Asymptotically almost periodic solutions of limit and almost periodic linear difference systems

Asymptotically almost periodic solutions of limit and almost periodic linear difference systems

Martin Chvátal

Masaryk University, Faculty of Science, Department of Mathematics and Statistics, Kotláˇrská 2, CZ-611 37 Brno, Czech Republic

Received 28 August 2015, appeared 2 November 2015 Communicated by Stevo Stevi´c

Abstract. In this paper, limit periodic and almost periodic homogeneous linear differ- ence systems are considered. We study the systems in which the coefficient matrices are taken from a given bounded group and the elements of the matrices are from an infinite field with an absolute value. We show a condition on limit periodic and almost periodic systems which ensures, that the considered systems can be transformed into new systems having certain properties. The new systems possess non-asymptotically almost periodic solutions. The transformation can be done by arbitrarily small changes.

Keywords: limit periodicity, almost periodicity, asymptotic almost periodicity, limit periodic sequences, almost periodic sequences, linear difference systems.

2010 Mathematics Subject Classification: 39A06, 39A10, 39A24, 42A75.

1 Introduction

We consider the homogeneous linear difference systems of the form

x_k+1= A_k·x_k, (1.1)

where A_k ∈ X. We suppose, that X is a bounded group of square matrices over an infinite field. The cases, when sequences{A_k}are limit periodic and almost periodic, are studied. We are interested in non-asymptotically almost periodic solutions of the considered systems. Our current research is motivated by the following two facts. The smallest class of systems (1.1), which can have at least one non-asymptotically almost periodic solution and which generalize the pure periodic case, is formed by the limit periodic systems. The most studied class is given by the almost periodic systems.

Our main motivation comes from papers [7,12,13,22,23,26]. Papers [22] and [23] (and also [20]) are devoted to unitary and orthogonal homogeneous linear difference systems (1.1). It is shown in [22,23], that, in any neighbourhood of any orthogonal or unitary system, there exists a system of the form (1.1) with a non-almost periodic solution. In papers [7,12,13,26],

BEmail: chvatal.m@mail.muni.cz

general systems of the form (1.1) are studied. In [7,13], it is supposed, thatXis a commutative group. In [12,26], transformable groups are studied. The results of these papers say that, in an arbitrary neighbourhood of any considered system (1.1), there exists a system of the same form without any almost periodic solution other than the trivial one. Our main goal is to complement these results. We investigate more general situations and show, that the systems of the form (1.1) with non-asymptotically almost periodic solutions form a dense subset of the set of all considered systems as well. To prove this result, we improve the method based on constructions introduced in papers [24,25].

The almost periodic (and also limit periodic) systems are studied closely. There are many papers from the field of almost periodic linear systems. In this paragraph, we point out the most relevant of them. In books [4,8,10,19,29], one can find the basic properties of limit periodic and almost periodic sequences and functions. The linear almost periodic equations, with regard to the almost periodicity of their solutions, are analyzed in, e.g., [1,30]. For general difference systems, criteria of the existence of almost periodic solutions are presented in [31,32]. Concerning linear almost periodic difference systems and their almost periodic solutions, we can refer to [5,6,30] (and also [11,14]). We refer to papers [2,15,18] for other properties of (complex) almost periodic systems. The findings about the skew-Hermitian and skew-symmetric differential systems, which correspond to ones from [22,23], can be found in [25] and [27], respectively. For almost periodic solutions of these systems, we can refer to [16,17,21] as well. Further, if one considers limit periodic homogeneous linear difference systems with respect to their almost periodic solutions, then the properties of such systems can be found in [7,13,28].

This paper is organized as follows. In the next section, we introduce the notation that is used in the whole paper, and we recall some elementary properties of infinite fields with absolute values. In Section3, we recall the definitions of limit periodicity, almost periodicity, and asymptotic almost periodicity. To define these notions, we recall the Bohr and also the Bochner concept. In the final section, we give the basic motivation explicitly and we formulate and prove the main theorem.

2 Preliminaries

Let F be an infinite field. Let| · |: F→_Rbe an absolute value on F. Then, the properties (i) |f| ≥0 and|f|=0⇔ f =0,

(ii) |f+g| ≤ |f|+|g|, (iii) |f·g|=|f| · |g|

hold for every f,g∈F, where symbol 0 stands for the real number and, at the same time, for the zero element of F. Note that we will later denote also the zero vector and the zero matrix by the same symbol. Letm∈_Nbe arbitrarily given. We denote the set of all square matrices of dimensionmwith elements in F by symbol Mat_m(F)and the set of all m×1 vectors with elements in F by symbol F^m. Using the absolute value, we can define the norms k · k on F^m, Matm(F)as the sums of the absolute values of the elements. We have

(i) kAk ≥0 andkAk=0⇔ A=0, (ii) kA+Bk ≤ kAk+kBk_,

(iii) kf ·Ak=|f| · kAk

for all f ∈F and A,B∈Matm(F)or A,B∈F^m. We denote the identity matrix in Matm(F)asI. The absolute value on F and the norms on F^m, Mat_m(F) induce the corresponding metrics.

For simplicity, each of these metrics will be denoted by symbol$(·_,·). Then, we consider the δ-neighbourhoods in these metrics as

O_δ(A) ={B|$(B,A)<δ}, where A,B∈F, F^m or Mat_m(F).

LetX ⊂ Mat_m(F)be a bounded group. In particular, for every matrix A ∈ X, there exists the inverse matrix A⁻¹∈ Xand a number H>0 satisfyingkAk ≤Hfor every A∈ X. Let us denote the set of all limit periodic and almost periodic sequences inXby symbolLP(X)and AP(X), respectively. For the notion of limit and almost periodicity, see Definitions3.1 and 3.2 below. InAP(X), we consider the metric

$({A_k},{B_k}) =sup

k

kA_k−B_kk.

For the reader’s convenience, the δ-neighbourhoods in this set are again denoted byO_δ. We putN0 =_N∪ {0}.

3 Limit, almost, and asymptotic almost periodicity

We recall the definitions of limit periodic, almost periodic, and asymptotically almost periodic sequences and we mention their properties, which we will need in the proof of the main theorem. The general metric space(M,$)is considered. First, we recall the definition of limit periodicity. Note that it can be defined in another equivalent manner (see [3]).

Definition 3.1. We say that a sequence{ϕ_k}_k∈N0 is limit periodic if there exists a sequence of periodic sequences {ϕⁿ_k}_k∈N0 ⊆ M, n ∈N, such that limn→_∞ϕⁿ_k = ϕ_k and the convergence is uniform with respect tok ∈_N0.

Next, we recall the concept of almost periodicity. It can be also defined in several equiv- alent ways. As a definition, we remind the so-called Bohr concept of almost periodicity. We also recall the so-called Bochner concept in the theorem below.

Definition 3.2. A sequence {ϕ_k}_k∈Z ⊆ M is called almost periodic if, for any ε > 0, there exists r(ε) ∈ _Nsuch that any set consisting of r(ε) consecutive integers contains at least one numberl∈ _Zsatisfying

$(ϕ_k+l,ϕ_k)<ε, k∈ _Z.

Theorem 3.3. Let {ϕ_k}_k∈Z ⊆ M be given. The sequence{ϕ_k}_k∈Z is almost periodic if and only if any sequence {l_n}_n∈N

0 ⊆ _Zhas a subsequence{l^¯_n}_n∈N0 ⊆ {l_n}_n∈N

0 such that, for anyε >0, there exists K(ε)∈_Nsatisfying

$

ϕ_k+l¯_i,ϕ_k+l¯_j

<ε, i,j>K(ε), k ∈_Z. (3.1) Proof. See, e.g., [24].

To complete this section, we also recollect the definition of asymptotic almost periodicity.

Definition 3.4. A sequence{ϕ_k}_k∈N0 ⊆ Mis called asymptotically almost periodic if, for any ε > 0, there exists r(ε) ∈ _N and m(ε) ∈ _N such that any set consisting ofr(ε) consecutive positive integers contains at least one numberl∈ _Nsatisfying

$(ϕ_k+l,ϕ_k)<ε, k>m(ε), k∈_N.

Note that, in Banach spaces, any asymptotically almost periodic sequence is the sum of an almost periodic sequence and a sequence, which vanishes at infinity. Similarly, as in the case of almost periodicity, we remind the equivalent concept of asymptotic almost periodicity.

Theorem 3.5. Let{ϕ_k}_k∈N

0 ⊆M be given. The sequence{ϕ_k}_k∈N

0 is asymptotically almost periodic if and only if any sequence{l_n}_n∈N

0 ⊆ _Z,lim_n→_∞l_n= _∞has a subsequence{l^¯_n}_n∈N0 ⊆ {l_n}_n∈N

0

such that, for anyε >0, there exists K(ε)∈_Nsatisfying

$

ϕ_k+¯li,ϕ_k+¯lj

<ε, i,j> K(ε), k∈_N0. (3.2) Proof. See [9].

4 Results

In the beginning of this section, we call up the most relevant known results. By doing this, one can see, how our result complements our motivations.

Theorem 4.1. Let X ⊆ _Matm(_F)be a commutative group. Let, for every non-zero vector u ∈ _F^m, there existξ >0such that, for everyδ>0, there exist matrices M₁, . . . ,M_l ∈ X satisfying

M_i ∈ O_δ(I), i∈ {1, . . . ,l}, kM_l· · ·M₁·u−uk>ξ. (4.1) Let ε > 0 and a non-zero vector u ∈ F^m be arbitrary. For any {A_k}_k∈N0 ∈ LP(X), there exists {S_k}_k∈N0 ∈ O_ε({A_k}_k∈N0)∩ LP(X)such that the solution of

x_k+1 =_Sk·x_k, k∈_N0, x0= _u is not almost periodic.

Proof. See [13].

Theorem 4.2. Let X ⊆ Mat_m(F) be a commutative group. Let there exist ξ > 0 such that, for every δ > 0, there exists l ∈ _N such that, for every u ∈ F^m fulfilling kuk ≥ 1, there exist ma- trices M₁, . . . ,M_l ∈ X with the property that (4.1) is valid. Let ε > 0 be arbitrary. Then, for every {A_k}_k∈N0 ∈ LP(X)and every sequence {u_n}_n∈N of non-zero vectors u_n ∈ F^m, there exists {S_k}_k∈N0 ∈ O_ε({A_k}_k∈N0)∩ LP(X)such that the solution of

x_k+1 =S_k·x_k, k∈_N0, x₀= u_n is not almost periodic for any n∈_N.

Proof. See [7].

Theorem 4.3. Let(F,$)be separable. LetX ⊆Mat_m(F)be a bounded group. Let there existξ >0 such that, for every δ > 0, there exists l ∈ _Nsuch that, for every u ∈ F^m fulfillingkuk ≥ 1, there exist matrices M₁, . . . ,M_l ∈ X with the property that

M₁∈ O_δ(I)_, M_i+₁ ∈ O_δ(M_i)_, i∈ {_{1, . . . ,}l−₁}_, kM_l·u−uk>_{ξ. (4.2)} Letε>0be arbitrary. Then, for every{A_k}_k∈N0 ∈ LP(X), there exists{S_k}_k∈N0 ∈ O_ε({A_k}_k∈N0)∩ LP(X)such that the system

x_k+1 =S_k·x_k, k∈_N0 does not have any non-zero asymptotically almost periodic solution.

Proof. See [28].

Theorem 4.4. Let(F,$)be separable. LetX ⊆Mat_m(F)be a bounded group. Let there existξ >0 such that, for every δ > 0, there exists l ∈ _Nsuch that, for every u ∈ F^m fulfillingkuk ≥ 1, there exist matrices M₁, . . . ,M_l ∈ X with the property that(4.2)is valid. Letε >0be arbitrary. Then, for every {A_k}_k∈Z∈ AP(X), there exists{S_k}_k∈Z ∈ O_ε({A_k}_k∈Z)such that the system

x_k+1 =S_k·x_k, k∈_N0 does not have any non-zero asymptotically almost periodic solution.

Proof. See [28].

For the reader’s convenience (see Theorem4.6 below), we recall the definitions of trans- formable and weakly transformable groups (for further informations, see, e.g., [12,26]).

Definition 4.5. We say that an infinite set X ⊆ Matm(F) is transformable, if it meets the following conditions:

(i) for all A,B∈ X, it holds

A·B∈ X, A⁻¹ ∈ X;

(ii) for any L ∈ (0,∞) and ε > 0, there exists p = p(L,ε) ∈ _N such that, for any n ≥ p (n ∈ N) and any sequence {C₀,C₁, . . . ,C_n} ⊂ X, L ≤ $(C_i, 0), i ∈ {0, . . . ,n}, one can find a sequence {D₁, . . . ,D_n} ⊂ X for which

D_i ∈ O_ε(C_i), i∈ {1, . . . ,n}, Dn· · ·D₁=C₀;

(iii) the multiplication of matrices is uniformly continuous onX and has the Lipschitz prop- erty on a neighbourhood of I inX;

(iv) for any L ∈ (0,∞), there exists Q = Q(L) ∈ (0,∞) such that, for every ε > 0 and C,D∈ X \ O_L(₀)_satisfyingC∈ O_ε(D), it is valid that

C⁻¹·D,D·C⁻¹ ∈ O_ε·Q(I).

The group X is weakly transformable if there exist a transformable groupX₀ ⊂ X, matrices X₁, . . . ,X_l ∈ X, andδ_X >0 such that the following conditions hold:

(i) anyU∈ X can be expressed asU=C(U)·X_j for someC(U)∈ X₀,j∈ {1, . . . ,l}; (ii) $(C·X_i,D·X_j)>δ_X for allC,D∈ X₀,i6= j,i,j∈ {1, . . . ,l}.

Theorem 4.6. Let (F,$) be complete. Let X ⊆ Mat_m(F)be weakly transformable. Let there exist a sequence{M_i}_n∈N ⊆ X₀ such that, for any non-zero vector u ∈ F^m, one can find i = i(u) ∈ _N satisfying M_i·u 6= u. Let ε > 0be arbitrary. If {A_k}_k∈Z ∈ AP(X), then there exists {S_k}_k∈Z ∈ O_ε({A_k}_k∈Z)such that the system

x_k+1 =S_k·x_k, k∈_Z, does not possess a non-trivial almost periodic solution.

Proof. See [12].

Before we formulate the main result of this paper, we recall some elementary properties of the bounded groupX. We use them in the proof of the main theorem.

Lemma 4.7. Let V_k ∈ X and M_k ∈ X, k∈ {_{0, . . . ,}K}, be given matrices. Then, there exist matrices T_k ∈ X, k∈ {0, . . . ,K}, such that:

(i) M_K· · ·M₀·V_K· · ·V₀=V_K·T_K· · ·V₀·T₀; (ii) M_K· · ·M₀·V_K· · ·V₀= T_K·V_K· · ·T₀·V₀

hold. Moreover, one can assume that T_k ∈ O_H2δ(I)if M_k ∈ O_δ(I), and T_k = I if M_k = I.

Proof. It is seen, that the matrices

TK= V_K⁻¹·MK·VK,

T_K−1= (V_K·V_K−1)⁻¹·M_K−1·V_K·V_K−1, ...

T₀= (V_K· · ·V₀)⁻¹·M₀·V_K· · ·V₀ satisfy the equality in the part (i). Analogously, the matrices

T_K= M_K,

T_K−1= (V_K)⁻¹·M_K−1·V_K, ...

T₀= (V_K· · ·V₁)⁻¹·M₀·V_K· · ·V₁ satisfy the equality in (ii). It holds

kV⁻¹·M_k·V−Ik ≤ kV⁻¹k · kM_k−Ik · kVk ≤H²· kM_k−Ik for everyV∈ X,k ∈ {0, . . . ,K}, which completes the proof.

Remark 4.8. Let A,B,C ∈ X. If kA−Ik > ξ and kC−Bk < ξ/(2H), then kA·B−Ck >

ξ/(2H)holds. It can be directly verified by the simple computation

ξ <kA−Ik=kA·BB⁻¹−BB⁻¹k ≤ kA·B−B+C−Ck · kB⁻¹k

≤ (kA·B−Ck+kB−Ck)·H.

Now, we can prove the announced result. We recall, thatXis a bounded group.

Theorem 4.9. Let X have the property that there exists ξ >0 such that, for everyδ > 0, there exist matrices M₁, . . . ,M_l ∈X with the property that

M_i ∈ O_δ(I), i∈ {1, . . . ,l}, kM_l· · ·M₁−Ik>ξ. (4.3) Then, for every {A_k}_k∈N0 ∈ LP(X) and an arbitrary positive number ε, there exists {T_k}_k∈N0 ∈ O_ε({A_k}_k∈N0)∩ LP(X)such that the fundamental matrix{X_k}_k∈N0 of

x_k+1 =T_k·x_k, k∈_{N0, (4.4)}

is not asymptotically almost periodic.

Proof. Let ε > 0 be arbitrary. We denote ζ = ξ/(2H). We use the following construction. In the first step of the construction, for

δ₁ = ¹ 2· ^ε

H³, (4.5)

there exist matrices M⁽₁¹⁾,M⁽₂¹⁾, . . . ,M⁽_l(¹_δ⁾

1)∈ O_δ1(I) (taken from (4.3)). Denote r₁ = 2·l(δ₁), i(1, 1) =0, p(1, 1) =r₁. Let us consider the matricesM⁽₀^1,1)= I,M⁽₁^1,1)= M⁽₁¹⁾, . . . ,M⁽_p^1,1_(1,1⁾₎₋₂= I, M⁽_p^1,1_(1,1⁾₎₋₁= M⁽_l(¹⁾

δ1). Then, there exist matrices ˜T_j^(1,1), j ∈ {0, . . . ,p(1, 1)−1}, satisfying (see Lemma4.7)

M⁽_p^1,1_(1,1⁾₎₋₁· · ·M₀^(1,1)·A_p(_1,1)−1· · ·A0 = A_p(_1,1)−1·T^˜_p^(1,1_(1,1⁾₎₋₁· · ·A0·T^˜₀^(1,1)

and ˜T₀^(1,1) = I, ˜T₁^(1,1) ∈ O_H2δ₁(I), . . . , ˜T_p^(1,1_(1,1⁾₎₋₂ = I, ˜T_p^(1,1_(1,1⁾₎₋₁ ∈ O_H2δ₁(I). We define the periodic sequence

T_k^(1,1) _k∈N

0 with the period p(_{1, 1}) in the following way. If kA_i(1,1)k > _{1 and} kA_{i(1,1)+r1−1}· · ·A₀−Ik < ζ, then we define T_j^(1,1) = T^˜_j^(1,1), j ∈ {0, . . . ,p(1, 1)−1}. In the other cases, we defineT₀^(1,1) =· · ·= T⁽_p^1,1_(1,1⁾₎₋₁= I. We denoteT_k¹= T_k^(1,1)andV_k¹ = A_k·T_k¹ for k∈_N0.

In the second step, there exists a positive integeri(2, 1) divisible by 4 satisfyingi(2, 1) >

p(_{1, 1})_{. For}

δ₂ = ¹ 4· ^ε

H³, there exist matrices (see (4.3))

M₁⁽²⁾,M⁽₂²⁾, . . . ,M⁽_l(²_δ⁾

2)∈ O_δ2(I).

Without loss of generality, we can assume that l(δ₂) ≥ l(δ₁). Denote r₂ = 16·l(δ₂)·l(δ₁), p(2, 1) = [i(2, 1) +r2]·p(1, 1). We consider the matrices

M₀^(2,1)= · · ·= M_i⁽₍^2,1_2,1⁾₎₋₁ = I,

M_i⁽₍^2,1_2,1⁾₎ = I, M⁽_i(^2,1_2,1⁾₎₊₁ = I, M⁽_i(^2,1_2,1⁾₎₊₂= M₁⁽²⁾,

M⁽_i(^2,1_2,1⁾₎₊₃= I, M_i⁽₍^2,1_2,1⁾₎₊₄= I, M_i⁽₍^2,1_2,1⁾₎₊₅= I, M_i⁽₍^2,1_2,1⁾₎₊₆ = M⁽₂²⁾, ...

M⁽_i(^2,1_2,1⁾_)+4(l(

δ2)−1)= I, M⁽_i(^2,1_2,1⁾_)+1+4(l(

δ2)−1)= I, M_i⁽₍^2,1_2,1⁾_)+2+4(l(

δ2)−1)= M⁽_l(²⁾

δ2), M⁽_i(^2,1_2,1⁾_)+3+4(l(δ

2)−1) =· · ·= M⁽_p^2,1_(2,1⁾₎₋₁= I.

Then, there exist matrices ˜T_j^(2,1), j∈ {0, . . . ,p(2, 1)−1}, satisfying (see Lemma4.7) M⁽_p^2,1_(2,1⁾₎₋₁· · ·M⁽₀^2,1)·V_p¹_(2,1)−1· · ·V₀¹ =V_p¹_(2,1)−1·T^˜_p^(2,1_(2,1⁾₎₋₁· · ·V₀¹·T^˜₀^(2,1) and

T˜₀^(2,1)=· · · =T^˜_i⁽₍^2,1_2,1⁾₎₋₁ = I,

T˜_i⁽₍^2,1_2,1⁾₎= I, T˜_i⁽₍^2,1_2,1⁾₎₊₁= I, T˜_i⁽₍^2,1_2,1⁾₎₊₂ ∈ O_H2δ2(I),

T˜_i⁽₍^2,1_2,1⁾₎₊₃= I, T˜_i⁽₍^2,1_2,1⁾₎₊₄= I, T˜_i⁽₍^2,1_2,1⁾₎₊₅= I, T˜_i⁽₍^2,1_2,1⁾₎₊₆ ∈ O_H2δ2(I), ...

T˜_i⁽₍^2,1_2,1⁾_)+4(l(

δ2)−1) = I, T˜_i⁽₍^2,1_2,1⁾_)+1+4(l(

δ2)−1) = I, T˜_i⁽₍^2,1_2,1⁾_)+2+4(l(

δ2)−1)∈ O_H2δ₂(I), T˜_i⁽₍^2,1_2,1⁾_)+3+4(l(δ

2)−1)=· · · =T^˜_p^(2,1_(2,1⁾₎₋₁ = I.

We define the periodic sequence {T_k^(2,1)}_k∈N0 with the period p(2, 1) in the following way.

If kV_i¹_(2,1)k > 1/4 and kV_i¹_(2,1)+r

2−1· · ·V₀¹−V_i¹_(2,1)−1· · ·V₀¹k < ζ, then we put T_j^(2,1) = T^˜_j^(2,1), j ∈ {0, . . . ,p(2, 1)−1}. Otherwise, we define T₀^(2,1) = · · · = T_p^(2,1_(2,1⁾₎₋₁ = I. We put V_k^(2,1) = V_k¹·T_k^(2,1),k∈_N0.

There exists a positive integer i(2, 2) divisible by 8 satisfyingi(2, 2) > p(2, 1). We define the periodic sequence {T_k^(2,2)}_k∈N0 with the period p(2, 2) = [i(2, 2) +r₂−r₁]·p(2, 1)in the following way. Let us consider the matrices

M⁽₀^2,2) =· · ·= M⁽_i(^2,2_2,2⁾₎₋₁ = I,

M_i⁽₍^2,2_2,2⁾₎ =· · ·= M⁽_i(^2,2_2,2⁾₎₊₃ = I, M⁽_i(^2,2_2,2⁾₎₊₄ = M₁⁽²⁾, M_i⁽₍^2,1_2,1⁾₎₊₅ =· · ·= M⁽_i(^2,2_2,2⁾₎₊₁₁ = I, M⁽_i(^2,2_2,2⁾₎₊₁₂ = M₂⁽²⁾,

... M⁽_i(^2,2_2,2⁾_)−3+8(l(δ

2)−1) =· · ·= M⁽_i(^2,2_2,2⁾_)+3+8(l(δ

2)−1)= I, M⁽_i(^2,2_2,2⁾_)+4+8(l(δ

2)−1)= M⁽_l(²_δ⁾

2), M⁽_i(^2,2_2,2⁾_)+5+8(l(

δ2)−1)=· · · = M⁽_p^2,2_(2,2⁾₎₋₁ = I.

We know that there exist matrices ˜T_j^(2,2), j∈ {0, . . . ,p(2, 2)−1}, satisfying (see Lemma4.7) M⁽_p^2,2_(2,2⁾₎₋₁· · ·M₀^(2,2)·V_p⁽₍^2,1_2,2⁾₎₋₁· · ·V₀^(2,1) =V_p⁽₍^2,1_2,2⁾₎₋₁·T^˜_p^(2,2_(2,2⁾₎₋₁· · ·V₀^(2,1)·T^˜₀^(2,2)

and

T˜₀^(2,2)=· · · =T^˜_i⁽₍^2,2_2,2⁾₎₋₁ = I,

T˜_i⁽₍^2,2_2,2⁾₎ =· · ·= T^˜_i⁽₍^2,2_2,2⁾₎₊₃ = I, T˜_i⁽₍^2,2_2,2⁾₎₊₄∈ O_H2δ₂(I), T˜_i⁽₍^2,2_2,1⁾₎₊₅ =· · ·= T^˜_i⁽₍^2,2_2,2⁾₎₊₁₁ = I, T˜_i⁽₍^2,2_2,2⁾₎₊₁₂∈ O_H2δ2(I),

...

T˜_i⁽₍^2,2_2,2⁾_)−3+8(l(δ

2)−1) =· · ·= T^˜_i⁽₍^2,2_2,2⁾_)+3+8(l(δ

2)−1)= I, T˜_i⁽₍^2,2_2,2⁾_)+4+8(l(δ

2)−1)∈ O_H2δ2(I), T˜_i⁽₍^2,2_2,2⁾_)+5+8(l(δ

2)−1) =· · ·= T^˜_p^(2,2_(2,2⁾₎₋₁= I. If kV_i⁽₍^2,1_2,2⁾₎k > 1/4 and kV_i⁽₍^2,1_2,2⁾_)+r

2−r₁−1· · ·V₀^(2,1)−V_i⁽₍^2,1_2,2⁾₎₋₁· · ·V₀^(2,1)k < ζ, then we put T_j^(2,2) = T˜_j^(2,2), j∈ {0, . . . ,p(2, 2)−1}. We defineT₀^(2,2)= · · ·=T_p^(2,2_(2,2⁾₎₋₁= I in the other cases. We put T_k² =T_k^(2,1)·T_k^(2,2),V_k^(2,2) =V_k^(2,1)·T_k^(2,2),V_k² =V_k^(2,2), k∈_N0.

We continue in the construction in the same way. Before the n-th step, we have {V_kⁿ⁻¹}_k∈N0 ≡ {A_k·T_k¹·T_k²· · ·T_kⁿ⁻¹}_k∈N0, where the sequence {T_k¹·T_k²· · ·T_kⁿ⁻¹}_k∈N0 has the period

p(n−1,n−1) = [i(n−1,n−1) +rn−1−rn−2]·p(n−1,n−2). We denote

α(x,y) =₂^(x−21)x+y_, x∈_N, y∈ {1, 2, . . . ,x}_{, (4.6)} δ_j = ¹

2·¹ j · ^ε

H³, j∈_N, (4.7)

δ_jj = H²·δ_j, j∈_N, (4.8)

r_j =

∏

α(s,s)·l(δ_s), j∈ _N. (4.9)

For the n-th step, there exists i(n, 1) ∈ _N divisible by α(n, 1) such that i(n, 1) >

p(n−1,n−1). Taken from (4.3), forδ_n, there exist matrices M⁽₁ⁿ⁾,M₂⁽ⁿ⁾, . . . ,M⁽_l(ⁿ_δ⁾

n)∈ O_δn(I), (4.10)

wherel(δ_n)can be taken in such a way thatl(δ_n)≥l(δ_n−1). We denote p(n, 1) = [i(n, 1) +rn]·p(n−1,n−1). We consider the matrices

M₀^(n,1)= · · ·= M_i⁽₍^n,1_n,1⁾₎₋₁ = I, M⁽_i(^n,1_n,1⁾₎ =· · ·= M⁽_i(^n,1_n,1⁾₎₊

α(n,1)/2−1= I, M⁽_i(^n,1_n,1⁾₎₊

α(n,1)/2 = M₁⁽ⁿ⁾,

M⁽_i(^n,1_n,1⁾_{)+α(n,1)/2+1} =· · ·= M⁽_i(^n,1_n,1⁾_{)+α(n,1)+α(n,1)/2−1} = I, M_i⁽₍^n,1_n,1⁾_{)+α(n,1)+α(n,1)/2}= M⁽₂ⁿ⁾,

... M⁽_i(^n,1_n,1⁾₎₋

α(n,1)/2+1+α(n,1)(l(δn)−1) =· · · =M⁽_i(^n,1_n,1⁾₎₊

α(n,1)/2−1+α(n,1)(l(δn)−1) = I, M_i⁽₍^n,1_n,1⁾₎₊

α(n,1)/2+α(n,1)(l(δn)−1)= M⁽_l(ⁿ⁾

δn), M⁽_i(^n,1_n,1⁾_{)+α(n,1)/2+1+α(n,1)(l(δ}

n)−1) =· · ·= M⁽_p^n,1_(n,1⁾₎₋₁= I.