : Suppose that AB(i 0 , j 0 ) = 0 and AB(i 0 , j 1 ) = 0. Notice that AB(i 0 , j 0 ) = 0 means the existence of j * s.t. A(i 0 , j * )B(j * , j 0 ) = 0, while AB(i 0 , j 1 ) = 0 implies A(i 0 , j)B(j, j 1 ) = 0 for any j; we also emphasize that A(i 0 , j * ) = 0 and A(i 0 , j * )B(j * , j 1 ) = 0 implies B(j * , j 1 ) = 0.

To compare ABU j 1 = i AB(i, j 1 ) with AB(i 0 , j 0 ) we write successively

ABU j 1 = A(i 0 ,j) =0
AU j B(j, j 1 ) +

A(i 0 ,j)=0 AU j B(j, j 1 ) (B(j, j 1 ) = 0 when A(i 0 , j) = 0) ≤ λ a A(i 0 , j * ) j B(j, j 1 ) = λ a A(i 0 , j * ) BU j 1 (B(j * , j 0 ) = 0 and B(j * , j 1 ) = 0) ≤ λ a A(i 0 , j * )λ b B(j * , j 0 ) = λ a λ b AB(i 0 , j 0 ) A(i 0 , j * )B(j * , j 0 ) AB(i 0 , j 0 )
and

ABU j 1 ≤ λ a λ b AB(i 0 , j 0 ): this proves that AB ∈ H 3 (λ a λ b ).
(ii) : Suppose that AB(i 0 , j 0 ) = 0 with A(i 0 , j * )B(j * , j 0 ) = 0; then, one has:

ABU j 0 = A(i 0 ,j) =0
AU j B(j, j 0 ) +

A(i 0 ,j)=0 AU j B(j, j 0 ) (A(i 0 , j * ) = 0) ≤ Λ a j A(i 0 , j)B(j, j 0 ) + λ a A(i 0 , j * ) j B(j, j 0 ) = Λ a AB(i 0 , j 0 ) + λ a A(i 0 , j * ) BU j 0 (B(j * , j 0 ) = 0) ≤ Λ a AB(i 0 , j 0 ) + λ a A(i 0 , j * )Λ b B(j * , j 0 ) = AB(i 0 , j 0 ) Λ a + λ a Λ b A(i 0 , j * )B(j * , j 0 ) AB(i 0 , j 0 ) , that is ABU j 0 ≤ AB(i 0 , j 0 ) Λ a + λ a Λ b : this proves that AB ∈ H 2 Λ a + λ a Λ b .
Given M a r × s matrix, we define

∆(AU j 0 ) ≤ ∆(AU j 1 ) =⇒ ∆(BAU j 0 ) ≤ ∆(BAU j 1 ); (b) : ∆(AU j 0 ) = ∆(AU j 1 ) =⇒ ∆(BAU j 0 ) = ∆(BAU j 1 ); (c) : ∆(AU j 0 ) = 0 =⇒ ∆(BAU j 0 ) = 0; moreover if A ∈ H 1 then (d) : BACU j = 0 =⇒ ∆(BACU j ) ∈ ∆(BAU 1 ), . . . , ∆(BAU d ) ; (e) : #Col(BAC) ≤ #Col(A); (f ) : BAC ∈ H 1 .
Proof. To begin with, any column of BA is a linear combination of the columns of B: more precisely, for any j = 1, . . . , d, one has [START_REF] Guivarch | Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence[END_REF] with (a-b-c) prove that BA ∈ H 1 and #Col(BA) ≤ #Col(A). Like in [START_REF] Guivarch | Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence[END_REF], each column of BAC is a linear combination of the columns of BA, with (26)

AU j = i A(i, j)U i = i∈I(AU j ) A(i, j)U i and thus (25) BAU j = i∈I(AU j ) A(i, j)BU i , which gives (a-b-c). If A ∈ H 1 , then
BACU j = i∈I(CU j ) C(i, j)BAU i .
Suppose that BACU j = 0 and let i 1 , . . . , i s be the indices in I(CU j ) s.

s k+1 ≤ n < s k+2 with k ≥ 1, ( 27
) 1 ≤ p ≤ k =⇒ Q n and Q sp • • • Q s k Q n ∈ H 2 (Λ ) where Λ = Λ 1 -λ ;
(by convention Q s 1 = Q 0 is the identity matrix); moreover, up to a change of the sequence (s 0 , s 1 , . . . ), it is licit to consider that

(28) n ≥ s 2 =⇒ Q n and Q sp • • • Q s k Q n ∈ H 1 ∩ H 2 (Λ ) ∩ H 3 (λ k ).
Proof. Let A = (A 1 , A 2 , . . . ) satisfy (C) w.r.t. 0 ≤ λ < 1 ≤ Λ and (s 0 , s 1 , . . . ). A direct application of part (ii) in Lemma 2.9 implies [START_REF] Kamae | A characterization of self-affine functions[END_REF]. To prove [START_REF] Kenyon | Measures of full dimension on affine-invariant sets[END_REF], define the sequence 1 = γ(0), γ(1), γ(2), . . . From now on and throughout, we assume (Lemma 3.1) the existence of 0 ≤ λ < 1 ≤ Λ and of a sequence 0 = s

s.t. γ(k + 1) = γ(k) + k; setting S k = s γ(k) , one gets 0 = S 0 = S 1 < S 2 < . . . . Given S k+1 ≤ n < S k+2 , let Q n := A S k +1 • • • A n = A S k +1 • • • A S k+1 • • • A n , so that Q n = A s γ(k) +1 • • • A s γ(k+1) • • • A n = Q s γ(k)+1 • • • Q s γ(k)+k • • • Q n . Now,
0 = s 1 < s 2 < • • • of integers s.t. for any n ≥ 0 and k = k(n) ≥ 0 s.t. s k+1 ≤ n < s k+2 , (29) n ≥ s 2 =⇒ Q n ∈ H 1 ∩ H 2 (Λ) ∩ H 3 (λ k ),
and moreover

(30) s 2 ≤ s p < s k+1 ≤ n < s k+2 =⇒ Q sp • • • Q s k Q n ∈ H 2 (Λ).
By definition, any d × d matrix M ∈ H 1 is associated in an unique way with disjoint nonempty subsets of {1, . . . , d} × {1, . . . , d} that are of the form I h (M ) × J h (M ), for 1 ≤ h ≤ κ(M ) := #Col(M ) and such that [START_REF] Page | Répartition d'état d'un opérateur de Schrödinger alétoire. Distribution empirique des valeurs propres d'une matrice de Jacobi, Probability measures on groups VII[END_REF]

(i, j) ; M (i, j) = 0 = κ(M ) h=1 I h (M ) × J h (M ) and I 1 (M ) I 2 (M ) • • • I κ(M ) (M ).
In particular, given any 1 ≤ h, h ≤ κ(M ) and (j, j ) ∈ J h (M ) × J h (M ), [START_REF] Ledrappier | Quelques propriétés des exposants caractéristiques[END_REF]

h = h =⇒ ∆(M U j ) = ∆(M U j ) = 0 while h > h =⇒ ∆(M U j ) > ∆(M U j ) = 0.
The definitions of 1 ≤ H ≤ d, of the index domains I h and J h (n) (n ≥ s 2 and 1 ≤ h ≤ H) and of the vectors V h -as introduced in Theorem A -will be given as a consequence of Theorem 3.11 below. Before that, we first consider the intermediate index domains I h (n) and J h (n), for 1 ≤ h ≤ H n as follows; by part b of Lemma 2.10, given 1 ≤ h ≤ κ(Q n ), the row-index set I(P n U j ) is the same for any j ∈ J h (Q n ); it may be empty if h exceed some value, say H n : so for any 1 ≤ h ≤ H n , we define

(33) J h (n) := J h (Q n ),
and I h (n) is by definition the common value of I(P n U j ) for any j ∈ J h (Q n ). By part a of Lemma 2.10, 3).

I 1 (Q n ) • • • I Hn (Q n ) implies I 1 (n) ⊃ • • • ⊃ I Hn (n) (see Figure
ϴ' n Q n ϴ n = ϴ'' n P n ϴ n = ϴ'' n P s k (ϴ' n ) --1 ϴ' n Q n ϴ n =
X = 0, define (35) h X (n) = min 1 ≤ h ≤ H n ; (h P n )X = 0 = min 1 ≤ h ≤ H n ; I(X) ∩ J h (n) = ∅ ; then, ∆(P n X) = ∆ (h X (n) P n )X with (36)
I(P n X) = I h X (n) P n X = I h X (n) (n).
The matrix P n is not likely to satisfy condition (H) and one can reasonably expect that δ(P n ) = +∞ for most of ranks n. However, it is relevant to look at

δ(h P n ) = max δ(P n U j 0 , P n U j 1 ) ; j 0 , j 1 ∈ J h (n) and max δ(h P n ) ; 1 ≤ h ≤ H n ;
we shall prove (see Lemma 3.5 below) that the latter maximum tends to 0 with an exponential rate depending on k = k(n). Before proving this convergence, we begin with crucial inequalities.

Lemma 3.4. Let s k+1 ≤ n < s k+2 (with k = k(n) ≥ 1) and X, Y ∈ S d such that both P n X, P n Y = 0; then, with M X,Y := max{Λ X , Λ Y }, one has (i) : h X (n) = h Y (n) =⇒ δ P n X, P n Y ≤ λ k 2ΛM X,Y + δ h X (n) P n ; (ii) : ∆(X) = ∆(Y ) =⇒ δ P n X, P n Y ≤ λ k 2ΛM X,Y + δ h X (n) P n M X,Y -1 M X,Y + 1 .
Proof. For s k+1 ≤ n < s k+2 with k ≥ 1 and X ∈ S d such that P n X = 0 we note h := h X (n). We first prove in [START_REF] Mukherjea | Addendum to "On the distribution of the limit of products of i. i. d. 2?2 random stochastic matrices[END_REF] below that each P n X(i) is closed to (h P n )X(i). By definition of h, there exists at least one index j 0 ∈ J h (n) such that X(j 0 ) = 0. On the one hand, because X(j 0 ) = 0, the definition of Λ X gives X ≤ Λ X X(j 0 ); on the other hand, recall that condition (29) means Q n ∈ H 2 (Λ) ∩ H 3 (λ k ): hence, for any h < r ≤ H n and j 1 ∈ J r (n), one has ∆(Q n U j 0 ) > ∆(Q n U j 1 ) = 0 and thus (Lemma 2.7) for any 1 ≤ i ≤ d (and n ≥ s 2 ):

P n (i, j 1 ) = j P s k (i, j)Q n (j, j 1 ) ≤ j P s k (i, j) λ k Λ Q n (j, j 0 ) = λ k ΛP n (i, j 0 ) ; therefore, Hn r=h+1 (r P n )X(i) = Hn r=h+1 j∈J r (n) P n (i, j)X(j) ≤ λ k ΛP n (i, j 0 ) X ≤ λ k ΛΛ X P n (i, j 0 )X(j 0 ),
which allows to write:

(h P n )X(i) ≤ P n X(i) = Hn r=h (r P n )X(i) ≤ (h P n )X(i) + Hn r=h+1 (r P n )X(i) ≤ (h P n )X(i) + λ k ΛΛ X P n (i, j 0 )X(j 0 ).
Because P n (i, j 0 )X(j 0 ) ≤ (h P n )X(i) one finally gets

(37) 1 ≤ P n X(i) (h P n )X(i) ≤ (h P n )X(i) + λ k ΛΛ X P n (i, j 0 )X(j 0 ) (h P n )X(i) ≤ 1 + λ k ΛΛ X .
Since h := h X (n), Lemma 3.3 and the definition of the δ-projective distance give

δ P n X, (h P n )X = max log P n X(i) • (h P n )X(i ) P n X(i ) • (h P n )X(i) ; i, i ∈ I h (n)
and since by a double application of (37),

P n X(i) • (h P n )X(i ) P n X(i ) • (h P n )X(i) ≤ P n X(i) (h P n )X(i) ≤ 1 + λ k ΛΛ X , it follows that (38) δ P n X, (h P n )X ≤ λ k ΛΛ X .
Now, let X, Y ∈ S d with P n X, P n Y = 0 and satisfying the additional condition h n (X) = h n (Y ) =: h. Using triangular inequality for δ(•, •) together with [START_REF] Olivier | Uniqueness of the measure with full dimension on sofic affine invariant subsets of the 2-torus[END_REF] gives

δ P n X, P n Y ≤ δ P n X, (h P n )X + δ (h P n )X, (h P n )Y + δ (h P n )Y, P n Y ≤ λ k 2ΛM X,Y + δ (h P n )B ,
where we have introduced the d × 2 matrix B = (X Y ). On the one hand, h P n satisfies (by definition) the condition (H) and thus, by Lemma 2.6, (39)

δ P n X, P n Y ≤ λ k (2ΛM X,Y ) + δ(h P n )τ (B).
This proves part (i) of the lemma, since τ (•) is bounded by 1. On the other hand, if one assumes that ∆(X) = ∆(Y ) then (Lemma 2.8) δ(B) ≤ log(M 2 X,Y ): because τ (B) = tanh(δ(B)/4), one deduces part (ii) from [START_REF] Olivier | Measures with full dimension on self-affine graphs[END_REF] and the lemma is proved.

We are now in position to prove the following key lemma which is the first step for proving assertions (i) and (ii) of Theorem A. Lemma 3.5. There exist two constants C > 0 and 0 < r < 1 for which the following properties hold for any s k+1 ≤ n < s k+2 (with k = k(n) ≥ 1): for any 1 ≤ h ≤ H n [START_REF] Olivier | On a class of sofic affine invariant subsets of the 2-torus related to an Erdös problem[END_REF] max δ(P n U j 0 , P n U j 1 ) ; j 0 , j

1 ∈ J h (n) = δ(h P n ) ≤ Cr k ; moreover, given X ∈ S d with P n X = 0 (41) j ∈ J h X (n) (n) =⇒ δ(P n U j , P n X) ≤ CΛ X r k .
Proof. To prove [START_REF] Olivier | On a class of sofic affine invariant subsets of the 2-torus related to an Erdös problem[END_REF] we proceed by induction over k ≥ 1. Consider r and C such that (in particular 1 2 ≤ r < 1 and C > 0). The case k = 1 means that 0 = s 1 < s 2 ≤ n < s 3 , so that

P n = P s 1 Q n = Q n and condition (29) -deduced from condition (C) -gives P n ∈ H 2 (Λ): for an arbitrary 1 ≤ h ≤ H n (Lemma 2.8) δ(h P n ) ≤ log(Λ 2 ) ≤ Λ 2 = C 4Λ ≤ C 4 ≤ Cr 1
and the induction is initialized for rank k = 1. Suppose [START_REF] Olivier | On a class of sofic affine invariant subsets of the 2-torus related to an Erdös problem[END_REF] satisfied for rank k ≥ 1, that is δ(h P n ) ≤ Cr k , for each 1 ≤ h ≤ H n and any s k+1 ≤ n < s k+2 . Let s k+2 ≤ n < s k+3 so that P n = P s k+1 Q n and take two columns of Q n , say X = Q n U j 0 and Y = Q n U j 1 with j 0 , j 1 ∈ J h (n) for an arbitrary 1 ≤ h ≤ H n . Notice that ∆(X) = ∆(Y ) with P s k+1 X, P s k +1 Y = 0 so that h X s k+1 = h Y s k+1 =: . Since k(s k+1 ) = k the induction hypothesis gives δ( P s k+1 ) ≤ Cr k ; moreover, condition [START_REF] Kenyon | Hausdorff dimensions of sofic affine-invariant sets[END_REF] implies that Λ X and Λ Y are bounded by Λ and thus M X,Y ≤ Λ: using part (ii) of Lemma 3.4

δ(P n U j 0 , P n U j 1 ) = δ(P s k+1 X, P s k+1 Y ) ≤ λ k (2ΛM X,Y ) + δ( P s k+1 ) M X,Y -1 M X,Y + 1 ≤ λ k (2Λ 2 ) + Cr k Λ -1 Λ + 1 = λ k (2Λ 2 ) + Cr k+1 -Cr k 1 Λ + 1 ; but (42) gives λ k (2Λ 2 ) ≤ r k (2Λ 2 ) = r k C 2Λ ≤ r k C Λ+1 = Cr k 1 Λ+1
so that δ(P n U j 0 , P n U j 1 ) ≤ Cr k+1 : the induction holds and ( 40) is established.

To prove [START_REF] Olivier | On the Gibbs properties of Bernoulli convolutions related to βnumeration in multinacci bases[END_REF] consider

s k+1 ≤ n < s k+2 for k ≥ 1 and X ∈ S d s.t. P n X = 0. Given Y = U j , for j ∈ J h X (n) (n) it is necessary (and sufficient) that h Y (n) = h X (n) =: h; moreover, Λ Y = 1
and M X,Y = Λ X . From [START_REF] Olivier | On a class of sofic affine invariant subsets of the 2-torus related to an Erdös problem[END_REF] there exists a constant C > 0 s.t. δ(h P n ) ≤ Cr k and thus using part (i) of Lemma 3.4 gives

δ(P n X, P n U j ) ≤ λ k (2ΛΛ X ) + δ(h P n ) ≤ (2ΛΛ X + C) r k ≤ C Λ X r k ,
where C = 2Λ + C : this proves (41).

Proof of Theorem A.

Recall that A = (A 1 , A 2 , . . . ) (where each A i is a d × d matrix with non negative entries) is supposed to satisfy condition (29) w.r.t. 0 ≤ λ < 1 ≤ Λ and the sequence of integers 0 = s

0 = s 1 < s 2 < • • • . The (compact) simplex S d of the nonnegative d × 1 vectors V s.t. V = 1 is
endowed with the topology induced from the normed topology on R d . The set S d (A) of the projective limit vectors of A is made of the W ∈ S d for which there exists a sequence 1 ≤ m 1 < m 2 < • • • of integers and a sequence (j 1 , j 2 , . . . ) in {1, . . . , d} such that X t = P mt U jt / P mt U jt → W as t → +∞; in other words, [START_REF] Olivier | Projective convergence of inhomogeneous 2 × 2 matrix products[END_REF] lim

t→+∞ X t -W = 0.
The following lemma ensures the convergence in [START_REF] Olivier | Projective convergence of inhomogeneous 2 × 2 matrix products[END_REF] to imply that X t is trapped in the face of the projective limit vector W : more precisely -for t large enough (44)

X t ∈ S I(W ) = {X ∈ S d ; I(X) = I(W )}. Lemma 3.6 (Trapping lemma). (i) : Let X n,j := P n U j / P n U j , for n ≥ s 2 (1 ≤ j ≤ d): then, (45)
X n,j (i) = 0 =⇒ X n,j (i) ≥ 1/Λ ;

(ii) : given any W ∈ S d (A), one has

(46) W (i) = 0 =⇒ W (i) ≥ 1/Λ ; (iii) : for any sequence 1 ≤ m 1 < m 2 < • • • (resp. j 1 , j 2 , . . . ) made of integers (resp. indices in {1, . . . , d}) and W 1 , W 2 , • • • ∈ S d (A), one has the equivalence (47) lim t→+∞ X mt,jt -W t = 0 ⇐⇒ lim t→+∞ δ (X mt,jt , W t ) = 0. Proof. (i) : For s k+1 ≤ n < s k+2 (k ≥ 1): because P n = Q s 2 • • • Q s k Q n , we know from condition (29)-(30) that X n,j (i) = 0 implies 1 = X n,j ≤ Λ • X n,j (i), so that X n,j (i) ≥ 1/Λ. (ii) : Let W ∈ S d (A) and 1 ≤ m 1 < m 2 < • • • s.t. X mt,jt -W → 0 as t → +∞; from part (i) one deduces that W (i) = 0 implies W (i) ≥ 1/Λ.
(iii) : If X mt,jt -W t → 0 as t → +∞ then, an immediate consequence of (i) and (ii) is that X mt,jt ∈ S I(Wt) (t large enough): hence, it is licit to apply the upper bound in (17) of Proposition 2.4 so that δ(X mt,jt , W t ) ≤ Λ • X mt,jt -W t and δ(X mt,jt , W t ) → 0. Conversely, if δ(X mt,jt , W t ) → 0, then δ(X mt,jt , W t ) is finite (t large enough) and X mt,jt ∈ S I(Wt) : the lower bound in (17) of Proposition 2.4 gives X mt,jt -W t ≤ d • δ(X mt,jt , W t ) and X mt,jt -W t → 0.

Lemma 3.7. The set S d (A) of the projective limit vectors is finite with

1 ≤ #S d (A) ≤ d.
Proof. Let C > 0 and 0 < r < 1 be the two constants given by Lemma 3.5 and fix ε > 0, q ≥ 1 such that CΛr q ≤ ε; moreover, assume X t := P mt U jt / P mt U jt → V , as t → +∞: according to [START_REF] Olivier | Projective convergence of inhomogeneous 2 × 2 matrix products II[END_REF] and by continuity of δ(•, •), it is licit to consider that lim t→+∞ δ(X t , V ) = 0. For m t ≥ s q+1 , write X t = P sq Y t / P sq Y t where

Y t = Q mt U jt if m t < s q+2 (i.e. k(m t ) = q) ; Q s q+1 • • • Q s k(m t ) Q mt U jt if m t ≥ s q+2 (i.e. k(m t ) ≥ q + 1).
By definition X t = 0 and (41) in Lemma 3.5 ensures the existence of a column index j t (actually j t ∈ J h sq (Y t )) for which [START_REF] Peres | Sixty years of Bernoulli convolutions[END_REF] δ(P sq U j t , X t ) = δ(P sq U j t , P sq Y t ) ≤ CΛ Yt r q .

Since (conditions ( 29)-( 30)

) both Q mt and Q s q+1 • • • Q s k(m t ) Q mt belong to H 2 (Λ)
, one gets that Λ Yt ≤ Λ and (48) implies δ(P sq U j t , X t ) ≤ ε. This last inequality being valid for any t such that m t ≥ s q+1 , there exists a column index j 0 , for which j t = j 0 for infinitely many t, meaning that δ(P sq U j 0 , V ) ≤ ε. Suppose -for a contradiction -the existence of at least d + 1 projective limit vectors, say V 1 , . . . , V d+1 , with ε sufficiently small so that δ (V , V ) ≥ 3ε as soon as = : the sets {X ∈ S d ; δ(X, V ) ≤ ε} are disjoint for = 1, . . . , d + 1 and each ones must contain a probability vector of the form P sq U j / P sq U j : this is a contradiction.

Lemma 3.8. Let 0 < δ 0 ≤ +∞ be the smallest δ-distance between two distinct projective limit vectors (and

δ 0 = +∞ if #S d (A) = 1); for 0 < ε ≤ δ 0 /3 there exists N (ε) ≥ 1 s.t. ( 49
) ∀n ≥ N (ε), 1 ≤ ∀h ≤ H n , ∃!W h (n) ∈ S d (A), j ∈ J h (n) =⇒ δ P n U j , W h (n) < ε ; moreover, δ P n U j , W h (n) < ε (≤ +∞) implies ∆(P n U j ) = ∆(W h (n)) and from the definition of J h (n), (50)
∆(W 1 (n)) ≥ • • • ≥ ∆(W Hn (n)).
Proof. Let 0 < ε ≤ +∞ be arbitrary given and define S ε d (A) the set of the vectors X ∈ S d for which there exists V ∈ S d (A) for which δ(X, V ) < ε. We claim the existence of a rank

N (ε) ≥ 1 s.t. each P n U j / P n U j , for 1 ≤ j ≤ d and n ≥ N (ε), belongs to S ε d (A)
. Suppose -for a contradiction -the existence of an increasing sequence m 1 < m 2 < • • • of ranks together with a sequence (j 1 , j 2 , . . . ) of indices with X t := P mt U jt / P mt U jt ∈ S d \ S ε d (A), for any t ≥ 1: taking a subsequence if necessary, this means the existence of at least one projective limit vector X * for which the normed convergence X t → X * holds as t → +∞. However, by the trapping lemma (see [START_REF] Parry | On the β-expansions of real numbers[END_REF] in Lemma 3.6) we know that X t -X * → 0 is equivalent to the projective convergence δ(X t , X * ) → 0 ensuring the existence of t 0 ≥ 1 s.t. δ(X t , X * ) < ε for any t ≥ t 0 : this is a contradiction, because X * ∈ S d (A), while δ(X t , V ) ≥ ε, for any t ≥ 1 and any V ∈ S d (A). Now, assume in addition that ε ≤ δ 0 /3, let q satisfy Cr q ≤ ε (with C, r given by Lemma 3.5) and choose N (ε) ≥ s q+1 : by the first part of the argument, for any n ≥ N (ε) and any 1 ≤ j ≤ d, there exists a unique projective limit vector W j (n) ∈ S d (A) for which δ P n U j , W j (n) < ε. If j 0 and j 1 are arbitrary taken in J h (n), it follows from (40) in Lemma 3.5 that δ(P n U j 0 , P n U j 1 ) ≤ ε, and W j 0 (n) = W j 1 (n): by definition W h (n) stands for this common projective limit vector.

Let L be the set of the X = (x 1 , . . . , x n ), with x i unspecified and where by abuse of notation #X := n ≥ 1 stands for the length of X. We shall consider the map Ξ :

L → L defined by setting Ξ(x 1 , . . . , x n ) = (x ϕ(1) , . . . , x ϕ(k * ) ),
where

ϕ(1) = 1 while ϕ(k + 1) is the minimum of the ϕ(k) < i ≤ n s.t. x i = x ϕ(k) , provided that (x ϕ(k) , . . . , x n ) = (x ϕ(k) , . . . , x ϕ(k) ); here k * is the minimum of the k ≥ 1 s.t. (x ϕ(k) , . . . , x n ) = (x ϕ(k) , . . . , x ϕ(k) ): for instance Ξ(1, 1, 1, 2, 3, 3, 3, 1, 1) = (1, 2, 3, 1) with k * = 4. According to Lemma 3.8 it is licit to define (51) H := min #Ξ W 1 (s k ), . . . , W Hs k (s k ) ; s k ≥ N (δ 0 /3) .
Remark 3.9. The definition of the index sets J h (n) and of the vectors V h (for 1 ≤ h ≤ H) -that are the main ingredients of Theorem A -are given in Theorem 3.11 below. Let's mention now that the argument for Theorem 3.11 stands on the simple remark that for X = (x 1 , . . . , x n ) any ordered lists and ϕ : {1, . . . , n} → {1, . . . , n} any nondecreasing map, one has

(52) #Ξ(x ϕ(1) , . . . , x ϕ(n) ) ≤ #Ξ(x 1 , . . . , x n ).
This is a motivation for the introduction of the maps 53) and ( 54) below and the associated Lemma 3.10.

ϕ n (•) : {1, . . . , H n } → {1, . . . , H s k } and ψ k (•) : {1, . . . , H s k+2 } → {1, . . . , H s k } in (
Given s k+1 ≤ n < s k+2 and 1 ≤ h ≤ H n , one has J h (n) = J h (Q n ) and:

j 0 , j 1 ∈ J h (n) =⇒ ∆(Q n U j 0 ) = ∆(Q n U j 1 ) =: X n,h .
We suppose that k ≥ 2 because J h (n) is defined for n ≥ s 2 and we need the map X → h X (s k ) -as introduced in (35) -to make sense. The map

ϕ n (•) : {1, . . . , H n } → {1, . . . , H s k } is defined by setting (53) ϕ n (h) = h X n,h (s k ) = min 1 ≤ ≤ H s k ; I(X n,h ) ∩ J (s k ) = ∅ . One defines ψ k (•) : {1, . . . , H s k+2 } → {1, . . . , H s k } analogously to ϕ n (•). Recall that P n = P s k Q n , while P s k+2 = P s k Q s k+1 Q s k+2 : clearly, for any 1 ≤ h ≤ H s k+2 , j 0 , j 1 ∈ J h (s k+2 ) =⇒ ∆(Q s k+1 Q s k+2 U j 0 ) = ∆(Q s k+1 Q s k+2 U j 1 ) =: Y k,h .
By definition

(54) ψ k (h) = h Y k,h (s k ) = min 1 ≤ ≤ H s k ; I(Y k,h ) ∩ J (s k ) = ∅ .
Lemma 3.10. For any s k+1 ≤ n < s k+2 (with k ≥ 2), one has the following propositions:

ϕ n (1) ≤ • • • ≤ ϕ n (H n ) and ψ k (1) ≤ • • • ≤ ψ k (H s k+2 ) ; (55) (j 0 , j 1 ) ∈ J ϕn(h) (s k ) × J h (n) =⇒ δ P s k U j 0 , P n U j 1 ≤ (CΛ)r k-1 ; (56) (j 0 , j 1 ) ∈ J ψ k (h) (s k ) × J h (s k+2 ) =⇒ δ P s k U j 0 , P s k+2 U j 1 ≤ (CΛ)r k-1 ; (57) moreover, if both inequalities s k ≥ N (δ 0 /3) and (CΛ)r k-1 ≤ δ 0 /3 are satisfied, then W 1 (n), . . . , W Hn (n) = W ϕn(1) (s k ), . . . , W ϕn(Hn) (s k ) ; (58) W 1 (s k+2 ), . . . , W Hs k+2 (s k+2 ) = W ψ k (1) (s k ), . . . , W ψ k (Hs k+2 ) (s k ) . (59) Proof. If 1 ≤ h 0 < h 1 ≤ H n and (j 0 , j 1 ) ∈ J h 0 (n) × J h 1 (n) then, by definition of the sets J h (n), ∆(Q n U j 0 ) > ∆(Q n U j 1 ) and thus by minimality, it follows that ϕ n (h 0 ) ≤ ϕ n (h 1 ). Suppose now that 1 ≤ h 0 < h 1 ≤ H s k+2 and (j 0 , j 1 ) ∈ J h 0 (s k+2 ) × J h 1 (s k+2 ), then ∆(Q s k+2 U j 0 ) > ∆(Q s k+2 U j 1 ), thus ∆(Q s k+1 Q s k+2 U j 0 ) ≥ ∆(Q s k+1 Q s k+2 U j 1 )
and by minimality, it follows that

ψ k (h 0 ) ≤ ψ k (h 1 ).
For part (56) -and (57) similarly -let

j 1 ∈ J h (n), where 1 ≤ h ≤ H n . It is licit to apply (41) in Lemma 3.5: indeed, P s k Q n U j 1 = 0 (by definition of J h (n)) and thus δ P s k U j 0 , P n U j 1 = δ P s k U j 0 , P s k Q n U j 1 ≤ CΛ r k-1
(here we used the fact that k(s k ) = k -1 and condition [START_REF] Kenyon | Hausdorff dimensions of sofic affine-invariant sets[END_REF] ensuring that Λ QnU j 1 ≤ Λ).

To prove (58) -and (59) similarly -assume

s k ≥ N (δ 0 /3) and (CΛ)r k-1 ≤ δ 0 /3. Consider j 0 ∈ J ϕn(h) (s k ) and j 1 ∈ J h (n), for 1 ≤ h ≤ H n ;
by definition of ϕ n (h) and according to (56) we know that δ(P s k U j 0 , P n U j 1 ) ≤ (CΛ)r k-1 ≤ δ 0 /3. However, since s k ≥ N (δ 0 /3), Lemma 3.8 ensures that both δ(P s k U j 0 , W ϕn(h) (s k )) and δ(P n U j 1 , W h (n)) are strictly upper bounded by δ 0 /3: for δ 0 being the minimal δ-distance between two different projective limit vectors, it is necessary that [START_REF] Seneta | Non-negative matrices and Markov chains[END_REF]; then, there exists a (surjective but not necessarily injective) finite sequence of probability vectors

W ϕn(h) (s k ) = W h (n) and (58) is proved. Theorem 3.11. Let 1 ≤ H ≤ d be defined in
V 1 , . . . , V H in S d (A) satisfying (60) ∆(V 1 ) ≥ • • • ≥ ∆(V H ) while V h-1 = V h (for any 1 < h ≤ H)
such that for any n (large enough) there exists a partition

{1, . . . , H n } = E 1 (n) • • • E H (n) into H nonempty sets E 1 (n), . . . , E H (n) for which ( , ) ∈ E h-1 (n) × E h (n) =⇒ < and ∈ E h (n) =⇒ W (n) = V h ; (61) in other words Ξ(W 1 (n), . . . , W Hn (n)) = (V 1 , . . . , V H ) and (with a rough representation) (62) (W 1 (n), . . . , W Hn (n)) = (V 1 , . . . , V 1 E 1 (n) , V 2 , . . . , V 2 E 2 (n) , . . . , V H , . . . , V H E H (n) ) ;
Proof. Let k 0 ≥ 1 be the minimal integer satisfying the three constraints (63) (a) :

s k 0 ≥ N (δ 0 /3) ; (b) : (CΛ)r k 0 -1 ≤ δ 0 /3 ; (c) : #Ξ W 1 (s k 0 ), . . . , W Hs k 0 (s k 0 ) = H.
The condition (63-a) is needed to ensure (Lemma 3.8) the existence of the projective limit vectors W 1 (n), . . . W Hn (n), for any n ≥ s k 0 : hence, by the condition (63-c) over k 0 together with the definition of H in (51

), it is licit to fix V 1 , . . . , V H ∈ S d (A) s.t. (64) Ξ W 1 (s k 0 ), . . . , W Hs k 0 (s k 0 ) = (V 1 , . . . , V H ).
By [START_REF] Raugi | Fonctions harmoniques sur les groupes localement compacts à base dénombrable[END_REF] in Lemma 3.8 we know that ∆(

V 1 ) ≥ • • • ≥ ∆(V H ), while (definition of Ξ) V h-1 = V h for any 1 < h ≤ H.
The theorem holds as soon as for any k ≥ k 0 and any 65), according to the specification in ( 61)).

s k+1 ≤ n < s k+2 , (65) Ξ W 1 (n), . . . , W Hn (n) = (V 1 , . . . , V H ) (then, the partition {1, . . . , H n } = E 1 (n) • • • E H (n) is completely determined by (
• To prove (65), we begin with an induction showing that for any k ≥ k 0

(66) #Ξ W 1 (s k ), . . . , W Hs k (s k ) = H.
The initialization being satisfied for k 0 -see (63-c) -let k ≥ k 0 for which (66) is satisfied as well: because (63-b) ensures (CΛ)r k 0 -1 ≤ δ 0 /3, it is licit to use (58) in Lemma 3.10, so that

(67) Ξ W 1 (s k+1 ), . . . , W Hs k+1 (s k+1 ) = Ξ W ϕs k+1 (1) (s k ), . . . , W ϕs k+1 (Hs k+1 ) (s k ) .
Using the fact (see (55) in Lemma 3.10) that ϕ s k+1 (•) is nondecreasing together with the definition of H in ( 51), one deduces from (67) -and the induction hypothesis over k -that

H ≤ #Ξ W 1 (s k+1 ), . . . , W Hs k+1 (s k+1 ) ≤ #Ξ W 1 (s k ), . . . , W Hs k (s k ) = H
and (66) is inductive.

• We now prove by induction that for any

k ≥ k 0 , (68)
Ξ W 1 (s k ), . . . , W Hs k (s k ) = (V 1 , . . . , V H ).
The initialization holds for k 0 due to definition of the projective limit vectors 68) holds: then, by ( 66) and ( 67),

V 1 , . . . , V H in (64). Let k ≥ k 0 s.t. (
(69) H = #Ξ W 1 (s k+1 ), . . . , W Hs k+1 (s k+1 ) = #Ξ W ϕs k+1 (1) (s k ), . . . , W ϕs k+1 (Hs k+1 ) (s k ) .
However, by the induction hypothesis Ξ(W 1 (s k ), . . . , W Hs k ) = (V 1 , . . . , V H ) and for ϕ s k+1 (•) being nondecreasing, (69) implies that

Ξ W ϕs k+1 (1) (s k ), . . . , W ϕs k+1 (Hs k+1 ) (s k ) = (V 1 , . . . , V H ) ;
finally, using again (67) one concludes

Ξ W 1 (s k+1 ), . . . , W Hs k+1 (s k+1 ) = (V 1 , . . . , V H ),
which means that (68) is inductive.

• We shall now prove that (65) holds, for any s k+1 ≤ n < s k+2 (k ≥ k 0 ). On the one hand, recall that (63-b) ensures (CΛ)r k 0 -1 ≤ δ 0 /3: hence, according to (58) in Lemma 3.10,

(70) W 1 (n), . . . , W Hn (n) = W ϕn(1) (s k ), . . . , W ϕn(Hn) (s k ) . Because (see (55) in Lemma 3.10) ϕ n (•) is nondecreasing, it follows from (66) that (71) #Ξ W 1 (n), . . . , W Hn (n) ≤ #Ξ W 1 (s k ), . . . , W Hs k (s k ) = H.
On the other hand (68) together with (59) in Lemma 3.10 give

(V 1 , . . . , V H ) = Ξ W 1 (s k+2 ), . . . , W Hs k+2 (s k+2 ) = Ξ W ψ k (1) (s k ), . . . , W ψ k (Hs k+2 ) (s k ) ;
in particular, this means the existence of a nondecreasing α : {1, . . . , H} → {1, . . . , H s k+2 } s.t.

(72)

W ψ k •α(1) (s k ), . . . , W ψ k •α(H) (s k ) = (V 1 , . . . , V H ).
We claim that (65) will be established (and the theorem as well) as soon has we have proved the existence of a nondecreasing β : {1, . . . , H} → {1, . . . , H n } for which (73)

W ϕn•β(1) (s k ), . . . , W ϕn•β(H) (s k ) = W ψ k •α(1) (s k ), . . . , W ψ k •α(H) (s k ) .
Indeed, (73) together with (72) gives

W ϕn•β(1) (s k ), . . . , W ϕn•β(H) (s k ) = (V 1 , . . . , V H ) and thus (68) implies Ξ W ϕn(1) (s k ), . . . , W ϕn(Hn) (s k ) = (V 1 , . . . , V H ): finally, with (70) one concludes Ξ W 1 (n), . . . , W Hn (n) = (V 1 , . . . , V H ).
• To prove the existence of the nondecreasing map β(•) satisfying (73), let 1 ≤ h 0 ≤ h 1 ≤ H s k+2 and take j i ∈ J α(h i ) (s k+2 ); because α : {1, . . . , H} → {1, . . . , H s k+2 } is a nondecreasing map, and from the definition of the sets J h (s k+2 ), we know that ∆(Q

s k+2 U j 0 ) ≥ ∆(Q s k+2 U j 1 ). Now, recall that Q s k+1 Q s k+2 = Q n A n+1 • • • A s k+2 ;
here it is crucial that Q n ∈ H 1 : indeed, according to part (d) of Lemma 2.10, we know that

∆ Q s k+1 Q s k+2 U j i = ∆ Q n A n+1 • • • A s k+2 U j i ∈ ∆(Q n U 1 ), . . . , ∆(Q n U d ) ensuring the existence of 1 ≤ j 0 , j 1 ≤ d s.t. (74) ∆(Q n U j 0 ) = ∆(Q s k+1 Q s k+2 U j 0 ) ≥ ∆(Q s k+1 Q s k+2 U j 1 ) = ∆(Q n U j 1 ). Given 1 ≤ β(h i ) ≤ H N s.t. j i ∈ J β(h i ) (n) = J β(h i ) (Q n ), since I h (Q n ) I h (Q n ) for any 1 < h < h ≤ H n , the inequality in (74) implies β(h 0 ) ≤ β(h 1 ), meaning that β : {1, . . . , H} → {1, . . . , H N } is nondecreasing. Furthermore, by definition of ϕ n in (53) (75) ϕ n • β(h i ) = h ∆(QnU j i ) (s k ) = min 1 ≤ ≤ H s k ; I ∆(Q n U j i ) ∩ J (s k ) = ∅
while by definition of ψ k in (54) (76) 74) allows to deduce from (75) and (76

ψ k • α(h i ) = h ∆(Qs k+1 Qs k+2 U j i ) (s k ) = min 1 ≤ ≤ H s k ; I ∆(Q s k+1 Q s k+2 U j i )) ∩ J (s k ) = ∅ ; however, the equality ∆(Q n U j i ) = ∆(Q s k+1 Q s k+2 U j i ) in (
) that ϕ n • β(h i ) = ψ k • α(h i ): the theorem is proved. Proof of Theorem A. Let 1 ≤ H ≤ d, {1, . . . , H n } = E 1 (n) • • • E H (n) the partition and V 1 , .
. . , V H the limit projective vectors as defined in Theorem 3.11. By definition

(77) J h (n) := ∈E h (n) J (n). (i) : Fix 1 ≤ h ≤ H and let (j 1 , j 2 , . . . ) be a sequence in {1, . . . , d} s.t. j n ∈ J h (n), i.e. j n ∈ J n (n) for n ∈ E h (n), for any n large enough; in particular W n (n) = V h . Hence, Lemma 3.8 implies (78) lim n→+∞ δ(P n U jn , V h ) = 0
and one concludes with part (iii) in Lemma 3.6 that

lim n→+∞ P n U jn P n U jn -V h = 0.
Part (i) of Theorem A is proved.

(ii) : Let s k+1 ≤ n < s k+2 (with k ≥ 1) and fix (j 0 , j 1 )

∈ J h-1 (n) × J h (n), so that ∆(Q n U j 0 ) > ∆(Q n U j 1 ) = 0. Recall that condition (29) means Q n ∈ H 2 (Λ) ∩ H 3 (λ k ): hence, Lemma 2.7 allows to write for any 1 ≤ i ≤ d (and n ≥ s 2 ): (79) P n (i, j 1 ) = j P s k (i, j)Q n (j, j 1 ) ≤ j P s k (i, j) λ k Λ Q n (j, j 0 ) = λ k Λ P n (i, j 0 ).
Given ( 1 , 2 , . . . ) s.t. 1 < n ≤ H n , for any n ≥ 1 and (j 1 , j 2 , . . . ), (j 1 , j 2 , . . . ) two index sequences, it follows from (79) that

∀n ≥ s 2 , (j n , j n ) ∈ J n-1 (n) × J n (n) =⇒ lim n→+∞ P n U j n / P n U jn = 0
and thus, for 1 < h ≤ H, the definition of J h (n) in (77) and the property (61

) of E n (h) gives ∀n ≥ s 2 , (j n , j n ) ∈ J h-1 (n) × J h (n) =⇒ lim n→+∞ P n U j n / P n U jn = 0. Part (ii) of Theorem A is proved. (iii) : Let X ∈ S d such that P n (X) = 0 for any n ∈ N, and let j n ∈ J h X (n) (n), where h X (n) := min{1 ≤ ≤ H n ; I(X) ∩ J (n) = ∅}: by inequality (41) in Lemma 3.5 one has (80) δ(P n X, P n U jn ) ≤ (CΛ X )r k (for k = k(n) s.t. s k+1 ≤ n < s k+2 ).
According to (77) and the property (61

) of E n (h), j n ∈ J h X (n) (n) ⊂ J h X (n) (n) and W h X (n) (n) = V h X (n) , where h X (n) := min{1 ≤ h ≤ H ; I(X) ∩ J h (n) = ∅} (
as defined in Theorem A). However, we know (Lemma 3.8) that

ε n := sup Hn h=1 δ P n U j , W h (n) ; j ∈ J h (n) → 0
as n → +∞ and because

δ P n U jn , V h X (n) = δ P n U jn , W h X (n) (n) ≤ ε n ,
it follows from (80) that

δ P n X P n X , V h X (n) = δ P n X, V h X (n) ≤ δ (P n X, P n U jn ) + δ P n U jn , V h X (n) ≤ (CΛ X )r k(n) + ε n .
For n large enough, δ P n X/ P n X , V h X (n) < +∞, meaning that ∆(P n X/ P n X ) = ∆(V h X (n) ), and with (17) in Proposition 2.4 we conclude

P n X P n X -V h X (n) ≤ dδ P n X P n X , V h X (n) ≤ d ε n + C • r k(n) Λ X .
Part (iii) of Theorem A is proved.

Heuristics for Theorem A -basic examples and applications

4.1. Example 1: Product of block-triangular matrices. This is a natural and relatively simple example of application of Theorem A: one suppose that each A n is lower block-triangular (resp. upper block-triangular) and that each block has only positive entries and size independent of n.

Corollary 4.1. (i) Suppose that

A n =      B n (1, 1) 0 • • • 0 B n (2, 1) B n (2, 2) • • • 0 . . . . . . . . . . . . B n (δ, 1) B n (δ, 2) • • • B n (δ, δ)     
where the entries of each submatrix B n (i, j) are positive. We suppose that the size d i × d j of the matrix B n (i, j) is independent of n, as well as the real

Λ ≥ 1 such that A n ∈ H 2 (Λ). Denoting by U (k) 1 , . . . , U (k) k
the canonical k-dimensional column vectors we suppose that

(81) ∀i, j, +∞ n=1 B 1 (i + 1, i + 1) • • • B n (i + 1, i + 1) B 1 (i, i) • • • B n (i, i)U (d i ) j < +∞.
Then the sequence (A 1 , A 2 , . . . ) satisfies condition (C) and Theorem A applies.

(ii) This conclusion remains true if we replace the hypothesis "A n lower block-triangular" by "A n upper block-triangular" and (81) by ∀i, j,

+∞ n=1 B 1 (i -1, i -1) • • • B n (i -1, i -1) B 1 (i, i) • • • B n (i, i)U (d i ) j < +∞. Proof. (i) The norm we use in this proof is M = max j L (k) M U ( ) j for any k × matrix M , where L (k) = 1 • • • 1 is the k-dimensional row vector with entries 1. Let (82) ε n = max 1≤i<δ 1≤j≤d i B 1 (i + 1, i + 1) • • • B n (i + 1, i + 1) B 1 (i, i) • • • B n (i, i)U (d i ) j (n ≥ N ),
and

S = 1 + Λ +∞ n=0
ε n < +∞ (where ε 0 = 1). We put B n (i) = B n (i, i) and we define

A n (i), C n (i) by (83) A n (i) =    B n (i, i) • • • 0 . . . . . . . . . B n (δ, i) • • • B n (δ, δ)    = B n (i) 0 C n (i) A n (i + 1)
.

Let us prove by descending induction on 1 < i ≤ δ that -for any n ∈ N and 1 ≤ j

≤ d i-1 (84) A 1 (i) • • • A n (i) ∈ H 2 (S δ-i Λ) and A 1 (i) • • • A n (i) ≤ ε n S δ-i B 1 (i -1, i -1) • • • B n (i -1, i -1)U (d i-1 ) j . Let i = δ, one has A 1 (δ) • • • A n (δ) = B 1 (δ, δ) • • • B n (δ, δ
) and this product of positive matrices belongs to H 2 (Λ) by Lemma 2.9 (ii). Using the definition of ε n in (82), the induction hypotheses are satisfied at the rank i = δ.

We suppose now that the induction hypotheses are satisfied at some rank i + 1 ≤ δ and we prove it at the rank i. From the second equality in (83), (85) A k (i) whose index is larger than

n k=1 A k (i) =       n k=1 B k (i) 0 n k=1 k-1 =1 A (i + 1)C k (i) n =k+1 B (i) n k=1 A k (i + 1)      
d i belong to H 2 (S δ-(i+1)
) and their norms are bounded by

ε n S δ-(i+1) B 1 (i, i) • • • B n (i, i)U (d i ) j (1 ≤ j ≤ d i ).
Consequently they belong to H 2 (S δ-i ). Moreover their norms are bounded by

ε n S δ-i B 1 (i-1, i-1) • • • B n (i-1, i-1)U (d i-1 ) j because B 1 (i,i)•••Bn(i,i)U (d i ) j B 1 (i-1,i-1)•••Bn(i-1,i-1)U (d i-1 ) j ≤ ε n ≤ S (1 ≤ j ≤ d i-1
). So it remains to look at the columns of

A 1 (i) • • • A n (i) with index j ≤ d i . Let D i = d i + • • • + d δ . Using the inequality L (D i+1 ) C k (i) ≤ ΛL (d i ) B k (i) (consequence of A k ∈ H 2 (Λ)) we obtain for any j, j ∈ {1, . . . , d i } (86) n k=1 A k (i)U (D i ) j = n k=1 B k (i)U (d i ) j + n k=1 k-1 =1 A (i + 1)C k (i) n =k+1 B (i)U (d i ) j ≤ n k=1 B k (i)U (d i ) j + n k=1 k-1 =1 A (i + 1) L (D i+1 ) C k (i) n =k+1 B (i)U (d i ) j ≤ n k=1 B k (i)U (d i ) j + n k=1 ε k-1 S δ-(i+1) k-1 =1 B (i)U (d i ) j ΛL (d i ) n =k B (i)U (d i ) j .
Clearly, for any d i -dimensional row vector L and any d i -dimensional column vector R,

min 1≤j ≤d i LU (d i ) j L (d i ) R ≤ LR. Applying this to L = L (d i ) k-1 =1 B (i) and R = n =k B (i)U (d i ) j we obtain (87) n k=1 A k (i)U (D i ) j ≤ n k=1 B k (i)U (d i ) j + n k=1 Λε k-1 S δ-(i+1) L (d i ) n =1 B (i)U (d i ) j ≤ S δ-i n =1 B (i)U (d i ) j .
Using the the definition of ε n in (82), we deduce that the second condition of (84) is fulfilled.

In order to check the first condition, we consider a entry in the j th column of

A 1 (i) • • • A n (i). From (85), either it is a entry of the j th column of B 1 (i) • • • B n (i), or it is at least equal to a entry of the j th column of C 1 (i)B 2 (i) • • • B n (i). From the second equality in (83) one has B 1 (i) C 1 (i)
∈ H 2 (Λ) and, from Lemma 2.9 (ii), the product matrix

B 1 (i) C 1 (i) B 2 (i) • • • B n (i) also
belongs to H 2 (Λ). So, the entry we have considered is at least equal to

B 1 (i) • • • B n (i)U (d i ) j /Λ.
Using (87) it is at least equal to

A 1 (i) • • • A n (i)U (D i ) j
/(S δ-i Λ), proving that the first condition of (84) is fulfilled. So( 84) is true for any n ∈ N, and we deduce that

A 1 • • • A n ∈ H 2 (S δ-1 Λ) ∩ H 3 (ε n S δ-2 Λ). For any n large enough, A 1 • • • A n ∈ H 2 (S δ-1 Λ) ∩ H 3 (1/2).
Notice that for any nonnegative integer N the hypotheses of Corollary 4.1, and in particular the condition (81), are satisfied by (A N +n ) n∈N because, denoting by m (resp. M ) the minimum (resp. the maximum) of the norms of

B 1 (i, i) • • • B N (i, i)U (d i ) j
for any i, j, we have

m B N +1 (i, i) • • • B n (i, i)U (d i ) j ≤ B 1 (i, i) • • • B n (i, i)U (d i ) j ≤ M B N +1 (i, i) • • • B n N (i, i)U (d i ) j
.

So we can define by induction the sequence (s 0 , s 1 , . . . ) such that (A n ) n∈N satisfies condition (C) w.r.t. λ = 1/2 and Λ = S δ-1 Λ: one put s 0 = s 1 = 0 and for any k, s k+1 is the smallest integer such that

∀n ≥ s k+1 , A s k +1 • • • A n ∈ H 3 (1/2). (ii) Let ∆ =      0 • • • 0 1 0 • • • 1 0 . . . . . . . . . . . . 1 • • • 0 0     
. If the A n are upper block-triangular, the ∆A n ∆ are lower block-triangular and the order of the diagonal blocs is reversed.

Example 2.

Let us look at the case of the 2 × 2 upper triangular matrices:

P n := a 1 b 1 0 d 1 • • • a n b n 0 d n with a n d n > 0 and b n ≥ 0.
The sequence of normalized columns of P n converges in S 2 , with more precisely Let

lim n→+∞ P n U j P n U j =                s s+1 1
n k = 1 + • • • + k and define P n = A 1 • • • A n ,
where

A n k =   1 1 1 0 0 0 0 0 2   while n = n k =⇒ A n =   1 3 1 0 4 0 0 0 1   .
Then, one gets

P n k =   1 1 3 • 2 k -2k -3 0 0 0 0 0 2 k   and P n k +k =   1 2 • 4 k -1 3 • 2 k -k -3 0 0 0 0 0 2 k  
and thus, provided that ∆(V ) ≥ (0, 1, 1),

lim k→+∞ P n k V P n k V =   3 4 0 1 4   while lim k→+∞ P n k +k V P n k +k V =   1 0 0   . 4
∀X ∈ S d , P n X P n X -X * ≤ Cγ n .
In the framework of Proposition 4.2, one deduces the matrix product P n to be asymptotically closed to a rank one matrix, in the sense that, there exists

d sequences n → α i (n) (1 ≤ i ≤ d)
each ones being made of positive real numbers and such that

P n ≈ α 1 (n)X * • • • α d (n)X * .
Here, the condition (C) is trivially fulfilled and Theorem A is applicable: according to Proposition 4.2 -and with the notations of Theorem A -one has H = 1 with V 1 = X * and J 1 (n) = {1, . . . , d}, for any n ≥ 1. The sequences n → α i (n) j α j (n) are not necessarily convergent. Consider for instance A i = β i B i where

B 2i-1 = 1/2 1/2 1/2 1/2 and B 2i = 1/3 2/3 1/3 2/3 so that P n = (β 1 • • • β n )B n . Suppose now that A i takes
(ω) = A(ω 1 ) • • • A(ω n ) =: A(ω 1 • • • ω n ) and U = U 1 + • • • + U d ). First consider that each A(i) is a 2 × 2 matrix of the form a(i) b(i) 1 -a(i) 1 -b(i)
with the additional condition that sup i | det(A(i))| < 1. Then (see [START_REF] Mukherjea | On the distribution of the limit of products of i.i.d. 2 × 2 random stochastic matrices[END_REF][START_REF] Mukherjea | Addendum to "On the distribution of the limit of products of i. i. d. 2?2 random stochastic matrices[END_REF][START_REF] Mukherjea | On the continuous singularity of the limit distribution of products of i.i.d. stochastic matrices[END_REF]), one gets

(88) V(ω) = p(ω) 1 -p(ω)
where

p(ω) = +∞ n=1 b(ω n ) det A(ω 1 • • • ω n-1 ).
To fix ideas let 1 ≤ β ≤ 2 and consider that a = 1 with the two matrices (89)

A(0) = 1/β 0 1 -1/β 1 and A(1) = 1 1 -1/β 0 1/β , so that (88) gives p(ω) = (β -1) +∞ n=1 ω n β n
(see Figure 4-right).

; 1[ x → p(x 1 x 2 • • • ), where 0 • x 1 x 2 • • •
is the binary expansion of x and p : {0, 1} N → [0 ; 1] is associated with A(0) and A(1) given in (90) and (89) respectively; in the left box plot, one recognizes the inverse of the Minkowski question mark function (also known as the Conway's function), while in the right box plot one finds a limit Rademacher function.

An other very classical case is related to the expansion of numbers by means of continued fraction: to see this, take a = 1 and the two matrices:

(90) A 0 = 1 0 1 1 and A 1 = 1 1 0 1 .
Also in this case, the top Lyapunov direction map about a sequence ω ∈ Ω = {0, 1} N , may be sketched by direct computation. Consider ω = 0 a 0 1 a 1 0 a 2 • • • , where a 0 , a 1 , • • • are integers (with a 0 ≥ 0 and a i > 0, for i ≥ 1); for any n ≥ 0 and ε ∈ {0, 1}, a simple induction gives

(91) A(1 a 0 0 a 1 1 a 2 • • • ε an ) = a 0 1 1 0 a 1 1 1 0 • • • a n 1 1 0 0 1 1 0 ε = q n q n-1 p n p n-1 0 1 1 0 ε
where by convention (q -1 , p -1 ) = (1, 0), while for n ≥ 0 (92)

p n q n = 1 a 0 + 1 . . . + 1 a n =: [[a 0 , . . . , a n ]]
converges toward an irrational real number x = [[a 0 , a 1 , . . .]] ∈ [0 ; 1]; we use (91), and put θ n (x) := q n-1 /q n , so that by the approximation p n ≈ xq n , one gets

A(1 a 0 0 a 1 1 a 2 • • • ε an )U A(1 a 0 0 a 1 1 a 2 • • • ε an )U ≈ 1 (1 + x)(1 + θ n (x)) 1 θ n (x) x xθ n (x) 1 1 = H(x) 1 -H(x)
where H(x) = 1/(1 + x); then, it is simple to deduce that (see Figure 4-left for illustration)

V(ω) = p(ω) 1 -p(ω)
where p(ω) = [[1, a 0 , a 1 , . . .]] (93) 4.6. Example 6. We consider that P n is the n-step product whose formal limit (as

n → +∞) is the infinite product RST ST 2 ST 3 S • • • ST k S • • • , where R :=     2 1 1 1 1 2 1 1 0 0 0 0 0 0 0 0     , S :=     11 0 0 0 7 4 0 0 7 4 0 0 7 2 1 1     , T :=     11 0 0 0 7 4 0 0 7 2 2 0 7 2 1 1     .
Here, condition (C) may be checked directly and the (non necessarily injective) map h → V h (with 1 ≤ h ≤ H and V h-1 = V h ) given by Theorem A is defined by

V 1 =     1/2 1/2 0 0     , V 2 =     3/7 4/7 0 0     , V 3 =     1/2 1/2 0 0     and H = 3.
It is rather simple to illustrate how the number of different (exponential) growth rates of P n U j may depend on n -while H = 3 remains constant -together with the fact that (for 2 ≤ h ≤ H) any column of P n that converges in direction to V h is negligible with respect to (at least) one column that converges in direction to V h-1 .

Consider the increasing sequence of integers n 0 = 1, n 1 = n 0 + 1, n 2 = n 1 + 2, . . . , n k = n k-1 + k, . . . , so that P n 0 = R, P n 1 = R(S), P n 2 = RS(T S), P n 3 = RST S(T T S), . . . Any integer n ≥ 1, may be written n = n k + r for some k ≥ 0 and 0 ≤ r ≤ k, so that (94)

P n =     5 • 11 n-1 -3 • 4 n-1 3 • 4 n-1 -2 r+1 2 r+1 -1 1 5 • 11 n-1 -4 • 4 n-1 4 • 4 n-1 -2 r+1 2 r+1 -1 1 0 0 0 0 0 0 0 0     .
For any n ≥ 1, the sets of indices

J h (n) (1 ≤ h ≤ H = 3) involved in Theorem A together with the estimates of P n U j for j ∈ J h (n) are respectively J 1 (n) = {1} with P n U 1 ≈ 11 n , J 2 (n) = {2} with P n U 2 ≈ 4 n and finally, J 3 (n) = {3, 4} with P n U 4 ≈ 1 and P n U 3 ≈ 2 ϕ(n) .
The function n → ϕ(n) may be obtained by a straightforward computation, which gives

ϕ(n) = r = n -1 + 1 2 1 2 √ 8n -7 - 1 2 1 2 √ 8n -7 + 1 2 .
In particular, for 5 for a representation of x → ϕ(x)). x

n = n k + k = 1 + k(k + 1)/2 + k, one gets P n U (n) 3 ≈ 2 √ 2n (see Figure
→ ϕ(x) = x -1 + 1 2 1 2 √ 8x -7 - 1 2 1 2 √ 8x -7 + 1 2 ,
while the dashed green and the dotted blue ones represent respectively

x → 1 2 √ 8x -7 - 1 2 and x → 1 2 √ 8x -7 - 1 2 .

Rank one property of normalized matrix products

Let A be a matrix in the space M d (C) of the d × d matrices with complex entries. We denote by A the adjoint matrix of A and we write its singular value decomposition

A = S    σ 1 . . .
= P n e nχ 1 (n) = S n (U 1 U 1 )T n + d i=2 e nχ i (n) e nχ 1 (n) S n (U i U i )T n .
Therefore, the condition e n(χ 2 (n)-χ 1 (n)) → 0 for i = 2, . . . d ensures that P n / P n 2 -B n 2 → 0 holds with B n := S n (U 1 U 1 )T n . For the converse implication, assume the existence of rank 1 matrices B 1 , B 2 , . . . such that P n / P n 2 -B n 2 → 0, and assume -for a contradiction -the existence of a sequence of integers 1

≤ n 1 < n 2 < • • • for which e n k χ 2 (n k ) /e n k χ 1 (n k ) → α 2 > 0.
By a compactness argument, it is possible to choose the n k in such a way that S n k U i , T n k U i and B n k as well as the reals e n k χ i (n k ) /e n k χ 1 (n k ) (i = 1, . . . , d) converge as k → +∞, with L i , R i , B and α i being their respective limits. From our assumption that P n / P n 2 -B n 2 → 0 together with (96) it follows that B = d i=1 α i L i R i : because {L i } d i=1 and {R i } d i=1 are both orthonormal families, the fact that BR i = α i L i , together with α 1 = 1 and α 2 > 0, means that B is at least of rank 2: however B must be a rank one matrix, a contradiction. Remark 5.2. (1) : Let σ : Ω → Ω be the full shift map on Ω = {0, . . . , a} N and let A : Ω → M d (C) be a borelian map. We already saw that (n, ω)

→ P n (ω) = A(ω)A(σ • ω) • • • A(σ n-1 • ω)
is a sub-multiplicative process whenever Ω is endowed with a σ-ergodic probability measure, say µ. To emphasize the dependance on ω we note χ k (n, ω) (1 ≤ k ≤ d) the n-step Lyapunov exponents associated with P n (ω) as in (95). By Oseledets Theorem, we know that χ k (n, ω) tends µ-a.s. to the k-th Lyapunov exponent χ k (µ). We apply Theorem 5.1 to P n = P n (ω) for ω a µ-generic sequence: hence, in that case, a sufficient condition for e nχ 2 (n,ω) /e nχ 1 (n,ω) → 0 is that χ 1 (µ) > χ 2 (µ); therefore, if the top-Lyapunov exponent χ 1 (µ) is strictly dominant then, there exists a sequence B 1 (ω), B 2 (ω), . . . of rank one matrices s.t.

lim n→+∞ P n (ω) P n (ω) -B n (ω) = 0.
(2) : Let A = (A 1 A 2 , . . . ) be a sequence in M d (C) and suppose that P n / P n -B n → 0, where P n = A 1 . . . A n and each B n is a rank 1 matrix. Then, there is no reason for the projective convergence of either the rows or the columns of B n . Consider for instance the matrices

A =   1 1 0 0 1 0 0 0 1   , B =   0 0 1 1 0 0 0 1 0   , C =   1 0 0 1 1 0 0 0 1   , D =   0 1 0 0 0 1 1 0 0  
and consider that P n is the right product of the n first matrices of the infinite product

A 2 0 BC 2 1 DA 2 2 BC 2 3 D • • • Then, for n k = 2 0 + 2 1 + • • • + 2 2k-1 + 2k and m k = 2 0 + 2 1 + • • • + 2 2k + 2k + 1 lim k→+∞ B n k =   0 1 3 0 0 0 0 0 2 3 0   and lim k→+∞ B m k =   2 3 0 0 0 0 0 1 3 0 0   .
About the divergence of the normalized infinite products of matrices, we prove the following proposition and corollary inspired from Elsner & Friedland argument in [12, Theorem 1]. Proposition 5.3. Let A = (A 1 , A 2 , . . . ) be a sequence in M d (C) and P n := A 1 • • • A n be the n-step product; if n → P n / P n converges (in M d (C)) then, the matrices that occur infinitely many times in A (i.e. the A such that #{n ; A n = A} = +∞) have a common left-eigenvector.

Proof. Suppose that P n / P n → P as n → +∞ and let

λ n = P n / P n-1 . If A n k = A for n 1 < n 2 < • • • then, λ n k P n k / P n k = P n k -1
A/ P n k -1 → P A as k → +∞: hence λ n k → P A and taking the limit as k → +∞ gives P A P = P A. Since P = 1, there exists at least one i s.t. 0 = U i P so that (U i P )A = U i (P A) = P A (U i P ).

ω = ω 1 ω 2 • • • ∈ Ω := {0, . . . , a} N define P n (ω) = A(ω 1 ) • • • A(ω n ).
Then, for any Borel continuous probability µ fully supported by Ω, the sequence n → P n (ω)/ P n (ω) diverges for µa.e. ω ∈ Ω.

Proof. For µ-a.e. ω ∈ Ω the sequence A(ω) = (A(ω 1 ), A(ω 2 ), . . . ) has infinitely many occurrences of each A(i): hence, by Proposition 5.3, the sequence P n (ω)/ P n (ω) must be divergent.

In the case of products of nonnegative matrices, a straightforward consequence of parts (i) and (ii) of Theorem A is the following theorem.

Theorem 5.5. Let P n = A 1 • • • A n be the n-step right product of a sequence A = (A 1 , A 2 , . . . ) made of d × d matrices with non negative entries, satisfying condition (C) and, for 1 ≤ h ≤ H, let J h (n) (resp. V h ) be the subset of {1, . . . , d} (resp. the probability vector) given by Theorem A; then, the following propositions hold: (i) : each limit point of P n / P n is a rank one matrix: actualy, for any n ≥ 1, there exists a probability vector R n such that

lim n→+∞ P n P n -V 1 R n = 0 ;
(ii) : recall that h P n is for the d × d matrix obtained from P n by replacing by 0 the entries whose column indices are not in J h (n); then, each limit point of n → h P n / h P n is a rank one matrix: actually, for any n ≥ 1, there exists a probability vector R n,h such that

lim n→+∞ h P n h P n -V h R n,h = 0
(moreover, for any 2 ≤ h ≤ H the ratio h P n / (h -1) P n → 0 as n → +∞).

6. Gibbs properties of linearly representable measures 6.1. Gibbs measures. Fix a ≥ 2 an integer; an element in {0, . . . , a} n is written as a word (the neutral element of {0, . . . , a} * being the empty word • /). The topology of the product space Ω := {0, . . . , a} N is compact and given by the metric (ω, ξ) → 1/a n(ω,ξ) , where n(ω, ξ) = +∞ if ω = ξ while n(ω, ξ) = min{i ≥ 1 ; ω i = ξ i }, otherwise. Actually, for any 1/a n+1 ≤ r < 1/a n , the (open-closed) ball of radius r and centered at

x 1 • • • x n of length n,
ω = ω 1 ω 2 • • • is the so called cylinder set of the sequences ξ ∈ Ω such that ξ 1 • • • ξ n = ω 1 • • • ω n , denoted by [ω 1 • • • ω n ]. We shall also consider the shift map σ : Ω → Ω s.t. σ • ω = σ(ω 1 ω 2 • • • ) = ω 2 ω 3 • • • ,
which is an expanding continuous transformation of Ω. (We write a sequence ω in Ω as one-sided infinite word ω 1 ω 2 • • • whose digits ω i are in {0, . . . , a}.) The notion of Gibbs measure we are concerned with, is the one originally introduced by Bowen in his Lecture Notes [START_REF] Bowen | Some systems with unique equilibrium states[END_REF]; here, it is not necessary to enter into the details of the underlying theory, since only a few elementary facts are needed. Suppose that µ is a Borel probability on Ω = {0, . . . , a} N which, for simplicity, we suppose to be fully supported by Ω. The n-step potential φ n : Ω → R associated with µ is

φ 1 (ω) = log µ[ω 1 ] if n = 1 and for n ≥ 2, (97) φ n (ω) := log µ[ω 1 • • • ω n ] µ[ω 2 • • • ω n ] .
The fact that µ is fully supported ensures the functions φ n to be well defined and continuous. Moreover, it is easily checked that

µ[ω 1 • • • ω n ] = exp 1 k=n φ k (σ n-k • ω) ,
this last identity leading to the following proposition.

Proposition 6.1. Suppose the φ n converge uniformly to φ on Ω, as n → +∞: then,

1 K n ≤ µ[ω 1 • • • ω n ] exp n-1 k=0 φ(σ k • ω) ≤ K n , where K n := exp n k=1 φ -φ n ∞
and µ is a weak Gibbs measure (c.f. Yuri [START_REF] Yuri | Zeta functions for certain non-hyperbolic systems and topological Markov approximations[END_REF]), because (use φ -

φ n ∞ → 0 as n → +∞) lim n→+∞ 1 n log K n = 0 ; moreover, if K := sup n {K n } < +∞, then µ is a Gibbs measure (in the sense of Bowen [5, Theorem 1.3]), since 1 K ≤ µ[ω 1 • • • ω n ] exp n-1 k=0 φ(σ k • ω) ≤ K.
6.2. Linearly representable measures. Now, suppose that A(0), . . . , A(a) are d × d matrices with nonnegative entries; moreover, R is a fixed probability vector and ω = ω 1 ω 2 • • • is a given sequence in Ω; then we usually note

A(ω 1 • • • ω n ) := A(ω 1 ) • • • A(ω n )
and by definition, the n-step probability vector about ω and R is

(98) Π n (ω, R) := A(ω 1 • • • ω n )R A(ω 1 • • • ω n )R .
Actually, the ratio in (98) is not necessarily defined for any ω ∈ Ω and we are led to introduce

(99) Ω R := ω ∈ Ω ; ∀n ≥ 1, A(ω 1 • • • ω n )R = 0 .
This set is shift invariant in the sense that σ(Ω R ) = Ω R and compact (in Ω), because for fixed n, the set of words

ω 1 • • • ω n ∈ {0, . . . , a} n such that A(ω 1 • • • ω n )R = 0 is obviously finite. We note A * := A(0) + • • • + A(a)
and suppose in addition that R satisfies the eigen-equation A * R = R (if A * R = αR for some α > 0, there is no loss of generality to replace A i by A i /α). Let L be a column vector with nonnegative entries for which L R = 1. Then, we define the map

µ : {0, • • • , a} * = +∞ n=0 {0, . . . , a} n → [0, 1] by setting (100) µ(ω 1 • • • ω n ) = L A(ω 1 • • • ω n )R
An application of Kolmogorov Extension Theorem ensures µ to extend to a unique Borel probability measure on Ω, that we abusively note µ: this measure is determined by the condition

µ[ω 1 • • • ω n ] = µ(ω 1 • • • ω n ) to be satisfied for any word ω 1 • • • ω n ∈ {0, • • • , a} * .
A measure defined by means of matrix products, like for instance in (100), is usually called linearly representable (see [START_REF] Boyle | Hidden Markov processes in the context of symbolic dynamics[END_REF] for more details).

The measure µ has support in Ω R defined in (99) and we shall consider it as a measure on Ω R instead; moreover, for the sake of simplicity, we shall now assume that µ is actually fully supported by Ω R , that is µ[ω 1 • • • ω n ] > 0 for any ω ∈ Ω R and any n ≥ 1 (by defintion of Ω R , this holds for instance if L has positive entries). In view of Gibbs properties, the main point is to start from Proposition 6.1 and look at the convergence (as n → +∞) of the n-step potential

φ n : Ω R → R such that, for any ω = ω 1 ω 2 • • • ∈ Ω R , φ n (ω) := log µ[ω 1 • • • ω n ] µ[ω 2 • • • ω n ] = log L A(ω 1 )Π n-1 (σ • ω, R) L Π n-1 (σ • ω, R) .
We emphasize on the fact that existence and continuity of the limit map

V : Ω R → S d with (101) V(ω) := lim n→+∞ Π n (ω, R)
does not prevent from the possibility that L A(ω 1 )V(σ • ω) = 0 for some ω ∈ Ω R (the map ω → V(ω) -provided it exists -should be related to the top Lyapunov direction map introduced in § 4.5). This remark leads to the following proposition.

Proposition 6.2. If Π n (ω, R) → V(ω), uniformly over Ω R as n → +∞ and A(i)V(ω) = 0, for any i ∈ {0, . . . , a} and any ω ∈ Ω R then, for any L with positive entries,

lim n→+∞ log L A(ω 1 )Π n-1 (σ • ω, R) L Π n-1 (σ • ω, R) = log L A(ω 1 )V(σ • ω) L V(σ • ω)
uniformly over Ω R : moreover, if A * R = R and L R = 1 then µ defined in (100) is a weak Gibbs measure.

Proof. If two real-valued continuous maps f n , g n converge uniformly on a compact set, to positive limits f and g respectively, then log f n /g n converges uniformly to log f /g.

6.3.

Conditions for the uniform convergence. Pointwise convergence of Π n (ω, R) (provided it holds), does not imply uniform convergence over Ω R . To see this, consider for instance

A(0) =   1 0 1 0 1 0 0 0 0   , A(1) =   1 0 1 1/2 0 0 0 0 0   and R =   1/3 1/3 1/3   .
Because A(0) and A(1) 2 = A( 11) are both idempotent matrices, Π n (ω, R) → (2/3, 1/3, 0) as n → +∞, this pointwise convergence being valid for any ω ∈ Ω R = Ω = {0, 1} N . However, the convergence is not uniform on Ω because Π n (ω, R) = (4/5, 1/5, 0), for any ω ∈ [0 n-1 1] with n ≥ 1. Similarly, for L = (1, 1, 1) the convergence, for n → +∞, of the potential

φ n (ω) = log L A(ω 1 . . . ω n )R L A(ω 2 . . . ω n )R is not uniform on Ω because (for n ≥ 2) either φ n (ω) = log(6/5) if ω ∈ [10 n-2 1] or φ n (ω) = 0
otherwise and yet, the limit φ of φ n is continuous, since φ ≡ 0. The following lemma gives a criterion for uniform convergence of Π n (ω, R).

Lemma 6.3. n → Π n (•, R) is uniformly convergent over Ω R if and only if for any ω ∈ Ω R (102) lim n→+∞ sup Π n+r (ξ, R) -Π n (ξ, R) ; ξ ∈ [ω 1 • • • ω n ] ∩ Ω R and r ≥ 0 = 0.
Proof. The direct implication is given by the Cauchy criterion. Conversely, suppose that (102) holds. Given ε > 0 and ω ∈ Ω R there exists a rank n(ω) ≥ 1 such that for any r, s ≥ n(ω)

ξ ∈ [ω 1 • • • ω n(ω) ] ∩ Ω R =⇒ Π r (ξ, R) -Π s (ξ, R) ≤ ε. Each cylinder [ω 1 . . . ω n(ω)
] is an open set containing ω. Because Ω R is compact, it is covered by finitely many of such cylinders, say [ω i 1 . . . ω i n(ω i ) ] for i = 1, . . . , N . Hence if r, s ≥ max i {n(ω i )} the inequality Π r (ξ, R) -Π s (ξ, R) ≤ ε holds for any ξ ∈ Ω R : this proves that Π n (•, R) is uniformly Cauchy and converges uniformly.

For a fixed R ∈ S d , we now consider the problem of uniform convergence of Π n (•, R) toward V(•) within the framework of Theorem A. The point is to deal with conditions (C) w.r.t. each sequence A(ω) = (A(ω 1 ), A(ω 2 ), . . . ) for ω running over the whole Ω R . We shall say that ω ∈ Ω R is (C)-regular (resp. (C)-singular) if A(ω) satisfies (resp. does not satisfy) condition (C). Actually, one possible -consistent -difficult point (as we shall see for instance in Section 9) is the existence of s ∈ {0, • • • , a} so that s = ss • • • (and thus each sequence in σ

-n {s}) is (C)- singular. For ω = ω 1 ω 2 • • • ∈ Ω R and 1 ≤ i ≤ j, we use the notation ω i,j = ω i+1 . . . ω j ;
in particular ω i,i is the empty word • /. Proposition 6.4 (Pointwise convergence). For R ∈ S d and ω ∈ Ω R , the convergence of Π n (ω, R) to a limit vector V(ω) ∈ S d holds if ω satisfies at least one of the following conditions:

(i) : ω ∈ Ω R is (C)-regular, there exist an integer N , a real Λ ≥ 1 and a sequence of integers n → ψ(n), with ψ(n) ≤ n and ψ(n) → +∞ as n → +∞, s.t. for any n ≥ N , any r ≥ 0, Using the notations of Definition 1.1, let k = k(ψ(n)) be the integer such that s k+1 ≤ ψ(n) < s k+2 . We can replace the sequence n → ψ(n) by the sequence n → s k(ψ(n)) < ψ(n): this sequence also satisfies (S1) because the equality ∆ A(ω

(S1) : ∆ A(ω ψ(n),n+r )R = ∆ A(ω ψ(n),n )R .
ψ(n),n+r )R = ∆ A(ω ψ(n),n )R implies ∆ A(ω s k ,n+r )R = ∆ A(ω s k ,n )R .
Moreover with this new definition of the sequence n → ψ(n), by ( 27) of Lemma 3.1 the column vector

X r := A(ω ψ(n),n+r )R/ A(ω ψ(n),n+r )R belongs to H 2 (Λ ) with Λ = Λ/(1 -λ). Take n ≥ N , r ≥ 0; from (S1), we know that ∆(X r ) = ∆(X 0 ) and thus min h ; I(X r ) ∩ J h (ψ(n)) = ∅ = min 1 ≤ h ≤ H ; I(X 0 ) ∩ J h (ψ(n)) = ∅ =: h n (ω), that is (with notations in Theorem A) 1 ≤ h ψ(n) (X r ) = h ψ(n) (X 0 ) = h n (ω)
≤ H: we emphasize that h n (ω) depends on n and ω but not on r (this is due to the synchronization condition (S1)). For ω being supposed (C)-regular, we can apply part (iii) of Theorem A which gives real numbers ε n → 0 (depending on ω but not on r) s.t.

A(ω 0,ψ(n) )X r A(ω 0,ψ(n) )X r -V hn(ω) ≤ ε ψ(n) Λ (here V 1 , .
. . , V H are the probability vectors associated with A(ω) = (A(ω 1 ), A(ω 2 ), . . . ) by Theorem A). Therefore, using the triangular inequality, (103)

Π n+r (ω, R) -Π n (ω, R) ≤ A(ω 0,ψ(n) )X r A(ω 0,ψ(n) )X r -V hn(ω) + A(ω 0,ψ(n) )X 0 A(ω 0,ψ(n) )X 0 -V hn(ω) ≤ 2ε ψ(n) Λ and thus n → Π n (ω, R) is a Cauchy sequence of S d .
(ii) : Let ω = ws where w is either the empty word

• / or w = ω 1 • • • ω N for a N ≥ 1. Under condition (S2.
2), we know that A(w)B s R = 0 (with the convention that A(• /) is the d×d identity matrix) and it follows from (S2.1) that Π n (ws, R) → A(w)B s R/ (A(w)B s R as n → +∞.

The following Proposition 6.5 is an extrapolation of the above Proposition 6.4, in view of uniform convergence of Π n (•, R). It gives (among many others) two possible situations for which the Cauchy condition (102) of Lemma 6.3 holds. It is intended to illustrate how Theorem A applies to establish (when it holds) the uniform convergence of Π n (•, R). Apart its illustrative content, we shall make a more specific use of Proposition 6.5 when dealing with the uniform convergence part of Theorem 8.7 considered in § 9.3. (We stress that uniform convergence of Π n (•, R) over Ω R is anyway a technical question.) The two items of Proposition 6.5 also correspond to the (inevitably technical) ideas we already use in some previous papers, to prove the uniformity of the convergence. For proving this, the main difficulty is related to the set H 2 (Λ), that is why this set is involved in the conditions of Proposition 6.5 and not in Proposition 6.4. Proposition 6.5 (Uniform convergence). Let R ∈ S d for which the set of the (C)-regular sequences in Ω R is X = Ω R \ s∈S +∞ n=0 σ -n {s}, where S ⊂ {0, . . . , a}; then, the Cauchy condition (102) of Lemma 6.3 holds at ω ∈ Ω R if at least one of the following conditions is satisfied:

(i) : ω ∈ X, there exist an integer N , a real Λ ≥ 1 and a sequence of integers n → ψ(n), with ψ(n) ≤ n and ψ(n) → +∞ as n → +∞, s.t. for any n ≥ N , any ξ

∈ [ω 1 • • • ω n ] ∩ Ω R and r ≥ 0, (U1.1) : A(ξ ψ(n),n+r )R ∈ H 2 (Λ), (U1.2) : ∆ A(ξ ψ(n),n+r )R = ∆ A(ξ ψ(n),n )R .
(ii) : ω ∈ Ω R and ω = ws, where w is a (possibly empty) word on {0, . . . , a} and s ∈ S is a digit for which (U2) : there exist C s , D s ∈ S d s.t.

(U2.1) : lim n→+∞ A(s) n A(s) n -C s D s = 0, (U2.2) : A(w)C s = 0 ;
and (U3) : there exist an integer N , a real Λ ≥ 1 and a sequence of integers n → ϕ(n), with ϕ(n) ≤ n and ϕ(n) → +∞ as n → +∞ and such that for any n ≥ N , any ξ ∈ [ω 1 • • • ω n ] ∩ Ω R and r ≥ 0, there exist m ∈ {ϕ(n), . . . , n + r} for which the following three propositions hold:

(U3.1) : ξ ∈ [ω 1 • • • ω m ], (U3.2) : A(ξ m,n+r )R ∈ H 2 (Λ), (U3.3) : D s A(ξ m,n+r )R = 0.
Proof. (i) : Let ω ∈ X for which (U1.1) and (U1.2) are satisfied with the integer N , the constant Λ and the sequence n

→ ψ(n). Take n ≥ N , ξ ∈ [ω 1 • • • ω n ] ∩ Ω R , r ≥ 0 and define X r (ξ) := A(ξ ψ(n),n+r )R/ A(ξ ψ(n),n+r )R . Notice that X 0 (ξ) = X 0 (ω) =: X 0 ; then, from (U1.2) we know that ∆(X r (ξ)) = ∆(X 0 ) and thus min h ; I(X r (ξ)) ∩ J h (ψ(n)) = ∅ = min h ; I(X 0 ) ∩ J h (ψ(n)) = ∅ =: h n (ω), that is (with notations in Theorem A) 1 ≤ h ψ(n) (X r (ξ)) = h ψ(n) (X 0 ) = h n (ω)
≤ H: we emphasize that h n (ω) does depend on n and ω but neither on ξ ∈ [ω 1 • • • ω n ] ∩ Ω R nor on r (this is due to the synchronization condition (U1.2)). For ω being supposed (C)-regular, we can apply part (iii) of Theorem A which gives real numbers ε n → 0 (depending on ω but not on r) s.t.

Π n+r (ξ, R) -V hn(ω) = A(ω 0,ψ(n) )X r (ξ) A(ω 0,ψ(n) )X r (ξ) -V hn(ω) ≤ ε ψ(n) Λ Xr(ξ)
(here V 1 , . . . , V H are the probability vectors associated with A(ω) = (A(ω 1 ), A(ω 2 ), . . . ) by Theorem A). Therefore, using the triangular inequality,

Π n+r (ω, R) -Π n (ω, R) ≤ A(ω 0,ψ(n) )X r (ξ) A(ω 0,ψ(n) )X r (ξ) -V hn(ω) + A(ω 0,ψ(n) )X 0 A(ω 0,ψ(n) )X 0 -V hn(ω) ≤ ε ψ(n) Λ Xr(ξ) + Λ X 0 ;
however, by (U1.1) one has Λ Xr(ξ) , Λ X 0 ≤ Λ and thus the uniform Cauchy condition (102) of Lemma 6.3 holds for ω ∈ X for which (U1.1)-(U1.2) are satisfied: this proves part (i).

(ii) : Let ω = ws ∈ Ω R , where s ∈ S and w is either the empty word • / (and

k 0 := 0) or w = ω 1 • • • ω k 0 for a k 0 ≥ 1. Consider n ≥ 1 and r ≥ 0 arbitrary given; for ξ ∈ Ω R s.t. ξ 1 • • • ξ n = ω 1 • • • ω n = ws n we note ζ = ξ if w = • / or ζ = σ k 0 • ξ otherwise. We
A(ξ 0,n+r )R A(ξ 0,n+r )R -B(w)C s = B(w)A(ζ 0,m )Y B(w)A(ζ 0,m )Y -B(w)C s = B(w) C s D s Y + M m Y U B(w)C s D s Y + U M m Y -C s (recall that U B(w)C s = 1) = B(w) C s D s Y + M m Y -D s Y C s -U M m Y C s ) D s Y + U M m Y = B(w) M m Y -U M m Y C s D s Y + U M m Y . On the one hand, M m Y -U M m Y C s ≤ 2 M m .
On the other hand, to lower bound

D s Y + U M m Y we use conditions (U3.2) and (U3.3) ensuring Y ∈ H 2 (Λ) and D s Y = 0. Indeed, Y (i) = 0 implies Y (i) ≥ 1/Λ and D s Y ≥ D s /Λ, where D s = min{D s (i) = 0}. Finally D s Y + U M m Y ≥ D s /Λ -M m ,
and thus (for n large enough so that

D s /Λ -M m > 0), Π n+r (ξ, R) -Π n (ξ, R) = A(ξ 0,n+r )R A(ξ 0,n+r )R - A(ξ 0,n )R A(ξ 0,n )R ≤ 4 B(w) • M m D s /Λ -M m .
By definition, ϕ(n)-k 0 ≤ m while ϕ(n) → +∞ as n → +∞: because M m → 0 as m → +∞, the Cauchy condition (102) of Lemma 6.3 is satisfied.

The Kamae measure 1

Let T 2 = R/Z × R/Z (identified with [0 ; 1[×[0 ; 1[) be the 2-torus and T : T 2 → T 2 s.t. T (x, y) = ({3x}, {2x}) ; (here {x} stands for the fractional part of the real number x). A T -invariant carpet of T 2 is a compact subset K of T 2 which is T -invariant in the sense that T (K) = K. The symbolic model of T is given by the full shift map σ : Σ → Σ, where Σ = (0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (2, 1)

N .

It is associated with the representation map Rep

: Σ → T such that Rep (x 1 , y 1 )(x 2 , y 2 ) • • • = +∞ k=1 x k 3 k , +∞ k=1 y k 2 k . Any subshift Z ⊂ Σ (i.e. Z is a compact subset of Σ which is shift-invariant in the sense that σ(Z) = Z) is associated with the T -invariant carpet K = Rep(Z): we call Z the symbolic model of K. (Conversely any T -invariant carpet of T 2
, has a symbolic model.) The fractal geometry of these subsets of the 2-torus, has been widely studied in the framework of the socalled Variational Principle for Dimension (see for instance [START_REF] Mcmullen | The Hausdorff dimension of general Sierpiński carpets[END_REF][START_REF] Bedford | Crinkly curves, Markov partitions and dimension[END_REF][START_REF] Kenyon | Measures of full dimension on affine-invariant sets[END_REF]). In this paragraph we shall focus our attention on what we call the Kamae carpet (see [START_REF] Kamae | A characterization of self-affine functions[END_REF] and [39, § 6]), that is

K = Rep(Z)
where Z is the sofic system specified by the adjacency scheme

A =    A (0,0) =   1 0 0 0 0 0 1 0 0   , A (1,0) =   0 0 0 0 0 0 0 0 0   , A (2,0) =   0 0 0 0 1 0 0 1 0   , A (0,1) =   0 0 0 0 1 0 0 0 0   , A (1,1) =   0 0 1 0 0 1 0 0 1   , A (2,1) =   1 0 0 0 0 0 0 0 0     
(An equivalent form of A is given by the adjacency graph in Fig. 6 : see [START_REF] Denker | Ergodic Theory on Compact Spaces[END_REF] and [START_REF] Lind | An Introduction to Symbolic Dynamics and Coding[END_REF] for the theory of shift systems). By the theory of sofic systems, ξ = (x 1 , y 1 )(x 2 , y 2 ) The adjacency matrix of A, that is

• • • ∈ Z if and only if A (x 1 ,y 1 ) • • • A (xn,yn) > 0 for any n ≥ 1.
A * = i,j A (i,j) =   2 0 1 0 2 1 1 1 1   ,
has a spectral radius ρ A * = 3. Hence, the topological entropy of σ : Z → Z is h σ (Z) = log 3: the Parry-Bowen measure over Z is the unique shift-ergodic measure η supported by Z and whose Kolmogorov-Sinaï entropy h σ (η) coincides with the topological entropy h σ (Z): in other words:

h σ (η) = h σ (Z) = log 3
(for the concepts of entropy -not decisive in the sequel -see [START_REF] Denker | Ergodic Theory on Compact Spaces[END_REF]). Let y : Σ → {0, 1} N =: Y be the map

(x 1 , y 1 )(x 2 , y 2 ) • • • → y (x 1 , y 1 )(x 2 , y 2 ) • • • = y 1 y 2 • • • ;
it is easily seen that y makes2 σ : Y → Y a factor of σ : Σ → Σ (y is continuous surjective and y • σ = σ • y). Because y(Z) = Y, the restriction y : Z → Y is also a factor map. Actually, the Gibbs properties of the y-axis projection ηy = η • y -1 of η over Y, plays a crucial role w.r.t.

the fractal geometry of K (see [38, Theorem A]). The measure ηy -that we call the Kamae measure -turns out to be linearly representable, since for any binary word

ξ 1 • • • ξ n , (104) ηy [y 1 • • • y n ] = 1 3 n A(y 1 • • • y n ) , where A(0) = A (0,0) + A (1,0) + A (2,0) and A(1) = A (0,1) + A (1,1) + A (2,1) , so that A(0) =   1 0 0 0 1 0 1 1 0   and A(1) =   1 0 1 0 1 1 0 0 1   .
To analyze the Gibbs property of ηy , we need prove the following proposition.

Π n (y, U ) := A(y 1 • • • y n )U A(y 1 • • • y n )U and U =   1 1 1   .
converge toward a limit function V : Y → S 3 uniformly over Y.

Proposition 7.1 may be establishes by elementary direct methods. However, to illustrates the technics developed in the present paper, we shall remain in the framework of Theorem A, using in particular Lemma 6.3 and the technics involved in Propositions 6.4 and 6.5.

Proposition 7.2. The set of sequences y ∈ Y (= {0, 1} N ) which are (C)-regular -in the sense that A(y) = (A(y 1 ), A(y 2 ), . . . ) satisfies condition (C) -is

Y := Y \ +∞ n=0 σ -n { 0} ∩ Y \ +∞ n=0 σ -n { 1} .
Proof. To begin with, A(0) is an idempotent matrix and for a ≥ 0 one has the matrix identities (105)

A(0)A(1) a A(0) =   a + 1 a 0 a a + 1 0 2a + 1 2a + 1 0   =: B a ;
moreover, a simple induction shows that for a 1 , a 2 , . . . a given sequence of positive integers

(106) B a 1 • • • B an = B αn where α n = 1 2 (2a 1 + 1) • • • (2a n + 1) - 1 2
and this proves A(y) satisfies (C), for any y ∈ Y . The matrix A(0) is idempotent and does not belong to H 1 : hence, 0 is a (C)-singular sequence. Likewise, 1 is (C)-singular, since (107)

A(1) n =   1 0 n 0 1 n 0 0 1   .
The simple convergence of Π n (y, U ) toward a limit V(y) ∈ S d , holds for any y ∈ Y. For y ∈ Y , the existence of the limit vector V(y) may be obtained by application of Theorem A (actually Corollary A). A more straightforward approach gives the expression of V(y): indeed, since 01 must factorize y infinitely many times, it is necessary that

y = 1 a 0 0 b 0 1 a 1 0 b 1 1 a 2 • • • = 1 a 0 0 * 1 a 1 0 * 1 a 2 • • • ,
where a i and b i are positive integers, with the possible exception that a 0 ≥ 0 (and the convention that 1 0 = • /); here, we also write 0 * for a non empty arbitrary block of 0. Decomposing y in blocks of the form 0 * 1 a i 0 * (for a i > 0) and using the fact that A(0) is idempotent together with (105)-( 106)-(107), one finds:

(108) V(y) = 1 a 0 + 1 •   1 0 a 0 0 1 a 0 0 0 1     1/4 1/4 1/2   .
The simple convergence also holds about sequence of the form y = w 0 and y = w 1, since

V(w 0) = A(w)A(0)U A(w)A(0)U and V(w 1) = A(w)(U 1 + U 2 ) A(w)(U 1 + U 2 ) .
Proof of Proposition 7.1. Let a ≥ 0 (and recall that

1 0 = • /): then, for any y ∈ [1 a 0] (109) n ≥ a + 1 =⇒ Π n (y, U ) =   θ(a) θ(a) 1 -2θ(a)   where θ(a) = 1 4 1 + 2a 1 + a .
The Cauchy condition (102) of Lemma 6.3 holds for y ∈ [1 a 0 0], since for n ≥ a 0 + 1 and r ≥ 0,

z ∈ [y 1 • • • y n ] =⇒ Π n+r (z, U ) -Π n (z, U ) = 0.
For the remaining case of y = 1, take z ∈ Y s.t.

z 1 • • • z n = 1 • • • 1. Then, either z 1 • • • z n+r = 1 • • • 1 and we use (107) to write Π n+r (z, U ) -Π n (z, U ) ≤ 2 2n + 2 2n + 3   1/2 1/2 1/(2n + 2)   -   1/2 1/2 0   , or z 1 • • • z n+r = 1 a 0w, so that using (109) Π n+r (z, U ) -Π n (z, U ) ≤   θ(n) θ(n) 1 -2θ(n)   -   1/2 1/2 0   + 2n + 2 2n + 3   1/2 1/2 1/(2n + 2)   -   1/2 1/2 0   ;
the Cauchy condition (102) of Lemma 6.3 is again satisfied about the (C)-singular sequence y = 1.

The function V being continuous, the value of V(y) we have obtained in (108) for y = 1 a 0 0 * 1 a 1 0 * 1 a 2 • • • remains valid for any y ∈ [1 a 0 0]. Finally, notice that for any y ∈ Y and n → +∞, one has

φ n (y) = log ηy [y 1 • • • y n ] ηy [y 2 • • • y n ] → φ(y) = log 1 3 • U A(y 1 )V(σ • y) U V(σ • y) = log 1 3 • U A(y 1 )V(σ • y) .
Proposition 7.3 (see [START_REF] Olivier | Uniqueness of the measure with full dimension on sofic affine invariant subsets of the 2-torus[END_REF]). The Kamae measure ηy is an ergodic weak Gibbs measure of φ : 8. Gibbs structure of a special Bernoulli convolution 8.1. Bernoulli convolutions. Let 1 < β < 2 be an arbitrary real number and define µ to be the distribution of the random variable X : {0, 1} N → R such that

{0, 1} N → R such that φ(y) =            log 1 3 if y ∈ [0 2 ] or y = 1, log 1 3 + log 2a+1 a+1 if y ∈ [01 a 0], a ≥ 1, log 1 3 + log a+1 a if y ∈ [1 a 0], a ≥ 1, log 2 3 if y = 0 1.
X(ξ 1 ξ 2 • • • ) = (β -1) +∞ n=1 ξ n β n
where {0, 1} N is weighted by the Bernoulli probability P of parameter (p 0 , p 1 ) (the normalization coefficient (β -1) is introduced so that 0 ≤ X(ξ 1 ξ 2 • • • ) ≤ 1). Equivalently, µ may be defined as a Bernoulli convolution, that is an Infinite Convolved Bernoulli Measure (ICBM). Indeed, the law of the random variable

X n : {0, 1} N → R such that X n (ξ) = (β -1)ξ n /β n is p 0 • δ 0 + p 1 • δ xn ,
where x n = (β -1)/β n (and δ a is the Dirac mass concentrated on {a}): because X 1 , X 2 , . . . form a sequence of independent random variables, one gets:

µ := +∞ * n=1 p 0 • δ 0 + p 1 • δ xn .
The present Section is based on previous works [START_REF] Feng | Multifractal Analysis of Weak Gibbs Measures and Phase Transition -Application to some Bernoulli Convolutions[END_REF][START_REF] Olivier | On the Gibbs properties of Bernoulli convolutions related to βnumeration in multinacci bases[END_REF][START_REF] Olivier | Infinite Convolution of Bernoulli Measures, PV numbers and related problems in the dynamics of Fractal Geometry[END_REF][START_REF] Olivier | On a class of sofic affine invariant subsets of the 2-torus related to an Erdös problem[END_REF], concerned with the Gibbs properties of Bernoulli convolutions when β has specific algebraic properties. The framework of these papers stands the arithmetic properties of the Pisot-Vijayaraghavan (PV) numbers: β is a PV-Number if it is a real algebraic integer with β > 1, each ones of its Galois conjugates having a modulus stricly bounded by 1. We shall develop here the approach in [START_REF] Olivier | On the Gibbs properties of Bernoulli convolutions related to βnumeration in multinacci bases[END_REF] and [START_REF] Olivier | Infinite Convolution of Bernoulli Measures, PV numbers and related problems in the dynamics of Fractal Geometry[END_REF] used to deal with the multinacci numbers: the multinacci (number) of order n (for n ≥ 2) is the PV-number whose minimal polynomial is X n -(X n-1 + • • • + X + 1); the large Golden Number Φ is the multinacci of order 2, that is Φ > 1 and Φ 2 = Φ + 1 -the opposite of its Galois conjugate being the small Golden Number 1/Φ. The so-called Erdős measure corresponds to the case of β = Φ and (p 0 , p 1 ) = (1/2, 1/2): it is issued from An Erdős Problem dating back to the 1939 paper by Erdős [START_REF] Erdös | On a family of symmetric Bernoulli convolutions[END_REF] (see [START_REF] Peres | Sixty years of Bernoulli convolutions[END_REF] for a review, historical notes and detailed references).

The question of the Gibbs nature of the Bernoulli convolutions was raised by Sidorov & Vershik in [START_REF] Sidorov | Ergodic Properties of the Erdös Measure, the Entropy of the Goldenshift and Related Problems[END_REF]: partial answers do exist (for instance for the multinacci numbers [START_REF] Olivier | On the Gibbs properties of Bernoulli convolutions related to βnumeration in multinacci bases[END_REF][42]) but, even for β within the class of the PV numbers, a general approach is lacking. In other to enlighten the difficulties of the question, we shall consider a PV-number -not belonging to the multinacci family -that is β ≈ 1.755 s.t.

(110)

β 3 = 2β 2 -β + 1.
In the subsequent analysis, the algebraic properties of the number β are not apparent: actually they are implicit w.r.t. the fact that the vertex set (namely V) of a certain adjacency graph (see Lemma 8.3) is finite: let's mention the underlying result ensuring that property for a general PV-number is known as Garsia separation Lemma (see [START_REF] Garsia | Entropy and Singularity of infinite convolutions[END_REF] and [41, Proposition 2.6]).

8.2. β-shifts. For the sake of simple exposition, we consider in this paragraph that 1 < β < 2. The Parry expansion [START_REF] Parry | On the β-expansions of real numbers[END_REF] of the real number x ∈ [0 ; 1[ in base β is the unique sequence

1 2 • • • ∈ {0, 1} N such that, for any n ≥ 1, (111)
. n ∈ Ω (n) β , I 1 ... n := x ∈ [0 ; 1[ ; ε 1 (x) • • • ε n (x) = 1 . . . n .

8.3.

A special case. We now assume that β ≈ 1.755 is the PV-number defined by (110) and µ stands for the uniform (i.e. (p 0 , p 1 ) = (1/2, 1/2)) Bernoulli convolution associated with this β.

Lemma 8.1. The β-shift is the subshift of {0, 1} N determined by the finite type condition:

(112) 1 2 • • • ∈ Ω β ⇐⇒ k k+1 = 11 =⇒ k+2 k+3 = 00
Proof. Let = 1 2 • • • be the Parry expansion of a given real number x; the condition (111) is equivalent to

β n (x -s n ) = β n (x -s n-1 ) -n ∈ [0 ; 1[, hence n = β n (x -s n-1
) -with the convention that s 0 = 0. Now (111) implies

n+1 β n+1 + n+2 β n+2 + n+3 β n+3 + n+4 β n+4 ≤ x -s n < 1 β n = 1 β n+1 + 1 β n+2 + 1 β n+4
(the last equality being equivalent to (110)); hence, the condition n+1 n+2 = 11 with n+3 n+4 = 00 leads to an impossible strict inequality in any cases.

Conversely, let = 1 2 • • • ∈ {0, 1} N be a sequence satisfying the r.h.s. conditions in (112); for any n ≥ 0, the sequence σ n • is either equal to 1100 or begins by either (1100) i 0 or (1100) i 10 for some integer i ≥ 0 (recall that given w any word, then, for any word w ∈ {0, 1, 2} * one has the partition

J w = J w0 J w1 J w2 .
The two subsequent lemmas give a way to compute the µ-measure of the intervals J w , for any w ∈ {0, 1, 2} * by means of the matrices (114)

A(0) :=          
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0

          A(1) :=          
0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

          A(2) :=          
1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

         
(considered in Section 9 in view of an application of Theorem A). Lemma 8.3. Consider the vertex set

V = v 1 = 0, v 2 = 1, v 3 = 1 -(β -1) 2 , v 4 = -(β -1) 2 , v 5 = β -1, v 6 = β -(β -1) 2 , v 7 = β(β -1)
together with the matrices Then, for any n ≥ 1 and any word

M (0) := 1 2 A(0), M (1)
ξ 1 • • • ξ n ∈ {0, 1, 2} n (115)      µ 1 β v 1 + J ξ 1 •••ξn . . . µ 1 β v 7 + J ξ 1 •••ξn      = M (ξ 1 • • • ξ n )R.
Proof. For any real number γ we evaluate

µ 1 β γ + J ξ 1 •••ξn
in the three cases when ξ 1 is either 0, 1 or 2. First, consider that ξ 1 = 0, so that

(β -1) +∞ k=1 ω k β k ∈ 1 β γ + J ξ 1 •••ξn ⇐⇒ (β -1) +∞ k=1 ω k+1 β k ∈ 1 β γ + J ξ 1 •••ξn with γ = γβ -ω 1 β(β -1) and since ω 1 ∈ {0, 1}, (116)
µ 1 β γ + J ξ 1 •••ξn = 1 2 γ ∈V 0 (γ) µ 1 β γ + J ξ 1 •••ξn , where V 0 (γ) := γβ -uβ(β -1) ; u ∈ {0, 1} .
We proceed in the same way for ξ 1 = 1:

(117) µ 1 β γ + J ξ 1 •••ξn = 1 4 γ ∈V 1 (γ) µ 1 β γ + J ξ 1 •••ξn , where V 1 (γ) := γβ 2 + β -(uβ + v)β(β -1) ; u, v ∈ {0, 1} .
Finally, in the case of ξ 1 = 2, one gets

(118) µ 1 β γ + J ξ 1 •••ξn = 1 16 γ ∈V 2 (γ) µ 1 β γ + J ξ 1 •••ξn , where V 2 (γ) = γβ 4 + β 3 + β 2 -(uβ 3 + vβ 2 + wβ + x)β(β -1) ; u, v, w, x ∈ {0, 1} .
However, the measure µ being supported by the unit interval [0 ; 1], one has: then, a direct computation gives

Lemma 8.4. γ ∈] -1 ; β[ =⇒ µ 1 β γ + [0 ; 1[ = 0.
V = v 1 = 0, v 2 = 1, v 3 = 1 -(β -1) 2 , v 4 = -(β -1) 2 , v 5 = β -1, v 6 = β -(β -1) 2 , v 7 = β(β -1)
.

For e ∈ {0, 1, 2}, we define A (e) to be the 7 × 7 incident matrix s.t. for any 1 ≤ i, j ≤ 7, the (i, j) entry of A (e) is 1 if and only if v i e -→ v j (the corresponding incidence relations are represented by the adjacency graph in Figure 8). The key point is that

A (0) = A(0), A (1) = A(1), A (2) = A(2)
and one deduces from (116), ( 117) and (118) that for any ξ ∈ {0, 1, 2} N and any n ≥ 1

     µ 1 β v 1 + J ξ 1 •••ξn . . . µ 1 β v 7 + J ξ 1 •••ξn      = M (ξ 1 • • • ξ n )R, where R =      µ 1 β v 1 + [0 ; 1[ . . . µ 1 β v 7 + [0 ; 1[     
.

In order to compute R we consider the case n = 1 and we sum for ξ 1 ∈ {0, 1, 2}. Since (Proposition 8.2) the sets J ξ 1 form a partition of [0 ; 1[, we obtain that R is an eigenvector of M (0) + M (1) + M (2). Moreover, because the sum of the two first entries in R is

µ 0 + 1/β[0 ; 1[ + µ 1/β + 1/β[0 ; 1[ = µ 1/β[0 ; 2[ = µ([0 ; 1]) = 1,
the computation of this eigenvector gives the expected value for R.

H k (ξ) for ξ ∈ [1] ∪ [0] and k ≥ 0 is positive. Because ξ 1 = 2, we consider the largest 1 ≤ m ≤ n for which 2 m is a prefix of ξ 1 . . . ξ n , so that ξ 1 • • • ξ n = 2 m ξ m+1 . . . ξ n where ξ m+1 = 2 if m < n. From ( 121) and (119) together with

U M (2) m ≤ 3m 2 4m • U ≤ 3n 2 4m • U and U 2 M (1)M (0)M (2) m-1 = 1 2 4m-1 • U 5 , one deduces ν (J ξ 1 •••ξn ) µ (J ξ 1 •••ξn ) = U M (2) m Π n-m (σ m • ξ, R) U 2 M (1)M (0)M (2) m-1 Π n-m (σ m • ξ, R) ≤ 3n 2 • U Π n-m (σ m • ξ, R) U 5 Π n-m (σ m • ξ, R) ,
that is, with Π n-m (σ m • ξ, R) = 1 and the definition of both H n-m and inf{H * }

(126) ξ ∈ [2] =⇒ ν (J ξ 1 •••ξn ) µ (J ξ 1 •••ξn ) ≤            3n • U R 2 • U 5 R if m = n 3n • U Π n-m (σ m • ξ, R) 2 • H n-m (σ m • ξ) ≤ 3n 2 • inf{H * } if m < n
It follows from ( 123)-( 124)-( 125)-( 126) that (122) holds, proving the theorem.

9. An advanced application of Theorem A: proof of Theorem 8.7

This section is devoted to the full proof of Theorem 8.7 stated and used in the previous section. The main problem is concerned with the application of Theorem A w.r.t. the matrix products of the three 7 × 7 matrices, A(0), A(1), A(2) given in (114). In view of Theorem A, the major ingredient we shall need is the following proposition. The proof of Proposition 9.1 depends on the introduction of a language W ⊂ {0, 1, 2} * . 9.1. The language W. The initial idea is to find (if possible) a language W ⊂ {0, 1, 2} * (i.e. each W ∈ W is a word with digits in {0, 1, 2} and whose length |W | is finite) with two properties as follows. First (see Lemma 9.3 for the exact statement), W generates the sequences in X (i.e. the ω ∈ {0, 1, 2} N for which σ k • ω never belong to { 0, 2}) in the sense that such a ω may be written as a concatenation of words, that is ω = W 0 W 1 W 2 . . . , where W 1 W 2 . . . are in W and W 0 is a (possibly empty) strict suffix of a word in W. The second condition is the existence of 0 ≤ λ 0 < 1 ≤ Λ 0 < +∞ such that A(W ) ∈ H 2 (Λ 0 ) for any W ∈ W (Lemma 9.2), and A(W ) ∈ H 1 ∩ H 3 (λ 0 ) when W is a concatenation of a (fixed) number of words in W (Lemma 9.4). To construct W, notice that for s either equal to 0 or 2, the condition that A(s k ) = A(s) k ∈ H 2 (Λ) for any k implies Λ = +∞ (this is the reason why 0 and 2 are (C)-singular); taking this remark into account, we begin to fix 1 ∈ W, the other words W ∈ W being of the form W s k , for some words W = • / specified later, while s = 0 or 2, and for any k ≥ 1: more precisely, the (non empty) prefixes W of the words W ∈ W will be determined in order to ensure that any finite word not ending by 1 has a suffix of the form W s k , or W s k with W suffix of W , while A(W ) = A(W s k ) belongs to H 2 (Λ 0 ), for a Λ 0 < +∞ independent of W (the determination of λ 0 being obtained in a second step). For instance, let's compute successively A(2 k ), A(02 k ), A(102 k ), A(0102 k ), . . . , which gives

A(2 k ) =           1 0 0 0 k 0 1 0 0 0 0 0 0 0 1 0 0 0 k -1 0 1 0 0 0 1 k 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0           A(02 k ) =           1 0 0 0 k 0 1 1 0 0 0 k -1 0 1 0 0 0 1 k + 1 0 0 0 0 0 0 0 0 0 1 0 0 0 k 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0           A(102 k ) =           0 0 0 1 k + 1 0 0 0 0 0 0 1 0 0 1 0 0 0 k 0 1 1 0 0 0 k 0 1 0 0 0 1 k + 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0           A(0102 k ) =           0 0 0 1 k + 1 0 0 1 0 0 0 k 0 1 1 0 0 1 2k + 1 0 1 0 0 0 0 0 0 0 0 0 0 1 k + 1 0 0 0 0 0 1 k + 1 0 0 0 0 0 0 1 0 0           A(10102 k ) =           1 0 0 1 2k + 1 0 1 0 0 0 1 k + 1 0 0 0 0 0 1 k + 1 0 0 0 0 0 1 k + 1 0 0 1 0 0 1 2k + 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0           . Observe that if A(W ) ∈ H 2 (Λ), for any W = 2 k , with k ≥ 1 (and likewise for W = 02 k , 102 k or 0102 k ) then Λ = +∞, while for W ∈ +∞ k=1 {10102 k } it is simple to check that A(W ) ∈ H 2 (7) (for indeed 2 ≤ 7 • 1, 5 ≤ 7 • 1, 7k + 5 ≤ 7 • (k + 1), 7k + 5 ≤ 7 • (2k + 1)
are valid inequalities, for any k ≥ 1). Using this algorithm, leads to the language:

W := 1 ∪ +∞ k=1 010 k , 110 k , 210 k , 20 k , 002 k , 00102 k , 10102 k , 20102 k , 1102 k , 2102 k , 202 k , 12 k ,
together with the fact that A(1) ∈ H 2 (2), while for any k ≥ 1

A(010 k ) ∈ H 2 (8) A(110 k ) ∈ H 2 (8) A(210 k ) ∈ H 2 (6) A(20 k ) ∈ H 2 (7) A(002 k ) ∈ H 2 (12) A(00102 k ) ∈ H 2 (13) A(10102 k ) ∈ H 2 (7) A(20102 k ) ∈ H 2 (6) A(1102 k ) ∈ H 2 (8) A(2102 k ) ∈ H 2 (6) A(202 k ) ∈ H 2 (5) A(12 k ) ∈ H 2 (9)
These computations also ensure that A(W ) ∈ H 3 (5), for any W ∈ W.

Lemma 9.2. A(W ) ∈ H 2 (13) ∩ H 3 (5), for any W ∈ W. Lemma 9.3. Given ω = ω 1 ω 2 • • • ∈ {0, 1, 2} N one has (i) : for any n ≥ 1, either ω 1 • • • ω n is a strict suffix W 0 (n) of a word in W or there exists a possibly empty strict suffix W 0 (n) of a word in W and W 1 (n), . . . , W kn (n) ∈ W s.t. ω 1 • • • ω n = W 0 (n)W 1 (n) • • • W kn (n) ;
(ii) : if ω ∈ X (i.e. σ k • ω never belongs to { 0, 2}) then, there exist W 1 , W 2 , . . . in W and W 0 a possibly empty strict suffix of a word in W, s.t.

ω 1 ω 2 • • • = W 0 W 1 W 2 • • • Proof. (i) : Let n ≥ 1 : by definition of W, either ω 1 . . . ω n is a strict suffix of a word of W or ω 1 . . . ω n has a suffix W ∈ W.
In the later case, there exists a word W such that ω 1 . . . ω n = W W . If W is neither empty nor a strict suffix of a word in W, we repeat the procedure with W and so on and so forth with a finite induction.

(ii) : We use the decomposition in part (i) for any n ≥ 1: for the word W 0 (n) is a strict suffix of a word of W, there exists k ≥ 1 such that W 0 (n) ∈ {• /, 10 k , 0 k , 0102 k , 102 k , 02 k , 2 k }. Since ω ∈ X, the set {W 0 (n) ; n ≥ 1} is necessarily finite. Hence, there exists a word W 0 and an infinite set E 0 ⊂ {1, 2, . . . } such that W 0 (n) = W 0 , for any n in E 0 . The words W 0 , W 1 , . . . are defined by induction: given k ≥ 0, let W 0 , . . . , W k be k+1 words for which W 0 (n) = W 0 , . . . , W k (n) = W k for any n in an infinite set E k ⊂ {1, 2, . . . }. Since ω ∈ X, there cannot exist infinitely many words W ∈ W such that W 0 . . . W k W is a prefix of ω 1 . . . ω n for a n ≥ 1. Hence, there exists W k+1 s.t. W 0 (n) = W 0 , . . . , W k+1 (n) = W k+1 for any n in an infinite set E k+1 ⊂ E k : the induction holds, so that ω

1 ω 2 • • • = W 0 W 1 W 2 • • • .
The language W has two important drawbacks w.r.t. condition (C). First, for many W in W the matrices A(W ) belongs to H 1 , but (unfortunately) there are words in W for which this is not true: for instance, A(010

) =           0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0          
(we shall see (Lemma 9.5) that A(W ) ∈ H 1 as soon as W is factorized by a concatenation of at least 13 words in W). Secondly, the constant λ = 5 in Lemma 9.2 is larger than 1. The following lemma is crucial (its proof is given in § 9.2 below). Lemma 9.4. If W ∈ {0, 1, 2} * is factorized by a concatenation of 130 (= 10 • 13) words in W, then:

A(W ) ∈ H 1 ∩ H 3 (3/4) .
Proof of Proposition 9.1. The fact that 0 and 2 are (C)-singular may be checked directly.

To prove that an arbitrary given ω ∈ X is necessarily (C)-regular we begin with Lemma 9.3 to write ω = W 0 W 1 W 2 • • • , where W 0 is a possibly empty strict suffix of a word in W and each W 1 , W 2 , . . . are words in W. Let 0 = s 0 = s 1 and for k ≥ 1 define:

s k+1 = 130k i=0 |W i |.
This leads to the recoding ω

= Ws 2 Ws 3 • • • , where Ws 2 = W 0 W 1 • • • W 130 = ω 1 • • • ω s 2 , while for k ≥ 2, Ws k+1 = W 130(k-1)+1 • • • W 130k = ω s k +1 • • • ω s k+1 .
The latter notations are consistent with setting W0 = • / and for n ≥ 1, k ≥ 0 and s k+1 ≤ n < s k+2

Wn = ω s k +1 • • • ω n = Ws k+1 W n where W n := ω s k+1 +1 • • • ω n ; in other words, ω 1 • • • ω n = Wn when n < s 3 , while for n ≥ s 3 , ω 1 • • • ω n = Ws 2 • • • Ws k Wn = Ws 2 • • • Ws k Ws k+1 W n . (For s k+1 ≤ n < s k+2 ,
= s0 = s1 < s2 < • • • for condition (C) is defined in consequence, that is s k = | W2 • • • Ws k | and for each s k+1 ≤ n < s k+2 (with k ≥ 0) Wn = ωs k +1 • • • ωn = Ws k+1
W n is also factorized by a concatenation of 130 words in W; the importance of the number 130 comes from crucial Lemma 9.4. Finally, it remains to consider the case where neither 01 nor 21 is a factor of W 2 . . . W 10 : in particular, this means that W 2 • • • W 10 = 1 k W with k ≥ 0 and W ∈ {0, 2} * (it is here where the 13 words in W will prove to be needed). Notice that A(1), . . . , A(1111) do not satisfy (i-ii-iii), while

With the notations of Definition 1.1, one has P

n = A(ω 1 • • • ω n ) = P s k Q n ,
A(11111) =           3 0 3 2 1 0 0 0 0 0 0 0 0 0 1 0 3 3 1 0 0 1 0 1 3 2 0 0 2 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0          
does satisfy this property: hence, W has a factor W (= 11111) with A(W ) satisfying (i-ii-iii), as soon as k ≥ 5. Now, suppose that k ≤ 4, so that W 6 • • • W 10 ∈ {0, 2} * (indeed the worst case arises for k = 4 with W 2 W 3 W 4 W 5 = 1111). The definition of W implies that each word W 6 , W 7 , W 8 , W 9 , W 10 must be either 20 k , 002 k or 202 k (for some k ≥ 1). Then, it is necessary that 02 is a factor of both W 6 W 7 and W 8 W 9 : this implies that W 6 • • • W 10 has a factor of the form 0 x 2 y 0 z 2 t 0 u with x, y, z, t, u ≥ 1. On the one hand, if t = 1 then z = 1, 2, 3, . . . and 0 x 2 y 0 z 2 t 0 u is factorized by either 2020, 20020 or 00020; on the other hand, if t ≥ 2, the word 0 x 2 y 0 z 2 t 0 u is factorized by 220. One concludes this case, since W is factorized by a word W which is either equal to 2020, 20020, 00020 or 220 and which (see § 11.3 in Appendix 11) satisfies (i-ii-iii).

We now conclude with the general case of W ∈ {0, 1, 2} * which is factorized by

W 1 • • • W 13 with W i ∈ W. Let W be the factor of W 1 • • • W 13 s.t. A(W ) satisfies (i-ii-iii) and notice that A(W ) ∈ H 1 .
Because A(W ) satisfies (i), a direct verification shows that both A(W i) and A(iW ) satisfy (i) for any i = 0, 1, 2: hence, by a finite induction, A(W ) also satisfies (i). The fact that (ii-iii) holds for A(W ) follows from Lemma 2.10: because A(W ) ∈ H 1 and #Col(A(W )) ≤ 2, part (e) of Lemma 2.10 ensures that #Col(A(W )) ≤ 2 and part (f) that A(W ) ∈ H 1 . 9.2.2. The adjacency graph (Γ 1 ). We begin to define the infinite adjacency graph (Γ 0 ) whose vertex set Vert(Γ 0 ) is made of the nonzero 7 × 1 vectors (represented for convenience by) the 7uplets (x 1 , . . . , x 7 ) having nonnegative integral entries and obtained from one of the basis vector U 1 , . . . , U 7 by successive left multiplication with the matrices A(0), A(1) and A(2). For instance, (0, 0, 1, 1, 0, 0, 0)

ωn -→ • • • ω 1 -→(x, 0, y, z, t, 0, 0) is a path in (Γ 0 ) so that A(ω 1 • • • ω n )(0,
0, 1, 1, 0, 0, 0) = (x, 0, y, z, t, 0, 0). Setting V ∼ V whenever ∆(V ) = ∆(V ) gives an equivalent relation on Vert(Γ 0 ): the quotient space Vert(Γ 0 )/ ∼ is made of finitely many classes of equivalence, each ones being uniquely represented by a 7-uplet (a 1 , . . . , a 7 ) ∈ { * , 0, 1, 2, . . . } 7 : here (a 1 , . . . , a 7 ) represents the set {V k } N k=1 (1 ≤ N ≤ +∞) of all the nonzero 7 × 1 vectors in a given class of equivalence, with a i either equal to sup k {U i V k } or * if either sup k {U i V k } is finite or infinite respectfully. This leads to introduce the adjacency graph (Γ 1 ) whose vertex set is Vert(Γ 1 ) = Vert(Γ 0 )/ ∼ an whose paths are obtained by quotient projection from the paths in (Γ 0 ): the representation of (Γ 1 ) is given in Figure 13 of Appendix 10. The set Vert(Γ 1 ) may be written as a partition V 1 V 2 , where V 2 := ( * , 0, * , * , * , 0, 0), ( * , * , * , 0, * , * , 0), ( * , * , * , 0, * , * , * ), ( * , * , * , * , * , 0, 0) By an abuse of notations the vertices in Vert(Γ 1 ) may be considered as usual 7 × 1 vectors, with for instance ( * , 0, * , * , * , 0, 0) = +∞ and ∆( * , 0, * , * , * , 0, 0) = (1, 0, 1, 1, 1, 0, 0); but we may also note (x 1 , . . . , x 7 ) ∈ (y 1 , . . . , y 7 ), when (y 1 , . . . , y 7 ) is considered as a class of equivalence. The set of column vectors T 2 considered in Lemma 9.5 is ∆(V 2 ).

≤ i ≤ 7 A(W )U j = 0 =⇒ ∃V ∈ T 2 , ∆(A(W )U j ) ≤ V ; (b) : if V ∈ V 1 then A(i)V ≤ (2, 2, 2, 2, 2, 2, 2), for any i = 0, 1, 2; (c) : if V ∈ V 2 then A(i)V ∈ V 2 ,
for any i = 0, 1, 2;

(d) : in the subgraph (Γ 1 ) of (Γ 1 ) with vertices in V 2 (see Figure 12), the words W ∈ {0, 1, 2} * of length |W | ≥ 3 are synchronizing: in other words, if V and V are two d × 1 vectors with nonnegative entries s.t. both ∆(V ) and ∆(V ) belong to T 2 , then

|W | ≥ 3 =⇒ ∆(A(W )V ) = ∆(A(W )V ) ;
(e) : let V be a vector having positive entries; then, for any word W ∈ {0, 1, 2} * ,

W = • / =⇒ ∆(A(W )V ) ∈ T 2 ;
(f ) : let V be a vector having positive entries; then, for any word W, W ∈ {0, 1, 2} * ,

|W | ≥ 3 =⇒ ∆(A(W )V ) = ∆(A(W W )V ).
Proof. Parts (a), (b) and (c) are verified directly on (Γ 1 ) while (d) is clear from the subgraph (Γ 1 ) (see Figure 12): the words in {00, 01, 11, 2} being synchronizing, each word

W ∈ {0, 1, 2} * with |W | ≥ 3 is synchronizing as well, i.e. if W = ω 1 • • • ω n ∈ {0, 1, 2} n for n ≥ 3 and if V 1 ω 1 -→ • • • ωn -→ V 2 and V 1 ω 1 -→ • • • ωn -→ V 2
are two paths in (Γ 1 ) then it is necessary that V 2 = V 2 . For parts (e) and (f), take a vector V with positive entries: then by direct verification one gets ∆(A(i)V ) ∈ T 2 , for i = 1, 2, 3 and (e) follows from (c); part (f) is a consequence (e) and of the synchronization property in (d). 9.2.3. The adjacency graph (Γ 2 ). The final argument leading to Lemma 9.4 stands on a doubling property (Lemma 9.7) displayed by a second adjacency graph (Γ 2 ), related to the graph (Γ 1 ) introduced in § 9.2.2, but defined in a slightly different way. Each V ∈ Vert(Γ 2 ) is a d × 1 nonzero vector with nonnegative integral entries and represents the set of d × 1 vectors V = 0 with nonnegative integral entries such that ∆(V ) = ∆(V ) and V ≥ V : then, the labelled edge V i -→ V in (Γ 2 ) means A(i)V ≥ V and ∆(A(i)V ) = ∆(V ). By definition T 2 ⊂ Vert(Γ 2 ), all the labeled edges of (Γ 2 ) are obtained by starting from one vertex in T 2 and making all the possible (left) multiplication by either A(0), A(1) or A(2). Among the infinitely many possibilities of an adjacency graph satisfying the previous specifications, we have chosen one -namely (Γ 2 ) -with finitely many vertices with the additional property that each V ∈ Vert(Γ 2 ) satisfies V ≤ (2, . . . , 2). (A representation of (Γ 2 ) is given in Figure 14 in Appendix 10). We shall use the important fact that most of the paths of (Γ 2 ) starting from a vertex in T 2 terminate on a vertex in 2T 2 , while all paths starting from a vertex in 2T 2 terminate on a vertex in 2T 2 . Lemma 9.7 (Doubling property). Let W = ω 1 • • • ω n ∈ {0, 1, 2} n be factorized by a concatenation of 3 13 words in W but not factorized by 001001; then, for any nonnegative vector V ,

∆(V ) ∈ T 2 =⇒ ∆(A(W )V ) ∈ T 2 and V ≥ a • ∆(V ) =⇒ A(W )V ≥ 2a • ∆(A(W )V ) . Proof. Suppose that W = ω 1 • • • ω n ∈ {0, 1, 2} * is factorized by ω i • • • ω j = W 1 • • • W 13 with W i ∈ W but
not factorized by 001001. Hence, the mirror word ω j • • • ω i is a concatenation of at least 13 words of the form 0 k , 1 or 2 k with k ≥ 1, without being factorized by 100100. We claim that a path in (Γ 2 ), labelled by ω j • • • ω i and starting from a vertex in T 2 terminates in 2T 2 : indeed (see Figure 14) the longest paths (longest w.r.t. the number of blocks of the form 0 k , 1 or 2 k ), not factorized by 100100 and whose terminal vertex does not belong to 2T 2 , are those joining (1, 0, 1, 1, 1, 0, 0) to (2, 2, 2, 0, 2, 2, 1) and labelled by words of the form 00010010212 k 010; these words are concatenations of the 11 words 000, 1, 00, 1, 0, 2, 1, 2 k , 0, 1, 0. If #Col(A(W )) = 1 then A(W ) ∈ H 3 (λ) for any λ ≥ 0. We now assume that #Col(A(W )) = 2 so that it is licit to consider any arbitrary column index, say j , s.t. 0 < ∆ A(W )U j < ∆ A(W )U j 0 . One deduce from (132) and the synchronization property (part (d) of Lemma 9.6) that (133)

T 2 ∆ A(ω 4 • • • ω n )U j < ∆ A(ω 4 • • • ω n )U j 0 ∈ T 2 .
However -in view of graph (Γ 1 ) -this is possible only if We claim that the following implication holds for any j: To complete the proof of Lemma 9.4 it remains to establish (135), for which the graph (Γ 2 ) is crucial. The key point is #Col(A(W )) = 2 implies W cannot be factorized by 001001: indeed, on the contrary -for A(001001) being a rank one matrix -the matrix A(W ) would be also of rank one and #Col(A(W )) would be equal to 1. In particular each word W 9 , W 8 , . . . , XW 1 is not factorized by 001001 and applying Lemma 9.7 yields:

A(ω 4 • • • ω n )U j ∈ Vert(Γ 1 ) \ V 2 = V 1 ,
A(W 10 Y )U j ≥ 1 • ∆ A(W 10 Y )U j ∈ T 2 A(W 9 )A(W 10 Y )U j = A(W 9 W 10 Y )U j ≥ 2 • ∆ A(W 9 W 10 Y )U j ∈ T 2 A(W 8 )A(W 9 W 10 Y )U j = A(W 8 W 9 W 10 Y )U j ≥ 2 2 • ∆ A(W 8 W 9 W 10 Y )U j ∈ T 2 . . . A(XW 1 )A(W 2 • • • W 10 Y )U j = A(XW 1 • • • W 10 Y )U j ≥ 2 9 • ∆ A(XW 1 • • • W 10 Y )U j ∈ T 2 .
Hence (135) is proved as well as Lemma 9.4. To establish the uniform convergence of Π n (ω, V ) for ω ∈ {0, 1, 2} N , we want to apply Propositon 6.5, which is the suitable form of Theorem A for the uniform convergence (Recall that we note ω i,j = ω i+1 . . . ω j for any ω = ω 1 ω 2 • • • ∈ {0, 1, 2} N and 0 ≤ i ≤ j, in particular ω i,i = • /.) • First, consider ω ∈ X. To check the conditions (U1.1) and (U1.2) in part (i) of Propositon 6.5, let n ≥ 1 and consider ξ ∈ [ω 1 • • • ω n ] (i.e. ξ 0,n = ω 0,n ). Part (ii) of Lemma 9.3 ensures that ω = ω 1 ω 2 • • • = W 0 W 1 W 2 • • • , where W 0 is a possibly empty strict suffix of a word in W, and W 1 , W 2 , . . . are words in W. Let p = p(n) ≥ 0 be the maximal rank such that W 0 • • • W p = ω 0,ψ (n) is a prefix of ω 0,n . For n large enough, there exists an integer ψ(n), with 0 ≤ ψ(n) ≤ ψ (n) ≤ n and s.t. W 0 . . . W p-3 = ω 0,ψ(n) and W p-2 W p-1 W p ξ ψ (n),n+r = ξ ψ(n),n+r , where r ≥ 0 is arbitrarily given. Notice that n → ψ(n) only depends on ω and that ψ(n) → +∞ as n → +∞. Consider the vector (136) X r = A(ξ ψ(n),n+r )V = A(W p-2 W p-1 W p ξ ψ (n),n+r )V ;

here, the key point is the synchronization property (f) in Lemma 9.6 which implies ∆(X r ) = ∆(X 0 ), for any r ≥ 0, proving that (U1.2) holds. As for (U1.1), it remains to prove the following lemma.

Lemma 9.8. The exists a constant Λ ≥ 1, such that, A(ξ ψ(n),n+r ) ∈ H 2 (Λ), for any r ≥ 0.

Proof. According to part (i) of Lemma 9.3), it is possible to write ξ ψ(n),n+r = W 0 . . . W q where W 0 is a possibly empty strict suffix of a word in W, and W 1 , . . . , W q are words in W. For W 0 being a strict suffix of a word of W, there exists k ≥ 1 s.t.

W 0 ∈ {• /, 10 k , 0 k , 02 k , 2 k , 0102 k , 102 k } and since W 0 is also a prefix of W p-2 W p-1 W p ξ n,n+r , it is necessary that W 0 ∈ {• /, 10, 100, 0, 00, 2, 0102}.

We note λ 0 and Λ 0 s.t. A(W ) ∈ H 2 (Λ 0 ) ∩ H 3 (λ 0 ), for any W ∈ {• /, 10, 100, 0, 00, 2, 0102}. Moreover (Lemma 9.2) A(W i ) ∈ H 2 (13) ∩ H 3 (5) for any 1 ≤ i ≤ q. With q = 130 • a + b (0 ≤ b < 130) being the euclidean division of q by 130, one writes

W 0 • • • W q = W 0 W 1 • • • W b W 1 • • • W a ,
where each word W i is a concatenation of 130 words in W: part (ii) of Lemma 2.9 implies

W 1 • • • W b ∈ H 2 (Λ )

and W

i ∈ H 2 (Λ ) for Λ = 13 129 k=0 5 k while Lemma 9.4 ensures W i ∈ H 3 (3/4). Finally (part (ii) of Lemma 2.9 again), one concludes that A(W 0 . . . W q ) ∈ H 2 (Λ), where (1,1,0,0,1,0,0) (0,0,1,1,0,0,0) (0,0,0,0,0,1,0)

(1,0,1,0,0,0,0) 2 1 (0,0,1,0,0,0,0) (0,0,0,0,1,0,0) (0,0,0,1,0,0,0)

2

(1,0,0,0,0,0,0)

(1,0,0,0,1,0,0) (0,1,0,0,0,0,0) (0,0,0,0,0,0,1) (0,1,1,0,0,1,0) (0,0,0,0,1,0,1) (0,1,1,0,0,0,0) (0,0,1,0,1,1,0) (2,0,1,0,1,0,0) (1,0,1,0,1,1,1) ( ,0, , , ,0,0) 1,2 ( , , ,0, , ,0) ( , , ,0, , , ) 0 (1,2,1,0,1,0,0) A representation of (Γ 1 ); for instance the vertex (1, 0, 2, 1, 0, 0, 0) corresponds to (1, 0, 1, 1, 0, 0, 0), (1, 0, 2, 1, 0, 0, 0) while ( * , 0, * , * , * , 0, 0) = (x, 0, y, z, t, 0, 0) ; x, y, z, t = 1, 2, . . . ; one has the partition Vert(Γ 1 ) = V 1 V 2 , where V 2 := ( * , 0, * , * , * , 0, 0), ( * , * , * , 0, * , * , 0), ( * , * , * , 0, * , * , * ), ( * , * , * , * , * , 0, 0) ; notice that (x 1 , . . . , x 7 ) ∈ V 1 = Vert(Γ 1 ) \ V 2 implies (x 1 , . . . , x 7 ) ≤ (2, 2, 2, 2, 2, 2, 2).

(2,2,2,0,2,2,1) (2,2,2,0,2,1,0) (2,1,2,0,2,2,1)

(2,1,2,2,2,0,0)

(2,1,2,0,2,1,0) (2,0,2,2,1,0,0) (2,1,1,1,2,0,0)

Figure 1 .

 1 Figure 1. The sequence 0 = s0 = s1 < s2 < • • • determines the definition of the d × d matrices Qn s.t. Ps k Qn = Pn for any n ≥ 0 and s k+1 ≤ n < s k+2 (with k = k(n) ≥ 0); notice that Pn = Qn for 0 ≤ n < s3 (i.e. k(n) = 0 or 1), with P0 = Q0 being the d × d identity matrix.

Definition 2 . 3 .

 23 For ∅ = I ⊂ {1, . . . , d}, we denote by S I the set of vectors X ∈ S d s.t. I(X) = I. Proposition 2.4. Let ∅ = I ⊂ {1, . . . , d}: then, (i) : for X, Y ∈ S d , one has δ(X, Y ) < +∞ ⇐⇒ ∆(X) = ∆(Y ); (ii) : the restriction δ(•, •) : S I × S I → [0 ; +∞[ defines a metric on S I (when I = {1, . . . , d}, the metric δ(•, •) over S I (or S I ) coincides with Hilbert projective metric δ H (•, •) already mentioned); (iii) : if X, Y ∈ S I , then

  as 1 ≥ a ≥ b > 0. Fix ∅ = I ⊂ {1, . . . , d} with #I ≥ 2 (otherwise the result is trivial): we then consider X, Y ∈ S I . First, up to a permutation of X and Y , we assume X

  one has δ(AB) = δ(A B ) = δ(A B ): because A has positive entries and B is allowable, one deduces by (22) that δ(A B ) ≤ δ(A )τ (B ) (the case where B has only one row is trivial:

Figure 2 .

 2 Figure 2. Illustration of inequality (23).

  M ) := # ∆(M U 1 ), . . . , ∆(M U s ) \ 0 . Lemma 2.10. Let A, B and C be three d × d matrices with non negative entries; then (a) :

Figure 3 .

 3 Figure 3. How the index sets I h (n) and J h (n) are obtained for each 1 ≤ h ≤ Hn ≤ κ(Qn): here, Θn is a permutation matrix which rearranges the columns in Qn and Pn so that both (I(QnΘnUj)) d j=1 and (I(PnΘnUj)) d j=1 , form a non-increasing sequence of row-index sets. Similarly, Θ n (resp. Θ n ) is a permutation matrix which rearranges the rows in Qn (resp. Pn) so that (I((Θ n Qn) * Uj)) d j=1 (resp. (I((Θ n Pn) * Uj)) d j=1 ), form a non-decreasing sequence of row-index sets.

M

  k means the identity matrix if b < a. Since we have supposed that the induction hypotheses are true at the rank i+1, the columns of n k=1

s+1 with s = +∞ n=1 a 1 4 . 3 .

 143 . . . a n-1 b n d 1 . . . d n-1 d n , if j = 2 and s < +∞ 1 0 if either j = 1 or s = +∞. If s < +∞ and lim n→+∞ 1 n log(d 1 . . . d n ) exists (resp. if s < +∞ and lim n→+∞ 1 n log(a 1 . . . a n ) exists), it is the first (resp. the second) Lyapunov exponent of the sequence of product matrices (P 1 , P 2 , . . . ). Example 3. Corollary A does not hold for the following products of triangular matrices.

4 . 5 .

 45 values in a finite set, say {A(0), . . . , A(a)}: the condition that γ := sup n {τ (A i )} < 1 is automatically satisfied and Proposition 4.2 holds. Example 5. We shall illustrate the possible underlying fractal nature of the top Lyapunov direction as introduced in Corollary A. Let A(0), . . . , A(a) be fixed d × d matrices with nonnegative entries and for ω = ω 1 ω 2 • • • ∈ Ω := {0, . . . , a} N , consider the sequence A(ω) = (A(ω 1 ), A(ω 2 ), . . . ). Provided it makes sense, the top Lyapunov direction map is V : Ω → S d s.t.V(ω) := lim n→+∞ P n (ω)U P n (ω)U (here we recall that P n

Figure 4 .

 4 Figure 4. Representation of the map [0; 1[ x → p(x 1 x 2 • • • ), where 0 • x 1 x 2 • • •

Figure 5 .

 5 Figure 5. The solid red graph represent the exponential growth rate of C (n) 3 :

Corollary 5 . 4 .

 54 Let A(0), . . . , A(a) ∈ M d (C) having no common left-eigenvector and for any

(

  ii) : ω = ws, for s ∈ {0, . . . , a} (with s possibly (C)-singular) and there exists a d × d matrix B s with B s = 1, s.t. (S2.1) : lim n→+∞ A(s) n A(s) n -B s = 0, (S2.2) : A(w)B s R = 0 . Proof. (i) : Let ω ∈ Ω R be (C)-regular and satisfy (S1) w.r.t. the sequence n → ψ(n).

  suppose that (U2) and (U3) are satisfied by ω and, assuming that ϕ(n) ≥ k 0 , we put n = n-k 0 , m = m-k 0 . By definition of C s and D s , conditions in (U2) ensure A(s) m = C s D s + M m with M n → 0 as n → +∞, while A(w)C s = 0: in particular, it is licit to put B(w) := A(w)/ A(w)C s . Defining Y := A(ζ m ,n +r )R/ A(ζ m ,n +r )R and U := U 1 + • • • + U d , one gets the following identities:

Figure 6 .

 6 Figure 6. The adjacency graph (left) defining the sofic system related to Kamae's sofic affine subset of T 2 (right).

Proposition 7 . 1 .

 71 The continuous functions Π n (•, U ) : Y → S 3 (n ≥ 1), where, for any y ∈ Y,

Figure 7 .

 7 Figure 7. Graph plot of [0 ; 1[ y → φ(y 1 y 2 • • • ), where 0.y 1 y 2 • • • is the dyadic expansion of y.

s n ≤ x < s n + 1 β n where s n =

  is licit to define ε k (x) := k . The set Ω β of the β-admissible sequences is the closure (in {0, 1} N ) of the set of Parry expansions of real numbers in [0 ; 1[. The set Ω β is compact and left invariant by the full shift map σ : {0, 1} N → {0, 1} N in the sense that σ(Ω β ) = Ω β : Ω β is called the β-shift. The set Ω * β (resp. Ω (n) β ) of β-admissible words (resp. β-admissible words of length n) is made of finite (resp. of length n) factors of β-admissible sequences (considered as one-side infinite words in Ω β ). We shall use the partition of [0 ; 1[ by β-adic intervals of order n, that we write [0 ; 1[= w I w , where w runs over Ω (n) β and for 1 . .

Proposition 8 . 2 .

 82 w k := w • • • w with k factors and w 0 stands for the empty word • /). For m > n the real numbers s m = m k=1 k /β k and sn = n k=1 k /β k satisfy 0 ≤ s m -s n < 1/β n +∞ k=1 ξ k /β k where ξ 1 ξ 2 • • • ∈ 1100, (1100) i 0 1, (1100) i 10 1 .A simple computation yields 0 ≤ s m -s n < 1/β n , so that 1 . . . m 0 is the Parry expansion of s m ; the condition m > n being arbitrary means that = 1 2 • • • is β-admissible. Lemma 8.1 implies that each β-admissible sequence = 1 2 • • • may be decomposed from left to right in an infinite concatenation of words, say = w 1 w 2 • • • , where (113) w i ∈ w(0) = 0, w(1) = 10, w(2) = 1100 . Let J • / := [0 ; 1[ and for any ξ 1 • • • ξ n ∈ {0, 1, 2} n , the concatened word w(ξ 1 ) • • • w(ξ n ) being a binary β-admisible word, define J ξ 1 •••ξn := I w(ξ 1 )•••w(ξn) ;

Because of Lemma 8 . 4 ,

 84 we define for each e ∈ {0, 1, 2} the relation γ e -→ γ ⇐⇒ γ ∈ V e (γ)∩] -1 ; β[ and by definition, V is the subset of ] -1 ; β[ containing 0 and the γ ∈] -1 ; β[ for which there exists a sequence e 0 , . . . , e n in {0, 1, 2} and 0 = γ 0 , • • • , γ n = γ in ] -1 ; β[ (with n ≥ 0) s.t. γ 0 e 0 -→ γ 1 , . . . , γ n-1 en -→ γ n ;

Figure 8 .

 8 Figure 8. Adjacency graph corresponding to the relation vi e -→ vj (e = 0, 1, 2) between the elements in V = {v1, . . . , v7}, which are the 7 vertices of the graph.

Proposition 9 . 1 .

 91 The sequences ω ∈ {0, 1, 2} N which are (C)-regular -in the sense that A(ω) = (A(ω 1 ), A(ω 2 ), . . . ) satisfies condition (C) -form the dense set (127)X := {0, 1, 2} N \ +∞ n=0 σ -n { 0} ∩ {0, 1, 2} N \ +∞ n=0 σ -n { 2} .

Figure 9 .

 9 Figure 9. Lexicographical construction (from right to left) of the words in W (here k ≥ 1).

Figure 10 .

 10 Figure 10. Each ω ∈ X may be recoded by writing ω1ω2 • • • = Ws 2 Ws 3 • • • , where each word Ws k is factorized by a concatenation of 130 words in W; the sequence 0 = s0 = s1 < s2 < • • • for

Figure 11 .

 11 Figure 11. How to obtain the words in L 01L (left) and in L 21L (right).

Lemma 9 . 6 (

 96 Synchronization lemma). The following proposition holds (a) : for any W ∈ {0, 1, 2} * and any 1

Figure 12 .

 12 Figure 12. Subgraph (Γ 1 ) of (Γ 1 ) with vertices (a 1 , . . . , a 7 ) ∈ V 2 : each word in W ∈ {00, 01, 11, 2} is synchronizing (in this subgraph) in the sense that each path coded by W ends on the same vertex: it follows that a word W ∈ {0, 1, 2} * with |W | ≥ 3 is synchronizing as well.

9. 2 . 4 .

 24 Proof of Lemma 9.4.Consider W := ω 1 • • • ω n = XW 1 • • • W 10 Y ,where X, Y ∈ {0, 1, 2} * and each W 1 , . . . , W 10 is a concatenation of 13 words in W (making W 1 • • • W 10 a concatenation of 130 words in W). The assertion A(W ) ∈ H 1 is proved in Lemma 9.5. From this lemma, #Col(A(W 10 Y )) ≤ 2 and there exists at least one column index j 0 such that ∆ A(W 10 Y )U j 0 ∈ T 2 and ∆ A(W 10 Y )U j ≤ ∆ A(W 10 Y )U j 0 for any j: hence, by part (c) of Lemma 9.6(132) ∆ A(W 10 Y )U j 0 , . . . , ∆ A(XW 1 • • • W 9 W 10 Y )U j 0 = ∆ A(W )U j 0 ∈ T 2 .

  W 10 Y )U j ∈ T 2 =⇒ A(W )U j ≥ 2 9 • ∆ A(W )U j . Take any j s.t. ∆(A(W 10 Y )U j ) ∈ T 2 and j with 0 < ∆(A(W )U j ) < ∆(A(W )U j ). It follows from (134) and (135) that A(W )U j ≤ 7 • 3 3 • 2 ≤ 7 • 3 3 • 2/2 9 A(W )U j (i), for any index 1 ≤ i ≤ 7 s.t. A(W )U j (i) = 0: this means A(W ) ∈ H 3 7 • 3 3 • 2/2 9 ⊂ H 3 (3/4).

9. 3 .

 3 Proof of Theorem 8.7. Let V ∈ S 7 with positive entries. By part (e) of Lemma 9.6 the subshift Ω V -i.e. the compact shift-invariant subset of {0, 1, 2} N made of the sequences ω s.t. A(ω 1 • • • ω n )V = 0, for any n ≥ 1 -coincides with {0, 1, 2} N . Moreover, according to Proposition 9.1, the (C)-regular ω -in the sense that A(ω) = (A(ω 1 ), A(ω 2 ), . . . ) satisfies condition (C) -form the dense subset {0, 1, 2} N that isX := {0, 1, 2} N \ +∞ n=0 σ -n { 0} ∩ {0, 1, 2} N \ +∞ n=0 σ -n { 2} .

10 .

 10 Appendix: The adjacency graphs (Γ1) and (Γ2)

1 Figure 13 .

 113 Figure 13. A representation of (Γ 1 ); for instance the vertex (1, 0, 2, 1, 0, 0, 0) corre-

14 .

 14 Representation of the adjacency graph denoted by (Γ 2 ) : the word-codes W ∈ {0, 1, 2} * of the "longest" paths not entering 2T 2 (red vertices) and not factorized by 100100 are of the form W = 00010010212 k 010 (k ≥ 1); they are represented in blue in the adjacency graph. The green triangle shows the "forbidden" cycle 100100100 • • • .

  Preparatory lemmas. We shall consider that A = (A 1 , A 2 , . . . ) is a given sequence of d×d matrices with nonnegative entries and satisfying condition (C). A key point of the argument leading to Theorem A, is to reenforce condition (C) (see condition in (29) and (30) below) without further assumptions about the sequence A.Lemma 3.1. Suppose that A = (A 1 , A 2 , . . . ) satisfies condition (C) w.r.t. 0 ≤ λ < 1 ≤ Λ < +∞ and the sequence of integers 0 = s 0 = s 1 < s 2 < . . . ; then for any

t. ∆(BAU i k ) = 0; because BA ∈ H 1 , one may suppose that ∆(BAU i 1 ) ≥ • • • ≥ ∆(BAU is ) and by

[START_REF] Hartfiel | Nonhomogeneous Matrix Products[END_REF]

, one deduces ∆(BACU j ) = ∆(BAU i 1 ): this is the content of part (d). Part (d) together with BA ∈ H 1 shows that BAC ∈ H 1 , proving (f). Part (d) also ensures #Col(BAC) ≤ #Col(BA) and (e) holds, for we already know that #Col(BA) ≤ #Col(A). 3. Proof of Theorem A 3.1.

  (i.e. h P n is obtained from P n by replacing by zero each entry P n (i, j), for (i, j) ∈ I h (n)×J h (n)); hence, each h P n satisfies condition (H) together with the identity Given s 1 < s k+1 ≤ n < s k+2 (i.e. k(n) = k ≥ 1) and X ∈ S d with P n

		P n =	Hn h=1	h P n .
	Lemma 3.3.
	Definition 3.2. Given any 1 ≤ h ≤ H n ,
	(34)	h P n :=		P n (i, j)U i U j
		(i,j)∈I h (n)×J h (n)

  .[START_REF] Bougerol | Products of Random Matrices with Applications to Schrődinger Operators[END_REF]. Example 4: positive matrices. Suppose that A 1 , A 2 , . . . is an infinite sequence of d × d matrices each ones having positive entries. We note I d := {1, . . . , d}, so that (see Definition 2.3) S I d is the subset of S d whose element are the probability vectors with positive entries. Fix X ∈ S d arbitrarily (i.e with possibly zero entries) and suppose that γ := sup i {τ (A i )} < 1; then, as an immediate consequence of Lemma 2.6 the probability vectors P 1 X/ P 1 X , P 2 X/ P 2 X , . . . form a Cauchy sequence in the (complete, non compact) metric space (S I d , δ(•, •)) and thus has a limit X * ∈ S I d . More precisely, one has the following proposition. Proposition 4.2 (Folklore). Let A = (A 1 , A 2 , . . . ) be a sequence of d × d matrices having positive entries such that γ := sup i {τ (A i )} < 1: then there exist X * ∈ S I d and C > 0 such that

  Recall that by definition A → A 2 := σ 1 defines the so-called spectral norm of A. In this paragraph, we consider that P = (P 1 , P 2 , . . . ) is a sequence in M d (C) and we write the singular value decomposition of P n . Here the i-th singular value of P n is denoted by e nχ i (n) and χ 1 (n) ≥ • • • ≥ χ d (n) may be thought as the ordered list of the n-step Lyapunov exponents.Proof. The norms on the space M d (C) being equivalent, and because we work with • 2 in place of • . According to the singular value decomposition of P n in (95),

		P n P n	-B n =	P n 2 P n	P n P n 2	-	P n P n 2	B n ,
	(96)	P n P n 2
							0	
					. . .	. . .	. . .	  T ;
					0 . . . σ d
				 e nχ 1 (n) . . .		0	
	(95)		P n = S n	 	. . .	. . .		. . .	  T n
					0	. . . e nχ d (n)

here S and T are unitary matrices and σ 1 ≥ • • • ≥ σ d is the ordered lists of the singular values of A.

Theorem 5.1. Let P = (P 1 , P 2 , . . . ) a sequence in M d (C); then there exists a sequence B = (B 1 , B 2 , . . . ) of rank 1 matrices s.t. P n / P n -B n → 0 if and only if e nχ 2 (n) /e nχ 1 (n) → 0.

  in the first case (see Figure11-right) W is factorized by a word W ∈ L 21L and (see § 11.2 in Appendix 11) each word in L 21L satisfies (i-ii-iii).

		00
		01			0
		02			10
	00	10	00		110
	10	110	10		111
	on the key correspondence (128) 20 1 2	111 112 12 20 22 Q 01 21	20 1 2	21	112 12 20 21 22	where we stress

n = A( Wn ).

like

  so that (part (b) of Lemma 9.6)(134) A(ω 4 • • • ω n )U j ≤ (2, 2, 2, 2, 2, 2, 2) and A(W )U j ≤ 3 3 • (2, 2, 2, 2, 2, 2, 2).

This section may be skipped by readers not familiar with the symbolic dynamics of self-affine sets.

We abusively use σ for the shift maps over symbolic spaces as soon as the digits are well specified.

A direct consequence is the exact value of the µ-measure of the intervals J w . Lemma 8.5. For any ξ ∈ {0, 1, 2} N and any n ≥ 1, (119)

It remains to prove that µ is a weak-Gibbs measure.

Theorem 8.6. µ is weak-Gibbs in the sense that there exists a continuous Ψ : {0, 1, 2} N → R and a sub-exponential sequence of positive constants K 1 , K 2 , . . . s.t for any ξ ∈ {0, 1, 2} N , (120) 1

We shall need the following theorem whose proof (see Section 9) depends on Theorem A.

Theorem 8.7. Let A(0), A(1), A(2) be the 7 × 7 matrices in (114): if the vector V has positive entries then, the sequence of the probability vectors

converges to a probability vector V(ω), uniformly over ω ∈ {0, 1, 2} N , as n → +∞; moreover,

Proof of Theorem 8.6. In view of Proposition 6.1, we stress that the uniform convergence of the n-step potential of µ does not hold: to overcome this difficulty we shall introduce an intermediate measure.

by Kolmogorov Extension Theorem, there exists a unique measure ν on [0 ; 1] such that, for any ξ 1 , . . . , ξ n ∈ W 0 , (121)

Recall that for any ξ ∈ {0, 1, 2} N and any n ≥ 1

converges uniformly on {0, 1, 2} N toward a continuous Ψ : {0, 1, 2} N → R: indeed, by Theorem 8.7 the functions ξ → Π n (ξ, R) form a sequence which converges uniformly for ξ ∈ {0, 1, 2} N toward V(ξ) ∈ S 7 s.t. M (a)V(ξ) = 0, for any ξ ∈ {0, 1, 2} N and a ∈ {0, 1, 2}. Therefore (Proposition 6.2), the measure ν is weak Gibbs w.r.t. Ψ in the sense that there exists a sub-exponential sequence of positive constants

Hence, the weak Gibbs property of µ (w.r.t. to Ψ) will be established if one is able to show that

First, from Lemma 8.5 together with the inequalities

one deduces that for any ξ ∈ {0, 1, 2} N and any n ≥ 1,

Concerning the upper bound we consider the different cases for ξ 1 either equal to 0, 1 or 2.

• Suppose that ξ 1 ∈ [START_REF] Bedford | Crinkly curves, Markov partitions and dimension[END_REF]. From ( 121) and ( 119) and the fact that

where

and k ≥ 0 is positive. Because ξ 1 = 0, we consider the largest 1 ≤ m ≤ n for which 0 m is a prefix of ξ 1 . . . ξ n so that ξ 1 . . . ξ k = 0 m ξ m+1 . . . ξ n where ξ m+1 = 0 if m < n. From (121) and (119) and the facts that

It remains to establish the existence of a constant Λ ≥ 1 (depending on ω) s.t.

(

Suppose s 2 ≤ s k+1 ≤ n < s k+2 ; by construction, Wn = Ws k+1 W n , where W n is a (possibly empty) strict prefix of Ws k+2 : actually, either W

, where 1 ≤ m < 130, while W n is a strict prefix of a word in W. However (Lemma 9.2) we know that A(W i ) ∈ H 2 (13) ∩ H 3 [START_REF] Bowen | Some systems with unique equilibrium states[END_REF], for any i > 0 and by part (ii) of Lemma 2.9),

, where Λ stands for the maximum of the Λ A(W ) over W being any prefix of a word in W. The inequality Λ < +∞ is valid, because either W ∈ W and A(W ) ∈ H 2 [START_REF] Erdös | On a family of symmetric Bernoulli convolutions[END_REF] or W ∈ W which is only possible for W ∈ {• /, 0, 00, 001, 0010, 01, 11, 10, 101, 1010, 2, 201, 2010, 21}. 9.2. Proof of Lemma 9.4. For Proposition 9.1 to be completely established, it remains to prove Lemma 9.4. The argument -given in § 9.2.4 -depends on a key lemma (Lemma 9.5) established in § 9.2.1 together with several properties of two adjacency graphs -(Γ 1 ) and (Γ 2 )that we consider in § 9.2.2 and § 9.2.3 respectively. 9.2.1. Key Lemma. The following lemma shows -in part (iii) -that A(W ) ∈ H 1 as soon as W is factorized by a concatenation of at least 13 words in W. Parts (i) and (ii) will be determinant in the final argument proving Lemma 9.4 in § 9.2.4, and use the following set of four column vectors (that we represent by convenience by four 7-uplets):

(131)

T 2 := (1, 0, 1, 1, 1, 0, 0), (

Lemma 9.5. If W ∈ {0, 1, 2} * is factorized by a concatenation of 13 words in W, then (i) :

Proof. We shall begin to verify that any • Consider now ω = w 0, where w is a (possibly empty) word on {0, . . . , a}, we shall verify the assertions (U2) and (U3) in part (ii) of Proposition 6.5. First, (U2.1) is satisfied, for one checks directly that lim n→+∞ A(0 n ) A(0 n ) -C 0 U 1 = 0, where C 0 = (0, 1/5, 1/5, 0, 1/5, 1/5, 1/5), and (U2.2) is satisfied because A(0)C 0 = C 0 and A(s

To verify (U3.1) -(U3.2) -(U3.3), consider n large enough in order that σ n-2 • ω = 0, let ξ ∈ [ω 1 • • • ω n ] and r ≥ 0. Let m be the integer s.t. ξ n-2,m+2 = 0 m-n+4 with n ≤ m + 2 ≤ n + r and maximal w.r.t this identity (in the notations of Proposition 6.5 we take ϕ

-is satisfied by definition. Then, there exists a word W , either empty or beginning by 1 or 2, such that ξ n-2,n+r = 0 m-n+4 W . According to part (i) of Lemma 9.3, we write W = W 0 . . . W p where W 0 is a possibly empty strict suffix of a word in W, and W 1 , . . . , W p are words in W.

hence W 0 defined to be either equal to 0W 0 or 00W 0 , belongs to W. If W 0 = • / we put W 0 = • /, so there exists 0 ≤ j ≤ 2 s.t. ξ n-2,n+r = 0 m-n+2 0 j W 0 W 1 • • • W p that is:

Condition (U3.3) holds by the following lemma (obtained by a direct systematic verification) Lemma 9.9. ∀w ∈ {0, 1, 2} * , U 1 A(w) = 0. Condition (U3.2) follows from the following lemma, that can be proved in the same way as (137): Lemma 9.10. There exists a constant Λ ≥ 1 s.t. for any 0 ≤ j ≤ 2, q ≥ 0, W 0 ∈ {• /} ∪ W and W 1 , . . . , W q ∈ W, one has A(0 j W 0 • • • W q ) ∈ H 2 (Λ).

• The verification of the conditions (U2) and (U3) about ω = w 2 is similar to the case of ω = w 0, using the fact that lim n→+∞ A(2 n ) A(2 n ) -C 2 U 5 = 0, where C 2 = (1/3, 0, 1/3, 1/3, 0, 0, 0).

The proof of Theorem 8.7 is complete.