Normalized image of a vector by an infinite product of nonnegative matrices

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Normalized image of a vector by an infinite product of nonnegative matrices

Alain Thomas

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Alain Thomas. Normalized image of a vector by an infinite product of nonnegative matrices. 2018.

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NONNEGATIVE MATRICES

ALAIN THOMAS

Abstract. We consider a probability-measure p^∗ on Md(C)^N which is an infinite product of a non-atomic Borel probability measure p of supportC. Setting Pn :=A1· · ·An for any (An)n∈N ∈ Md(C)^N, we prove that the sequence n7→Pn/kPnkdiverges p^∗-a.e.. Nevertheless given ad-dimensional column-vectorV, it seems that the sequencen7→PnV /kPnVkconverges in much cases: for instance ifMis a finite set of positived×dmatrices, thenn7→PnV /kPnVk converges for any (An)n∈N∈ M^N and any nonnegative column-vectorV; while, if we assume that no couple of matrices of Mhas a common left-eigenvector, thenn 7→Pn/kPnkdiverges except if (An)n∈N belongs to the countable setM0⊂ M^Nof the eventually constant sequences.

The purpose of this paper is to give in Theorem A some sufficient conditions on the sequence of nonnegative matrices (An)n∈N, forn7→PnV /kPnVkto converge for any positive column-vector V. We apply this theorem to the study of a sofic (i.e. linearly representable) measureµ, chosen such that Theorem A is the suitable way to prove its multifractal property.

1. Introduction

GivenA= (An)n∈N an infinite sequence ofd×dmatrices, we consider the right-products Pn:=A1· · ·An (n >0),

P_m,n :=A_m+1· · ·A_n (0≤m < n) and, by convention,P₀ andP_m,m are the identity matrixI_d of order d.

Among the numerous ways for studying the left or right products, the problem of the convergence of the sequencen7→P_nis solved by Daubechies and Lagarias [4, 5]. A probabilistic approach is exposed in the book of Bougerol and Lacroix [1] with a large range of results about the products of random matrices.

When the sequencen7→P_n does not converge, one may be interested in the convergence of n7→ _kP^Pn

nk. But we prove in Proposition 7.3(iii) that this sequence diverges with probability 1, if the probability-measure we use is the infinite product of a non-atomic Borel probability-measure of supportC.

An immediate consequence of [15, Theorem 1.1] is that the sequencen7→ _kP^Pn

nk converges to a rank one matrix if the matrices _kA^An

nk are positive and converge to a positive matrixAasn→ ∞ – we say that a matrix is positive (resp. nonnegative) when all its entries are positive (resp.

nonnegative). More generally, another approach consists to find a sequence of rank one matrices Rn such that limn→∞ Pn

kPnk −Rn

= 0. We prove in Proposition 7.6 that, given a sequence (M_n)n∈Nof complex-valued matrices, there exists a sequence (R_n)n∈N of rank one matrices such that limn→∞ Mn

kMnk−Rn

= 0 if and only if the singular valuesδ1(n)≥ · · · ≥δd(n) ofMnsatisfy limn→∞ δ2(n)

δ1(n) = 0.

1991Mathematics Subject Classification. 15B48, 28A12.

Key words and phrases. Infinite products of nonnegative matrices, multifractal analysis, Bernoulli convolutions.

1

Let us consider the case Mn =Pn = A1· · ·An with An nonnegative. By Corollary 7.7(i), if there exist a constant column-vector C and a sequence of row-vectors (L_n)n∈N such that limn→∞ Pn

kPnk −CL_n

= 0, then limn→∞ PnV

kPnVk = C for any positive vector V. However in the counterexample of Subsection 4.2, this constant column-vector C does not exists and the sequence n7→ _kP^PnV

nVk diverges, except if the first entry ofV is null.

About the choice of the norm we note that, for any sequence (Mn)n∈N of d1×d2 matrices, the convergence of n7→ _kM^Mn

nk does not depend on the norm because, given two normsN₁ and N₂ and ad1×d2 matrixM, one has _N^M

1(M) = _N^M/N2(M)

1(M/N2(M)) , so the convergence ofn7→ _N^Mn

2(Mn)

to a (necessarily nonnull) matrix M implies the convergence of n7→ _N^Mn

1(Mn) to _N^M

1(M). In the sequel we use, unless otherwise specified, the norm defined for any matrixM = (M(i, j))1≤i≤d1

1≤j≤d2

, by kMk=P

i,j|M(i, j)|.

The paper is organized as follows. We first give in Theorem 1.1 a method to express any infinite product of nonnegative matrices in terms of an infinite product of block-triangular matrices. More precisely, P_rk =P_r0ST₁· · ·T_kS⁻¹ with T_k block-triangular, (r_k)k≥0 increasing sequence of integers,S permutation matrix, chosen in such a way that each row of each block of T_kis either positive or null. This method is based on the following obvious property of the infinite products of nonnegative matrices: there exists (rk)k∈N such that the location of the nonnull entries of Pr_k is the same for any k, and consequently one can choose a permutation matrix S for T_k := S⁻¹P_rk−1,rkS to be block-triangular. To give an example with 8×8 nonnegative matrices:

suppose that the location of the nonnull entries in eachP_rk is

1.jpg 1.jpg

,

then necessarily the location of the nonnull entries in eachPrk−1,rk is

2.jpg 2.jpg

. Setting, for any column-vectorV =V(i)1≤i≤d and any matrixA= (A(i, j))1≤i≤d0

1≤j≤d

, I(V) :=

i; V(i)6= 0 and I(A) :=

(i, j) ; A(i, j)6= 0 ,

H(A) := #{I ⊂ {1, . . . , d} ; ∃j, I=I(AU_j)} (where {U₁, . . . , U_d} is the canonical basis), we prove in§2 the following theorem:

Theorem 1.1. Let A= (A_n)n∈N be a sequence of nonnegative d×dmatrices and let

(1) κ=κ(A) := max

m≥0 lim sup

n→∞

H(P_m,n) .

Then there exists an increasing sequence of nonnegative integers(r_k)k≥0, a permutation matrixS and some integersc0= 0< c1<· · ·< cκ =d such that

(2) S⁻¹Prk−1,rkS=







B^1,1_k 0 ... 0 B^2,1_k B^2,2_k ... 0 ... ... . .. ... B_k^κ,1 B_k^κ,2 ... B^κ,κ_k





=:Tk for anyk∈N,

where each B<sup>h,`</sup><sub>k</sub> has size (c<sub>h</sub>−ch−1)×(c<sub>`</sub>−c`−1), and moreover there exist I1, . . . , Iκ , subsets of {1, . . . , d}, such thath < `⇒I_h(A)6⊂I_`(A) and

(3) 1≤k≤k⁰ ⇒ I(T_k· · ·T_k⁰) =I(T₁) =

κ

[

h=1

I_h× {ch−1+ 1, . . . , c_h}.

We also prove in §3 that the submatrices B_k^h,h forh6=κ, are distinct from the null matrix if each A_n satisfy the following condition: a matrixA is said to satisfy condition (E) if

(4) ∀j, j⁰, I(AU_j)⊂ I(AU_j⁰) or I(AU_j⁰)⊂ I(AU_j).

Section 4 is devoted to the sequencen7→PnV /kP_nVkand to the proof of the main theorem (Theorem A). The ingredients for Theorem A are the coefficients Λ(A) andλ(A) defined for any d×dnonnegative matrix Aby

Λ(A) := max{kAU_jk/A(i, j) ; (i, j) such thatA(i, j)6= 0} (and Λ(0) := 1), (5)

λ(A) := max{kAU_j⁰k/A(i, j) ; (i, j, j⁰) such thatA(i, j)6= 0 =A(i, j⁰)} (andλ(0) := 0), (6)

and the following condition (C): one says that a sequenceA= (An)n∈Nofd×dmatrices satisfies condition (C) with respect to the increasing sequence (s_k)k≥0 if

Psk−1,sk satisfies (E) for any k∈N (7)

maxk∈N n∈[sk,sk+1)

Λ(P_sk−1,n)<∞ (8)

maxk∈N n∈[sk,sk+1)

λ(Psk−1,n)<1.

(9)

Theorem A. Let A = (An)n∈N be a sequence of d×d nonnegative matrices satisfying condi- tion (C) with respect to(s_k)k≥0 , and let κ=κ(A) be defined as in Theorem 1.1. Then, ifP_n is not eventually null, there exists a positive integer κ^∗ ∈ {κ−1, κ} and there exist2κ^∗ nonempty subsets of {1, . . . , d}, namely J₁⁽ⁿ⁾, . . . , J_κ⁽ⁿ⁾∗ (disjoint) and I1 ⊃ · · · ⊃ Iκ^∗ (independent of n), such that

(10) I(P_n) =

κ^∗

[

h=1

Ih×J_h⁽ⁿ⁾ for n large enough.

Moreover there exist κ^∗ column-vectors V₁, . . . , V_κ^∗ such that∀h, I(V_h) =I_h and (i) if (j_n)n∈N∈Q

n∈NJ_h⁽ⁿ⁾ then limn→∞P_nU_jn/kP_nU_jnk=V_h ; (ii) if (j_n)n∈N∈Q

n∈NJ_h⁽ⁿ⁾ and (j_n⁰)n∈N∈Q

n∈NJ_h+1⁽ⁿ⁾ then limn→∞kP_nU_j⁰_nk/kP_nU_jnk= 0 ; (iii) there exists a sequence of positive numbers (εn)n∈N with limit 0 such that, for any nonneg- ative normalized vector V = (V(i))1≤i≤d for which ∀n, P_nV 6= 0,

(11)

PnV

kP_nVk−V_hV(n)

≤ εn

min1≤i≤dV(i) where hV(n) := min{h; I(V)∩J_h⁽ⁿ⁾6=∅}(n∈N).

In particular, ifV is positive then limn→∞ PnV

kP_nVk =V₁ and I(P_nV) =I(V₁) for nlarge enough.

Note that, by Corollary 7.7(ii), if the condition of the main theorem (Theorem A) is satisfied then there exist a constant column-vector C and a sequence of row-vectors (L_n)n∈N such that limn→∞ Pn

kPnk−CLn

= 0.

The sofic measures (see Subsection 5.1) can be considered as a generalisation of the Markov chains; the relationship between the transition matrix of a Markov chain and the matrices used to compute the values of a sofic measure, is explicited in the proof of [30, Theorem 7]. For the application we make in Section 6, we choose a sofic measure defined by Bernoulli convolution (see [7, 27]); the involved set of matrices is a set M of three matrices of order 7 with much null entries; we have chosen this measure because Theorem A cannot be avoided to prove its multifractal property. The technical difficulty, for proving the convergence of n 7→ _kP^PnV

nVk for any (An)n∈N ∈ M^N, is due to the form of the set {(i, j) ; Pn(i, j) 6= 0}: for much sequences (A_n)n∈N∈ M^Nit does not have the formI×J but (I₁×J₁)∪(I₂×J₂) withI₁ 6=I₂ . Some more simple Bernoulli convolutions, that do not require Theorem A, are considered in [25], where we give a necessary and sufficient condition for the uniform convergence of the sequencen7→ _kP^PnV

nVk

when the matrices A_n belong to a finite set of 2×2 nonnegative matrices, and when V is a nonnegative 2-dimensional column-vector. See [24] for the necessary and sufficient condition for the pointwise convergence of n7→ _kP^PnV

nVk in the 2×2 case.

A large bibliography about infinite products of matrices can be found in the paper of Victor Kozyakin [21].

Acknowledgement. – This paper was written in collaboration with Eric Olivier (Institut de Math´ematiques de Marseille, UMR 7373). We 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 sequences of normalized product-matrices (see Section 7.2).

2. Proof of Theorem 1.1

We first define an increasing sequence (r_k)k≥0 in order to prove, through a few technical lemmas, that (2) holds for this sequence.

Lemma 2.1. Let A= (A_n)n∈N be a sequence of d×dnonnegative matrices, let κ=κ(A) and r=r(A) := min{m≥0 ; lim sup

n→∞ H(Pm,n) =κ}.

One can define by induction a sequence (rk)k≥0 as follows:

r₀=r₀(A) := min{n≥r ; H(P_r,n) =κ and I(P_r,n) =I(P_r,n⁰) for infinitely many n⁰}, (12)

r_k=r_k(A) := min{n > rk−1 ; H(P_rk−1,n) =κ and I(P_r,n) =I(P_r,r0)} (k∈N).

(13)

So, in brief one has

(14)

0≤r≤r0< r1 < . . .

I(P_r,r0) =I(P_r,r1) =I(P_r,r2) =. . .

H(P_r,r0) =H(P_r0,r1) =H(P_r1,r2) =· · ·=κ.

Proof. Since lim sup_n→∞H(Pr,n) =κ, the setE ={n > r ; H(Pr,n) =κ}is infinite. Since the setI(P_r,n)⊂ {1, . . . , d}² can take at most 2^d2 values, it takes the same value for infinitely many n∈E; in other words the set {n∈E ; I(P_r,n) =I(P_r,n⁰) for infinitely manyn⁰} is nonempty, and the integerr0 defined in (12) is its minimum.

Before defining r_k in function of rk−1 we prove the following property of the nonnegative matrices:

Lemma 2.2. For any nonnegative d×dmatrices A and B one has

(15) ∀j, I(ABU_j) = [

i∈I(BU_j)

I(AU_i), (in particular if I(BU_j) is empty then I(ABU_j) is empty), and

(16) H(AB)≤H(B).

Proof. The column-vector ABUj is a linear combination of the column-vectorsAUi : ABUj = P

iB(i, j)AU_i . SinceB(i, j) is positive if and only if i∈ I(BU_j), (15) follows.

The setI(BU_j) takesH(B) distinct values whenj∈ {1, . . . , d}. So by (15) the setI(ABU_j)

can take at most H(B) distinct values, and (16) follows.

By definition of r0 , one has H(Pr,r0) =κ and the set E⁰ ={n > r₀ ; I(P_r,n) =I(P_r,r0)}

is infinite. For any n ∈ E⁰ ∩(rk−1,∞) one deduce from (16) that H(P_rk−1,n) ≥ H(P_r,n) = H(Pr,r0) = κ. But if n is large enough one also has the reverse inequality H(Prk−1,n) ≤κ by definition ofκ. So the set{n∈E⁰∩(r_k,∞) ; H(P_rk−1,n) =κ} is nonempty, and the integerr_k

defined in (13) is its minimum.

Lemma 2.3. Let A be a d×dmatrix.

(i) There exists a unique family (Ih(A))1≤h≤H(A) of subsets of {1, . . . , d}and a unique partition (J_h(A))_1≤h≤H(A) of {1, . . . , d} into nonempty subsets, such that

(17) I(A) =

H(A)

[

h=1

Ih(A)×Jh(A) and such that U I1(A)

· · · U I_H(A)(A)

, where U(I) = (u1, . . . , ud) is defined for any I ⊂ {1, . . . , d} by

ui = 1⇔i∈I, and the lexicographical order is defined by

(u₁, . . . , u_d)(v₁, . . . , v_d)⇔ ∃i, u_i> v_i and (i⁰ < i⇒u_i⁰ =v_i⁰).

(ii) One has Ih(A)6⊂I`(A) whenever h < `.

(iii) IfA andB are two nonnegative d×d matrices, one has the equivalence:

(18)

H(AB) =H(B)⇔ {J₁(AB), . . . , J_H(AB)(AB)}={J₁(B), . . . , J_H(B)(B)} (equality of the sets).

Proof. (i) : The lexicographical order being total, one can define the sets Ih(A) and Jh(A) by {I₁(A), . . . , IH(A)}:={I ⊂ {1, . . . , d} ; ∃j, I=I(AU_j)}withU I1(A)

· · · U IH(A) , (19)

J_h(A) :={j , I(AU_j) =I_h(A)}.

(20)

(ii) : Let (u1, . . . , ud) =U Ih(A)

and (v1, . . . , vd) =U I`(A)

. Ifh < `, then (u1, . . . , ud) (v1, . . . , v<sub>d</sub>) and there exists isuch thatui > vi , implying I<sub>h</sub>(A)6⊂I<sub>`</sub>(A).

(iii) : For any h ∈ {1, . . . , H(B)} one has, by (15), I(ABU_j) = S

i∈Ih(B)I(AU_i) for any j ∈ Jh(B), and consequently the set I(ABUj) is the same for any j ∈ Jh(B). One deduce, in view of (20), that{J₁(B), . . . , J_H(B)(B)} is a refinement of{J₁(AB), . . . , J_H(AB)(AB)}}, hence

both partitions are equal if and only if H(AB) =H(B).

Lemma 2.4. Let A be a d×d matrix and σ a permutation of {1, . . . , d}, let S be the matrix defined by ∀j, SU_j =U_σ(j) . Then

(i) I(S⁻¹AS) = (σ^∗)⁻¹(I(A)), where σ^∗(i, j) := (σ(i), σ(j));

(ii) H(S⁻¹AS) =H(A).

Proof. (i) :

(i, j)∈ I(S⁻¹AS) ⇔ ^tUi tS A SUj 6= 0

⇔ ^tU_σ(i)A U_σ(j)6= 0

⇔ A(σ(i), σ(j))6= 0

⇔ σ^∗(i, j)∈ I(A)

⇔ (i, j)∈(σ^∗)⁻¹(I(A)).

(ii) : By (i) and (17), I(S⁻¹AS) = SH(A)

h=1 σ⁻¹(Ih(A))×σ⁻¹(Jh(A)). For h 6= ` one has I<sub>h</sub>(A)6=I<sub>`</sub>(A) henceσ⁻¹(I_h(A))6=σ⁻¹(I_`(A)), so H(S⁻¹AS) =H(A).

Definition 2.5. For any d×dmatrix A we put c_h =c_h(A) :=

h

X

`=1

#J_`(A) (1≤h≤H(A)) and c₀=c₀(A) = 0.

We denote by σ_A the permutation of {1, . . . , d} whose restriction to each set {c_h−1+ 1, . . . , c_h} is the increasing bijection from this set to J_h(A), and we denote by S_A the permutation matrix defined by ∀j, S_AUj =U_σA(j).

Proposition 2.6. Let A and B be two nonnegative d×dmatrices, let c_h=c_h(A),σ =σ_A and S =S_A.

(i) IfI(AB) =I(A) thenS⁻¹BS has the form

(21) S⁻¹BS=





B^1,1 0 ... 0

B^2,1 B^2,2 ... 0 ... ... . .. ... B^H(A),1 B^H(A),2 ... B^H(A),H(A)





where the submatrices B<sup>h,`</sup> have size (c<sub>h</sub>−c<sub>h−1</sub>)×(c<sub>`</sub>−c_`−1) and

(22) j∈ {c_h−1+ 1, c_h}, j⁰∈ {c_{`−1</sub>+ 1, c<sub>`}}, h < ` ⇒ I S⁻¹BSU_j)6⊂ I S⁻¹BSU_j⁰).

(ii) If I(AB) =I(A) and H(A) =H(B) then, choosing one element jh in each {c_h−1+ 1, ch}, one has I S⁻¹BS

=SH(A)

h=1 I S⁻¹BSUj_h

× {c_h−1+ 1, c_h}.

Proof. (i) : Suppose I(AB) =I(A). Given h ∈ {1, . . . , H(A)} and j ∈ J_h(A), we first prove that

(23) I(BU_j)⊂J_h^∗(A) := [

`≥h

J_`(A).

Indeed for any`∈ {1, . . . , H(A)}and anyi∈ I(BUj)∩J<sub>`</sub>(A), the relation (15) impliesI(AU_i)⊂ I(ABU_j) and consequentlyI<sub>`</sub>(A)⊂ I(AU<sub>j</sub>) =I<sub>h</sub>(A), and`≥h by Lemma 2.3(ii).

Now (23) implies I(B) ⊂ SH(A)

h=1 J_h^∗(A) ×J_h(A) and, using Lemma 2.4(i), one has the inclusionI(S⁻¹BS)⊂SH(A)

h=1 σ⁻¹(J_h^∗(A))×σ⁻¹(J_h(A)) =SH(A)

h=1 {c_h−1+ 1, d} × {c_h−1+ 1, c_h}, which is equivalent to (21).

To prove (22) by contraposition, we consider two indicesj∈ {c_h−1+1, c_h},j⁰ ∈ {c_{`−1</sub>+1, c<sub>`}} such that I S⁻¹BSUj) ⊂ I S⁻¹BSUj⁰). Using the permutation σ^∗ defined in Lemma 2.4(i),

this inclusion is equivalent to (i, j)∈ I S⁻¹BS

⇒(i, j⁰)∈ I S⁻¹BS (σ(i), σ(j))∈σ^∗ I S⁻¹BS

⇒(σ(i), σ(j⁰))∈σ^∗ I S⁻¹BS (σ(i), σ(j))∈ I(B)⇒(σ(i), σ(j⁰))∈ I(B),

so it is equivalent to I BU_σ(j)

⊂ I BU_σ(j⁰₎

. According to (15), this implies I ABU_σ(j)

⊂ I ABU_σ(j⁰₎

. SinceI(AB) =I(A), one also hasI AU_σ(j)

⊂ I AU_σ(j⁰₎

, that is,I_h(A)⊂I<sub>`</sub>(A), implying h≥`by Lemma 2.3(ii).

(ii) : Suppose that I(AB) = I(A) and H(A) = H(B). This implies H(A) = H(S⁻¹BS) by Lemma 2.4(ii). Now, by (22), the sets I S⁻¹BSU_j1

, . . . ,I S⁻¹BSU_jH(A)

are distinct and consequently, for any j ∈ {1, . . . , d}, the set I S⁻¹BSU_j

is one of the sets I S⁻¹BSU_jh , proving (together with (22)) that I S⁻¹BS

=SH(A)

h=1 I S⁻¹BSUj_h

× {c_h−1+ 1, c_h}.

Lemma 2.7. The assertion (2) holds if(r_k)k≥0 is the sequence defined in Lemma 2.1. Moreover, for any1≤`≤h≤κ, H B<sup>h,`</sup>_k

= 1 and the size of B_k^h,` is independent of k.

Proof. From (13), the matrices A =P_r,rk−1 and B = P_rk−1,rk satisfy I(AB) =I(A); one has H(A) =H(B) because bothH(A) =H(Pr,rk−1) =H(Pr,r0) and H(B) =H(Prk−1,rk) are equal toκ. So Proposition 2.6 applies. One hasH B^h,`_k

= 1 by Proposition 2.6(ii). The size of B_k^h,`

is #Jh(Pr,rk−1)

× #J`(Pr,rk−1)

, independent ofk becauseI(Pr,rk−1) =I(P_r,r0).

Now the matrices Tk = S⁻¹Prk−1,rkS do not necessarily satisfy the last condition of The- orem 1.1: indeed I(T_k· · ·T_k⁰) may depend on (k, k⁰), 1 ≤ k ≤ k⁰. So it remains to find a subsequence (r_i⁰)i≥0 = (rki)i≥0 for the matrices T_i⁰ :=S⁻¹P_r⁰

i−1,r⁰_iS to satisfy the condition that I(T_i⁰· · ·T_i⁰0) does not depend on (i, i⁰), 1 ≤ i ≤ i⁰. Since T_i⁰· · ·T_i⁰0 = T_ki−1+1· · ·T_k

i0 , the last assertion of Theorem 1.1 is a consequence of the following lemma:

Lemma 2.8. Let (M_k)k∈N be a sequence of nonnegative d×d matrices that have the block- triangular form

(24) Mk =







B^1,1_k 0 ... 0 B^2,1_k B_k^2,2 ... 0 ... ... . .. ... B^δ,1_k B^δ,2_k ... B^δ,δ_k







where, for any 1 ≤ ` ≤ h ≤ δ, H B<sub>k</sub><sup>h,`</sup>

= 1 and the size of B_k^h,` is independent of k. Then there exists an increasing sequence of nonnegative integers (k_i)i≥0 for I(M_ki−1+1· · ·M_ki0) to be independent of the couple (i, i⁰) such that 1≤i≤i⁰.

Proof. We first consider the case δ = 1. If {k ; M_k = 0} is infinite, it is sufficient to choose the integers k₀ < k₁ < k₂ < . . . such thatM_ki+1 = 0 for any i≥0. If {k ; M_k= 0}is finite, it is sufficient to choose some integers k0 < k1 < k2 < . . ., at least equal to max({k ; Mk = 0}), such that I(M_k0+1) = I(M_k1+1) = I(M_k2+1) = . . .: indeed, since H(M_k) = H(B_k^1,1) = 1, one deduce from (15) thatI(M_ki−1+1· · ·Mk_i0) =I(M_ki−1+1) for any (i, i⁰) such that 1≤i≤i⁰.

Let nowδ be an integer such that the assertion of Lemma 2.8 holds at the rank δ−1, and let the matricesM_k be as in (24). For anyk there exists two submatricesC_k and D_k such that

(25) M_k =

B^1,1_k 0 Ck Dk

and, according to the induction hypothesis, there exists an increasing sequence (ki)i≥0 for the matrices D_i⁰ :=D_ki−1+1· · ·D_ki to satisfy the condition

(26) I(D_i⁰· · ·D_i⁰0) = constant (1≤i≤i⁰).

Setting for anyi∈N

(27) _B0

i 0 C_i⁰ D_i⁰

:=M_ki−1+1· · ·M_ki , one has for 1≤i≤i⁰

(28)

M_ki−1+1· · ·M_k

i0 =_B⁰

i...B⁰_i0 0 S_i,h⁰ D_i⁰...D⁰_i0

, where S_i,h⁰ :=C_i⁰B⁰_i+1· · ·B_i⁰⁰ +D⁰_i· · ·D⁰_i⁰₋₁C_i⁰⁰+ X

i<j<i⁰

D_i⁰· · ·D⁰_j−1C_j⁰B_j+1⁰ · · ·B_i⁰⁰.

To computeI(M_ki−1+1· · ·M_k

i0) we use the following straightforward lemma:

Lemma 2.9. For any nonnegative matrices A and B,

I(A+B) =I(A)∪ I(B) if A and B have same size;

(29)

I(AB)⊂ I(A⁰B⁰) if AB, A⁰B⁰ exist and I(A)⊂ I(A⁰) and I(B)⊂ I(B⁰);

(30)

I(AB) =I(A⁰B⁰) if AB, A⁰B⁰ exist and I(A) =I(A⁰) and I(B) =I(B⁰);

(31)

I(AB) =I(A) if AB exists andH(A) =H(B) = 1 and B6= 0.

(32)

Proof. (29) is obvious sinceAandB are nonnegative; (30), (31) and (32) are true because (15)

holds when Aand B are nonnegative.

Using (26) and Lemma 2.9, since (27), (25) and (16) implyH _B0

i

C_i⁰

≤H _B

ki

C_ki

= 1 we have

(33)

I(B_i⁰· · ·B_i⁰0) =

0 ifB_i⁰0 = 0

I(B_i⁰) if∀j ≥i, B⁰_j 6= 0 I(S_i,h⁰ ) =







I(D⁰₁C_i⁰0) ifB_i⁰0 = 0 I(C_i⁰)∪ I

D₁⁰ X

i<j≤i⁰

C_j⁰

if∀j≥i, B_j 6= 0 I(D_i⁰· · ·D_i⁰0) =I(D⁰₁).

Setting I(i, i⁰) = I D⁰₁ X

i<j≤i⁰

C_j⁰

= [

i<j≤i⁰

I D₁⁰C_j⁰

and I(i) := [

i<j<∞

I D₁⁰C_j⁰

= [

i<i⁰<∞

I(i, i⁰), the sequence of sets i 7→ I(i) is non-increasing and the sequence of sets i⁰ 7→ I(i, i⁰) is non- decreasing. So there exists a set I, an integer m and, for any i ≥ m, an integer M(i) such that

(34) ∀i≥m, I(i) =I and ∀i⁰ ≥M(i), I(i, i⁰) =I.

If{j; B_j⁰ = 0}is infinite, there exists an increasing sequence of nonnegative integers (ij)j≥0

such that B_i⁰

j = 0 for any j and such that I(D₁⁰C_i⁰

j) does not depend on j. If {j ; B⁰_j = 0}

is finite, there exists an increasing sequence of nonnegative integers (ij)j≥0 such that B_i⁰ 6= 0 from the rank i= i₀ and such that (I(B_i⁰

j+1),I(C_i⁰

j+1)) does not depend on j; let (I⁰, I⁰⁰) :=

(I(B_i⁰

j+1),I(C_i⁰

j+1)) (j ≥0). In both cases we deduce from (33) that, if i−1 and i⁰ belong to

the set {i₀, i1, i2, . . .},

(35)

I(B⁰_i· · ·B⁰_i0) =

0 if{j ; B_j⁰ = 0} is infinite I⁰ if{j ; B_j⁰ = 0} is finite I(S_i,h⁰ ) =

I(D₁⁰C_i⁰0) if{j ; B_j⁰ = 0} is infinite I⁰⁰∪I if{j ; B_j⁰ = 0} is finite I(D⁰_i· · ·D⁰_i0) =I(D₁⁰).

Let 1≤j≤j⁰, letNow we put k_j⁰ :=k_ij (j≥0).Fori=ij−1+ 1 andi⁰ =i_j⁰ (j≥1) one has M_ki−1+1· · ·M_k

i0 =M_k⁰

j−1+1· · ·M_k⁰

j0, so one deduce from (28) and (35) thatI(M_k⁰

j−1+1· · ·M_k⁰

j0) does not depend on (j, j⁰), and the assertion of Lemma 2.8 holds for the sequence (k⁰_j)j≥0.

3. The condition (E)

Lemma 3.1. Let the d×d matrixA satisfiy the condition (E) defined in (4). Then (i) I1(A))· · ·)I_H(A)(A);

(ii) if σ is a permutation of {1, . . . , d} and S the matrix defined by ∀j, SU_j =U_σ(j) , then (36) ∀h∈ {1, . . . , H(A)}, I_h S⁻¹AS

=σ⁻¹(I_h(A)) and J_h S⁻¹AS

=σ⁻¹(J_h(A)).

Proof. (i) : By definition of the sets I_h(A), for any 1≤h < `≤H(A) one has I<sub>h</sub>(A)6⊂I<sub>`</sub>(A).

IfA satisfies (E), this is equivalent toI_h(A))I_`(A).

(ii) : (36) is a consequence of the unicity of the decomposition (17), becauseI S⁻¹AS

= SH(A)

h=1 σ⁻¹(I_h(A))×σ⁻¹(J_h(A)) by Lemma 2.4(i) and σ⁻¹(I₁(A) )· · ·)σ⁻¹(I_H(A)(A)) by(i).

Definition 3.2. For any matrixA= (A_i,j)1≤i≤d0

1≤j≤d

we put

H^∗(A) := #{I ⊂ {1, . . . , d} ; I 6=∅ and ∃j, I =I(AU_j)}=

H(A)−1 if∃j, AU_j = 0 H(A) otherwise.

Lemma 3.3. Let A and B be twod×dnonnegative matrices and suppose thatA satisfies(E).

Then

(i) for any j ∈ {1, . . . , d} such that I(ABU_j) 6= ∅, there exists i(j) ∈ I(BU_j) such that I(ABU_j) =I AU_i(j)

, and there exists ϕ:{1, . . . , H^∗(AB)} → {1, . . . , H^∗(A)} such that (37) ∀h∈ {1, . . . , H^∗(AB)}, ∀j∈J_h(AB), i(j)∈J_ϕ(h)(A) and I_h(AB) =I_ϕ(h)(A);

(ii) AB satisfies(E);

(iii) ϕ is increasing;

(iv) H^∗(AB)≤min(H^∗(A), H^∗(B));

(v) BAsatisfies (E);

(vi) if H^∗(AB) =H^∗(A) thenϕ is the identity: Ih(AB) =Ih(A) for any h∈ {1, . . . , H^∗(A)}.

(vii) if H^∗(BA) =H^∗(A) thenJ_h(BA) =J_h(A) for any h∈ {1, . . . , H^∗(A)}.

Proof. (i) : If A satisfies (E) one has, by definition of the sets Ih(A), (38) I₁(A))· · ·)I_H^∗_(A)(A)

For anyh ∈ {1, . . . , H^∗(AB)} and j∈Jh(AB) one has, by (15), I(ABU_j) =S

i∈I(BUj)I(AUi), hence in view of (38) there exists i(j)∈ I(BU_j) such thatI(ABU_j) =I AU_i(j)

. So one has

∀h∈ {1, . . . , H^∗(AB)}, ∀j∈J_h(AB), ∅ 6=I_h(AB) = I(ABU_j)

= I AU_i(j)

= I<sub>`</sub>(A) (` such thati(j)∈J_`(A)).

Since `only depends on h and belongs to{1, . . . , H<sup>∗</sup>(A)}, (37) holds by setting ϕ(h) :=`.

(ii) : By (37) one has I_h(AB) = I_ϕ(h)(A) whenever I_h(AB) 6= ∅; this implies that AB satisfies (E).

(iii) : Consequently, by definition of the setsIh(AB), (39) I1(AB))· · ·)I_H^∗_(AB)(AB).

Combining (39) and (37), one has

(40) I_ϕ(1)(A))· · ·)I_ϕ(H^∗_(AB))(A).

The relations (38) and (40) imply thatϕ is increasing.

(iv) : H^∗(AB) ≤H^∗(A) because ϕ is increasing from{1, . . . , H^∗(AB)} to {1, . . . , H^∗(A)}.

And H^∗(AB) ≤ H^∗(B) because, in the relation (15), the set I(BUj) can take at most H^∗(B) nonempty values.

(v) : By (15),I(BAU_j) =S

i∈I(AUj)I(BU_i) hence the assumption thatI(AU_j)⊂ I(AUj⁰) orI(AU_j⁰)⊂ I(AU_j) for any (j, j⁰) implies thatI(BAU_j)⊂ I(BAU_j⁰) orI(BAU_j⁰)⊂ I(BAU_j) for any (j, j⁰).

(vi) : This is true because 1≤ϕ(1)<· · ·< ϕ(H^∗(A))≤H^∗(A).

(vii) : For any j₁ ∈ J₁(A), . . . , j_H^∗_(A) ∈ J_H^∗_(A)(A) one has, by Lemma 3.1(i), I(AU_j1) )

· · · ) I(AU_jH∗(A)). Hence, by (15), I(BAU_j1) ) · · · ) I(BAU_jH∗(A)). Using the hypothesis H^∗(BA) =H^∗(A), this implies that I(BAU_jh) =I_h(BA) for anyh and, using the hypothesis

on the indices jh ,Jh(BA) =Jh(A) for any h.

Corollary 3.4. Let A = (A_n)n∈N be a sequence of nonnegative d×d matrices that satisfy condition (E). Then the matrices T_k defined in Theorem 1.1 have nonnull diagonal blocks B^h,h_k for any1≤h≤κ−1.

Proof. By Theorem 1.1, H(T_k) =κ and H B_k^h,`

= 1, hence there exist c₀ = 0< c₁ < · · · <

cκ =dsuch that, for any k∈Nand 1≤`≤h≤κ,

Jh(Tk) ={c_h−1+ 1, . . . , ch} and B_k<sup>h,`</sup> has size (ch−ch−1)×(c`−c`−1).

From Lemma 3.3(ii) and (v), the matrixT1=S⁻¹Pr0,r1S satisfies (E). We apply Lemma 3.3(i) to the matrices A = T₁ and B = T₂: for any h ∈ {1, . . . , H^∗(AB)} and j ∈ J_h(AB), one has B(j, i(j)) 6= 0 and i(j) ∈ J_ϕ(h)(A). According to Theorem 1.1 one has I(AB) = I(A) and, from Lemma 3.3(vi), ϕ is the identity; consequently i(j) belongs to J_h(A) = J_h(T₁) = {ch−1+ 1, . . . , c_h}as well asj, and the inequalityB(j, i(j))6= 0 with I(B) =I(T₁) implies that

the matrixB_k^h,h is nonnull.

4. The sequence n7→PnV /kP_nVk, proof of the main theorem

4.1. Example with d= 2. Let An = ^a_cⁿ_nd⁰_n

, one suppose an >0, cn >0, dn ≥0 and that the set of matrice M={A ; ∃n, A_n=A} is finite. Since

Pn=A1· · ·An=a1· · ·an 1 0 snrn

with







r_n = ^d_a^1···dn

1···a_n (n≥1) r₀ = 1

sn = Pn

i=1ri−1·_a^ci

i

two cases may arise:

?either lim

n→∞sn=∞, and in this case, for any nonnegative vectorV 6= 0, limn→∞ PnV

kPnVk = (⁰₁);

? eithersn tends to a finite limits, in this caserntends to 0 and, for any nonnull vector V,

n→∞lim PnV kP_nVk =

( _1/(1+s)

s/(1+s)

ifV(1)6= 0 (⁰₁) ifV(1) = 0.

4.2. Counterexample withd= 3. Suppose now that theA_nare 3×3, nonnegative and lower triangular. Then the product-matrix Pn has the form

_α

n 0 0 βnγn 0 δn εn ϕn

, and n 7→ _kP^PnV

nVk does not always converge because there is no reason for n7→ ^β_δⁿ

n to converge.

The simplest example of divergence, with nonnegative lower triangular not block-diagonal matrices, is given by the product-matrix

Pn=A1· · ·An= _{1 0 0}

1 1 0 0 0 1

2⁰_{1 0 0}

0 1 0 1 0 1

2¹_{1 0 0}

1 1 0 0 0 1

2²_{1 0 0}

0 1 0 1 0 1

2³

· · ·; more precisely A_n = _{1 0 0}

1 1 0 0 0 1

or A_n = _{1 0 0}

0 1 0 1 0 1

, depending on whether or not 2^2k ≤ n < 2^2k+1 withk≥0. Since

n= 2⁰+· · ·+ 2^2k−1 ⇒ Pn=

1 0 0

4k−1

3 1 0

2·^4k₃⁻¹ 0 1

!

n= 2⁰+· · ·+ 2^2k ⇒ P_n=

1 0 0

4k+1−1

3 1 0

2·^4k₃⁻¹ 0 1

! ,

the ratio of the (2,1)-entry by the (3,1)-entry converges to ¹₂ in the first case and to 2 in the second, hencen7→ _kP^Pn

nk diverges as well asn7→ _kP^PnV

nVk for any vectorV with nonnull first entry.

4.3. Properties of the coefficients Λ and λ. The following lemmas concerns some stability properties of the coefficients Λ(A) andλ(A) defined in (5) and (6) respectively.

Lemma 4.1. Let A and B be a d1×d2 and a d2 ×d3 matrix respectively. If A and B are nonnegative,

(i) Λ(AB)≤Λ(A) +λ(A)Λ(B);

(ii) λ(AB)≤λ(A)λ(B).

Proof. (i) : Suppose that (AB)(i0, j0)6= 0; this implies∃j_∗, A(i0, j∗)B(j∗, j0)6= 0 and one has:

kABU_j0k = X

A(i0,j)6=0kAU_jkB(j, j₀) +X

A(i0,j)=0kAU_jkB(j, j₀)

≤ Λ(A)X

jA(i₀, j)B(j, j₀) +λ(A)A(i₀, j∗)X

jB(j, j₀)

≤ Λ(A)(AB)(i₀, j₀) +λ(A)A(i₀, j∗)kBU_j0k

≤ Λ(A)(AB)(i₀, j₀) +λ(A)A(i₀, j∗)Λ(B)B(j∗, j₀)

≤ Λ(A)(AB)(i0, j0) +λ(A)A(i0, j∗)Λ(B)(AB)(i0, j0),

that is kABU_j0k ≤(AB)(i0, j0) Λ(A) +λ(A)Λ(B)

, proving that Λ(AB)≤Λ(A) +λ(A)Λ(B).

(ii) : Suppose that (AB)(i₀, j₀)6= 0 = (AB)(i₀, j₁); this implies ∃j_∗, A(i₀, j∗)B(j∗, j₀)6= 0 and ∀j, A(i₀, j)B(j, j₁) = 0; moreover, A(i₀, j∗) 6= 0 =A(i₀, j∗)B(j∗, j₁) implies B(j∗, j₁) = 0.

To compare kABU_j1k=P

iAB(i, j1) with (AB)(i0, j0) we write successively kABU_j1k = X

A(i0,j)6=0

kAU_jkB(j, j₁) + X

A(i0,j)=0

kAU_jkB(j, j₁)

= 0 + X

A(i0,j)=0

kAU_jkB(j, j1)

≤ λ(A)A(i₀, j∗)X

j

B(j, j₁)

≤ λ(A)A(i0, j∗)kBU_j1k

≤ λ(A)A(i0, j∗)λ(B)B(j∗, j0)

≤ λ(A)λ(B)(AB)(i₀, j₀),

proving that λ(AB)≤λ(A)λ(B).

An immediate consequence of Lemma 4.1 is the following:

Corollary 4.2. If M1, . . . , Mn are nonnegative d ×d matrices such that λ(Mk) ≤ λ and Λ(M_k)≤Λ for anyk, then

Λ(M1M2· · ·Mn)≤(1 +· · ·+λⁿ⁻¹)Λ (41)

Λ(M₁M₂· · ·M_n)≤Λ/(1−λ) if λ <1.

(42)

λ(M₁M₂· · ·M_n)≤λⁿ (43)

4.4. Proof of Theorem A. We shall consider that A= (A_n)n∈N is a given sequence of non- negative d×dmatrices that satisfy condition (C) with respect to the sequence (s_k)k≥0 .

1) Definition of the sets of indicesI_h and J_h⁽ⁿ⁾ satisfying the equality (10), and proof of (ii).

By hypothesis the matrices

(44) A⁰_k:=P_sk−1,sk

satisfy (E). We first apply Theorem 1.1 to the sequence (A⁰_k)k∈N : there exist an increasing sequence of nonnegative integers (rk)k≥0 and a permutation matrix S for the assertions of Theorem 1.1 to hold for the block-triangular matrices

(45) Tk=S⁻¹A⁰_rk−1+1· · ·A⁰_rkS.

By Corollary 3.4,B_k^h,h 6= 0 for any 1≤h < κ. Note that

(46) T_k=S⁻¹Ptk−1,t_kS with t_k:=sr_k (k≥0).

We first establish the following properties of the matrix Pt0,t1 :