Almost reducibility of linear difference systems from a spectral point of view

We prove that, under some conditions, a linear nonautonomous difference system is Bylov's almost reducible to a diagonal one whose terms are contained in the Sacker and Sell spectrum of the original system. We also provide an example of the concept of diagonally significant system, recently introduced by P\"otzche. This example plays an essential role in the demonstration of our results.


Introduction
Let us consider the non autonomous system of linear difference equations (1.1) x(n + 1) = A(n)x(n), where x(n) is a column vector of R d and the matrix function n → A(n) ∈ R d×d satisfies the following properties: where || · || denotes a matrix norm. The purpose of this article is to study the contractibility or almost reducibility to a diagonal system. Namely, the δ-kinematical similarity of (1.1) to (1.2) y(n + 1) = U (n)y(n), where U (n) = U D (n){I + U R (n)}, where U D (n) is a diagonal matrix and U R (n) has some smallness properties which will be explained later. such that the change of coordinates y(n) = F −1 (n)x(n) (resp. y(n) = F −1 (δ, n)x(n)) transforms (1.1) into (1.2).
The concept of almost reducibility was introduced by Bylov [5] in the continuous context and the following definition is a discrete version.
In the case when V (n) is a diagonal matrix it is said that (1.1) is almost reducible to a diagonal system and it was proved in [5] that any continuous linear system satisfy this property and the components of V (n) are real numbers.
The concept of almost reducibility to a diagonal system was rediscovered and improved by F. Lin in [11], who introduces the concept of contractibility. In this paper, we are introducing its discrete version.
Definition 3. The system (1.1) is contracted to the compact subset E ⊂ R + if is almost reducible to a diagonal system y(n + 1) = Diag(C 1 (n), . . . , C d (n))y(n), where C i (n) ∈ E for any n ∈ Z.
It is worth to emphasize that while Bylov's result only says that the diagonal components are real numbers, Lin provides explicit localization properties. This arises the following definition.
Definition 4. The compact set E ⊂ R + is said to be the contractible set of (1.1) if E is the minimal compact set such that the system (1.1) can be contracted.
In the continuous case, the concept of contractibility has been applied in some results of topological equivalence [12]. The major contribution of Lin's article [11] is to prove that the Sacker and Sell spectrum of A(n) (a formal definition will be given later) is the contractible set of (1.1). To the best of our knowledge, there are no results in the discrete case and the purpose of this article is to obtain conditions for the contractibility of (1.1) by following some lines of Lin's work.
1.1. Notation and terminology. The fundamental matrix of (1.1) is defined by The transition matrix X(n, k) = X(n)X −1 (k) is defined as follows: Vector and matrix norms will be respectively denoted by | · | and || · ||. As usual the infinite norm || · || ∞ is We will also assume the following convention for sums and products [7,Pag.3 , [17], [18], [19]). The system (1.1) has an exponential dichotomy on Z if there exist numbers K ≥ 1, ρ ∈ (0, 1) and a projector P 2 = P such that Definition 6. The Sacker-Sell spectrum (also called exponential dichotomy spectrum) of (1.1) is the set Σ(A) of λ > 0 such that the systems have not an exponential dichotomy on Z.
It is interesting to point out that Siegmund and Aulbach [2], [3] developed an spectral theory directly from (1.1) avoiding the technicalities from linear skewproduct flows, which are used in the original work of Sacker and Sell [20]. This approach is widely used in the current research and will simplify our work.
Definition 7. The system (1.1) has the full spectrum condition if 2.2. Main results. In order to contextualize our main results, let us consider the scalar difference equation studied in [2]: (2.5) x n+1 = a n x n , with a n = a if n ≤ −1 b if n ≥ 0 and 0 < a ≤ b.
We claim that (2.5) is contracted to any interval [c, d] with 0 < c ≤ a ≤ b ≤ d. Indeed, given a fixed δ > 0, we consider where F 0 = 0 and we can verify that (2.5) is δ-kinematically similar to y n+1 = a n {1 + b n }y n with b n = δ cos 2 (n) 1 + n 2 . The claim follows since a n ∈ [a, b] ⊆ [c, d] and |b n | ≤ δ. Morevoer, it is shown in [2] that the linear difference equation (2.5) has Σ(A) = [a, b] and we verified that (2.5) can be contracted to any closed interval containing [a, b]. In addition, [a, b] is the minimal interval where the system can be contracted. This fact illustrates our main result. Theorem 1. If the spectrum Σ(A) is a bounded set of R + , then is a contractible set of (1.1). Theorem 2. If (1.1) has the full spectrum condition, then is diagonalizable.
Remark 3. Note that a) A careful reading of the proof of Theorem 1 will show that the coefficient δ > 0 of Definition 3 has an upper bound dependent of Σ(A). b) In spite that Theorem 2 is not new, we point out the remarkable simplicity of our proof which follows the lines of the proof of Theorem 1.

Preparatory results
The kinematically similar between (1.1) and (1.2) will be denoted by A ≃ U . Let us recall that kinematical similarity is an equivalence relation and have several properties described in the following lemmatas: Lemma 2. If A ≃ B and (1.1) has an exponential dichotomy on Z, then (1.2) has also an exponential dichotomy with the same projector and constant ρ.
The proof of these results is a straightforward exercise and can be done similarly as in the continuous case.
The inverse contention can be proved analogously.
The following results give us useful properties of the spectrum Σ(A): and the result follows.
Proof. We will consider the case λ > b, the other one can be proved by the reader in a similar way. By using the discrete Gronwall's inequality [7,Ch.4], it is easy to deduce Let h = max{L + 2, λ} and note that (1 + L)/h ∈ (0, 1) and which implies that the system has an exponential dichotomy on Z with projector P h = I. By using Remark 2, we know that (2.2) also has an exponential dichotomy on Z with P λ = P h = I for any λ > b.
Remark 4. The result above gives explicit constants for the exponential dichotomy as K = 1 and θ = (1 + L)/h ∈ (0, 1). This fact will be useful in some future steps.
The following result has been proved by Siegmund in [21] in a more general case with conditions less restrictive than (P1)-(P2). Moreover, the exponential dichotomy considers variable projectors. Nevertheless, we provide a proof in order to make the article the most self-contained possible.
By using Lemma 1 from [15], we know that (3.4) is kinematically similar to In addition (see e.g., [10, p.281]), the subsystem has an exponential dichotomy on Z with the identity as a projector and (3.7) y 2 (n + 1) = A 2 (n)y 2 (n) has an exponential dichotomy on Z with the null projector. As a consequence of Lemmas 3 and 4, we know that for some j ∈ {1, . . . , ℓ}. Now, we will verify that j = ℓ and Σ( has not an exponential dichotomy on Z and note that 1 / ∈ Σ(A 2 ) since (3.7) has an exponential dichotomy on Z with the null projector. In addition, let us recall that 1 ∈ ( b ℓ−1 λ ℓ , a ℓ λ ℓ ). Now, if j < ℓ and µ ∈ ( b ℓ−2 λ ℓ , a ℓ−1 λ ℓ ), the system y 2 (n + 1) = µ −1 A 2 (n)y 2 (n) will have an exponential dichotomy with a projector Q and statements (a)-(c) from Remark 2 says that Rank(Q) must be lower than zero, obtaining a contradiction.
An important matter of spectral theory for differential [4] and difference [19] nonautonomous systems is to determine sufficient conditions ensuring diagonal significance in the Pötzche's sense [19], namely that the spectrum Σ(C) of an upper triangular system u(n + 1) = C(n)u(n) coincides with the union of the diagonal spectra Σ(c ii ). The following result provides an example of diagonal significance and will play a fundamental role in the proof of our main result.
Proof. Firstly, we will prove that Σ Σ(c ii ), then the diagonal system has an exponential dichotomy. Now, let us consider the upper triangular system By using · ∞ norm combined with β < 1, we can see that Due to roughness results for difference equations [1, Corollary 3] (see also [13, p.276],[16, Proposition 1]), we know that if δ is small enough, the system has an exponential dichotomy. By construction, this system is kinematically similar to (3.9) and Lemma 2 says that (3.9) has also an exponential dichotomy, which implies that λ / ∈ Σ(C) and Σ(C) ⊆ d i=1 Σ(C ii ) follows.
Secondly, we will prove that By Proposition 2, we have that the system (3.9) has exponential dichotomy with projection P = I. That is, the fundamental matrix of (3.9), namely X λ , satisfies X λ (n)X −1 λ (k) ∞ ≤ ρ n−k with n ≥ k. Let us recall that X λ (n) = X(n)λ −n , where X(n) is the fundamental matrix of the system x(n + 1) = C(n)x(n). Now, as C(n) is upper triangular, we can see with the help of (1.4) that which implies that and we conclude that each scalar difference equation x i (n + 1) = λ −1 c ii (n)x i (n) (i = 1, . . . , d) has an exponential dichotomy with projection 1, which implies that The case λ < a can be proved analogously, thus Proof. Let us choose λ / ∈ E and notice that the compactness of E allow to define α = inf x∈E |λ − x| > 0. By using Definition 4, we have that (1.1) is kinematically similar to where C i (n) ∈ E for any n ∈ Z and sup n∈Z B(n) < δ/||C||. Now, by Lemma 1 we know that (2.2) is δ-kinematically similar to (3.11) y(n + 1) = 1 λ Diag(C 1 (n), . . . , C d (n)){I + B(n)}y(n).
Since C i (n) ∈ E for any n ∈ Z and i = 1, . . . , d, without loss of generality, we can assume that By definition of α, we have that We can verify that the system has an exponential dichotomy on Z since By using roughness results, we have that (3.11) has an exponential dichotomy on Z. Now, due to kinematical similarity and Lemma 2, the system (2.2) has exponential dichotomy. In consequence, λ / ∈ Σ(A) and the Proposition follows.

4.1.
Proof of Theorem 1. The proof will be made in several steps: Step 1): (1.1) is kinematically similar to an upper triangular system: By Proposition 1 and hypothesis, there exists a positive integer ℓ ≤ d such that: By Proposition 3, we know that (1.1) is kinematically similar to ( i = 1, . . . , ℓ).
By using the method of QR factorization, we know that, for each i ∈ {1, . . . , ℓ}, the systems are kinematically similar to the m i × m i upper triangular systems Step 2) Exponential dichotomy of scalar difference systems: From now on, the diagonal terms of the upper triangular matrices C i described in (4.2) will be denoted by {c irr } mi r=1 , where i is a fixed element of {1, . . . , ℓ}. Now, by Proposition 4, we know that the upper triangular system (4.2) has the property of diagonal significance, that is By Proposition 2, for any δ ∈ (0, 2a 1 ), the scalar difference equation has exponential dichotomy with constant K = 1 and projector P = 1. In consequence, there exists θ ∈ (0, 1) such that are the fundamental matrices of (4.3) and (4.4) respectively.
Step 3) A technical result: Lemma 5. For any fixed i ∈ {1, . . . , ℓ}, there exist two sequences h i (n) and ∆ i (n) such that for any n ∈ Z and there exists 0 < M 2 < 1 < M 1 verifying Proof. See Appendix A.
As a consequence of this result, we construct the m i × m i matrix: where for any r ∈ {1, . . . , m i }, it follows that for any n ∈ Z.

M2
, and C + is defined as in (3.10). Now, we can see that (4.10) and (4.2) are δ-kinematically similar to where the rs-coefficient of Γ i (n) is Let us observe that Γ i (n) can be written as follows: where the rs-coefficient of R i (n) is defined by By (4.6), (4.7) and (4.8), we can verify that and by using (4.11) it follows that ||R i (n)|| ∞ < δ for any n ∈ Z.
As the matrix above has order d×d, the results follows since is a diagonal matrix.
Appendix A. Proof of Lemma 5 We will construct a strictly increasing and unbounded sequence {N p } +∞ p=0 satisfying N 0 = 0 such that the sequences h i ,∆ i : satisfy properties (4.6) and (4.7) on Z + ∪ {0}. The case for Z − is similar and be donde by the reader.
It is straightforward to see that (4.6) is always satisfied. In order to verify (4.7), we interchange n by k in the first inequality of (4.5), we have: By using induction, we will verify that for any µ > 1, there exists a sequence {N p } p such that ≤ µ for any n ≥ 0.
By using the first inequality of (A.1) and considering n = 2m in the inequality above, we have that is unbounded for any n > N 2m . Then, there exists N 2m+1 such that this product is lower than µ for any n ∈ {N 2m , · · · , N 2m+1 }. In addition, by inductive hypothesis combined with |U (n)U −1 (N 2m )| ≥ 1 (ensured by A.1), we can see that the product above is lowerly bounded by 1/µ. Finally, we have that converges to zero when n → +∞. Then, there exists N 2m+2 such that the product above is bigger than 1/µ. As before, by (A.5) combined with |S(n)S −1 (N 2m+1 )| ≤ 1 for any n ≥ N 2m+1 , we can deduce that the product above is lower than µ and (A.2) is proved.