DICHOTOMY SPECTRUM AND ALMOST TOPOLOGICAL CONJUGACY ON NONAUTONOMUS UNBOUNDED DIFFERENCE SYSTEMS

. We construct a bijection between the solutions of a linear system of nonautonomous diﬀerence equations which is uniformly asymptotically stable and its unbounded perturbation. The key idea used to made this bijection is to consider the crossing times of the solutions with the unit sphere. This approach prompt us to introduce the concept of almost topological conjugacy in this nonautonomous framework. This task is carried out by simplifying both systems through a spectral approach of the notion of almost reducibility combined with suitable technical assumptions.


1.
Introduction. The Hartman-Grobman Theorem shows the existence of a local and not explicit homeomorphism between the solutions of an autonomous finite dimensional (continuous or discrete) dynamical system and its linearization around an hyperbolic equilibrium. The generalization of this linearization result to a nonautonomous framework is a delicated task since there not exists a univocal definition of hyperbolicity as in the autonomous case. Nevertheless, some qualitative properties can be recovered by the properties of dichotomy, which induces a splitting between the stable and unstable directions of the solutions of a linear system.
In 1969, C. Pugh [21] studied a family of linear autonomous dynamical systems with bounded and Lipschitz nonlinear perturbations, thus allowing the construction of an explicit and global homeomorphism, making a strong difference with the Hartman-Grobman's result. This sheds light to work in the nonautonomous case. Indeed, K.J. Palmer [15] introduced the concept of topological equivalence for the continuous case and later G. Papaschinopoulos [17] studied a discrete case. This article will be focused in the discrete case and considers the nonautonomous linear system x(n + 1) = A(n)x(n), (1) and its pertubation w(n + 1) = A(n)w(n) + f (n, w(n)), where x(n) and w(n) are column vectors of R d , the matrix function n → A(n) ∈ R d×d is non singular and f : Definition 1.1 ( [17]). The systems (1) and (2) will be called topologically equivalent if there exists a map H : Z × R d → R d with the properties (i) For each fixed n ∈ Z, the map u → H(n, u) is a bijection.
(ii) For any fixed n ∈ Z, the maps u → H(n, u) and u → H −1 (n, u) = G(n, u) are continuous. (iii) If x(n) is a solution of (1), then H(n, x(n)) is a solution of (2). Similarly, if w(n) is a solution of (2), then G(n, w(n)) is a solution of (1).
A classification of the topological equivalence results can be made by considering three aspects, which are strongly related between them: a) the type of dichotomy of the linear system (1), b) the properties of the nonlinear perturbation f , and c) the character either explicit or non explicit of the resulting homeomorphism H.
In [17], Papaschinopoulos assumes that (1) has an exponential dichotomy on Z while the perturbation is such that |f (n, x)| ≤ µ and |f (n, x) − f (n,x)| ≤ γ|x −x| for any n ∈ Z and x,x ∈ R d , and proves that if γ is small enough, for any fixed n ∈ Z there exists a bijection H(n, ξ) = ξ − ∞ k=−∞ G(n, k + 1)f (k, w(k, n, ξ)), mapping solutions of (1) in solutions of (2), where G(n, k) defined by is the Green's function associated to (1) and k → w(k, n, ξ) is the solution of (2) passing through ξ at k = n.
The case when f is Lipschitz but unbounded cannot always be carried out by using the Green's function. This fact has been studied by K.J. Palmer [16] and F. Lin [14] in the continuous context by using the crossing time (a formal definition will be given later) of a solution with the unit sphere and its continuity properties. This fact cannot be replicated in a direct way to the discrete case due to the lack of continuity of the crossing time in this context. The first approach dealing with discrete crossing times was made by Barreira et.al [3], which allow to construct an homeomorphism between the solutions of two linear systems.
In order to study the topological equivalence between (1) and (2) a first step will be to consider the simpler families and w(n + 1) = C(n){I + B(n)}w(n) + g(n, w(n)), where C(n) is a diagonal matrix and B(n) is an upper triangular matrix satisfying ||B(n)|| < δ, where δ can be chosen arbitrarily small. This fact is motivated by a recent result of almost reducibility [6], which states that (1) can be transformed into (4) by a linear change of coordinates. The concept of reducibility is well known in the continuous case and we refer the reader to [8,9] for details.
In this paper, we continue the study of the discrete crossing times initiated by Barreira et al. and proved that if (4) is uniformly asymptotically stable and the perturbation g satisfies some suitable properties, then the crossing times are locally constant with a possible set of discontinuities having Lebesgue measure zero, which prompt us to introduce the following definition. Definition 1.3. The systems (1) and (2) will be called almost topologically equivalent if there exists a map H : Z × R d → R d with the properties (i) For each fixed n ∈ Z, the map u → H(n, u) is a bijection.
(ii) For any fixed n ∈ Z, the maps u → H(n, u) and u → H −1 (n, u) = G(n, u) are continuous with the possible exception of a set with Lebesgue measure zero. (iii) If x(n) is a solution of (1), then H(n, x(n)) is a solution of (2). Similarly, if w(n) is a solution of (2), then G(n, w(n)) is a solution of (1).
To the best of our knowledge, the Definition 1.3 seems not be introduced before in the literature. Now, when the above maps H and G are continuous in R d for any n ∈ Z, we recover the classical topological equivalence which has been studied in several works as [3], [17], [22].
2.1. Mathematical preliminaries. In order to establish the relation between the systems (1)-(2) and (4)-(5), we need to recall some definitions from the nonautonomous dynamical systems theory.
The concept of δ-kinematical similarity generalizes the well known notion of kinematical similarity and was proposed in the continuous framework by B.F. Bylov in order to introduce the concept of almost reducibility [5]. The following definition is a discrete version.

Definition 2.3 ([6]). The system (1) is almost reducible to
if for any δ > 0, the system (1) is δ-kinematically similar to The almost reducibility to a diagonal system (i.e. V is a diagonal matrix) has been studied in the continuous context. Indeed, in [5] B.F. Bylov proved that its coefficients are real numbers and F. Lin improved this localization in [13] by using the property of exponential dichotomy and its associated spectrum. In a discrete context, there exists a version of this localization result (see [6]), which employs the same tools combined with the diagonal significance property introduced recently by C. Pötzsche [20]. [2], [12], [23]). The exponential dichotomy spectrum of (1) is the set Σ(A) of λ > 0 such that the systems have not an exponential dichotomy on Z.
Remark 2. The exponential dichotomy spectrum allows a deeper understanding of assumptions (P1)-(P2). Indeed: ) to say that (1) has the property of bounded growth, namely ||X(n)X −1 (k)|| ≤ Cβ |n−k| for any n and k ∈ Z for some C ≥ 1 and β > 1, which implies (see e.g. Theorem 2.1 from [1]) that Σ(A) is a finite union of at most ≤ d compact intervals (P1) also implies that (1) is δ-kinematically similar to (4). Furthermore, in [6] we proved that the diagonal terms of C(n) are contained in Σ(A).
(P2) implies that Σ(A) ⊂ (0, 1). where In addition, under the same transformation, the system (2) is transformed in Remark 4. Some comments about (Q1)-(Q2): The assumption (Q1) is always verified when the Lipschitz constant L g of g in the system (5) satisfies some smallness conditions which can be described in terms The assumption (Q2) is a technical condition which allows to construct a bijection between the solutions of (5) and a linear diagonal autonomous system by using the crossing times described previously.
are almost topologically equivalent.
In order to state our last result, we will introduce the system where then the systems (12) and (16) verify (Q1). In addition if (12) and (16) also verify (Q2), then (1) and (2) are almost topologically equivalent.
3. Some basic results. The following proposition is a classical result of local continuity with respect to the initial conditions for difference equations.
Proposition 1. Let us consider the difference equation where F ∈ F 2 with L = L F , then for the solution x(n, n 0 , u) of (19) with x(n 0 , n 0 , u) = u we have that Proof. By considering that n > n 0 , it is easy to verify that By using the discrete Gronwall's inequality (see e.g., [10,Lemma 4.32]) we have that and the right inequality follows. Finally, we replace u and v respectively by x(n 0 , n + 1, u) and x(n 0 , n + 1, v), obtaining the left inequality.
The next result is an extension to the discrete framework of [14,Proposition 5].
Proposition 2. Assume that the system (1) has an exponential dichotomy on Z with K ≥ 1, 0 < ρ < 1 and P = I. Let us consider the nonlinear perturbation where F : Z × R d × B → R d and B is a non-empty set. Moreover, F satisfies the following conditions: If L F K < 1 − ρ then for any fixed κ ∈ B, the system (20) has a unique bounded solution Z(n, κ) described by such that sup n∈Z,κ∈B || Z(n, κ) ||< +∞.
We will prove by induction that ϕ j ∈ ∞ for any j ∈ N. Indeed, if ϕ j ∈ ∞ , it follows that F (n, ϕ j (n), κ) is bounded and (P2) implies that is the unique sequence in ∞ satisfying the recursivity stated above (we refer the reader to Lemma 2 from [7] for details).
On the other hand, by using KL F < 1 − ρ, it is easy to see that {ϕ j } is a Cauchy sequence, convergent to the fixed point Z(n, κ) defined by (21).
As we are considering a fixed κ ∈ B we have that sup n∈Z | Z(n, κ) |< C(κ). That is, Z(·, κ) ∈ ∞ but its bound C(κ) could be dependent of κ ∈ B. Nevertheless, we will prove that C(κ) has an upper bound independently of κ. Indeed, by combining properties (ii), (iii) with the exponential dichotomy of (1), we can deduce that
Remark 6. The crossing times N and M are inspired in an idea carried out by Barreira et al. in [3], which use them in order to construct a bijection between two linear systems.
Now, we will verify that the systems (5) and (14) are almost topologically equivalent through the maps H : and Proof. The proof follows by using the identities (23) and (24) from Remark 5.
The above result shows the maps H and G have a fixed points in the solutions of (5) and (14) at their respective crossing times.

Remark 7.
The assumption (Q2) was essential to prove (27) but it was not necessary to deduce (28). This follows from the fact that any solution of (14) with non-null initial condition passes for all the annuli {z ∈ R d : θ j < |z| ≤ θ j+1 } j∈Z in forward and backward time while this is not always true for the solutions of (5).
Lemma 4.4. For any fixed k ∈ Z, the map H is bijective being G its inverse.
We couple the above inequalities and obtain then |H(k, ξ)| < 1 and the invariance of {ξ ∈ R d : |ξ| < 1} under H and G follows.
The next results are devoted to study the continuity properties of the maps G and H. Lemma 4.6. For any fixed k ∈ Z, the map ξ → G(k, ξ) is continuous with the possible exception of a set of initial conditions with Lebesgue measure zero.
Proof. We will focus our attention in the map M (k, ξ). We consider the sets It is easy to see that ξ ∈ Γ k if only if M (k, ξ) is written as follows The above identity prompt us to construct the step map for any fixed k, where [·] denotes the ceiling function, whose discontinuities are in the level sets D j = ξ ∈ R d \ {0} : |ξ| = θ j for any j ∈ Z, and we know that for any ξ ∈ D j it follows that M (k, ξ) = k − 1 + j. Moreover, it is easy to see that M (k, ξ) is constant in the annuli Λ j : An important consequence of the above property is that for any fixed k ∈ Z the map G is Lipschitz on the annuli Λ j . In fact, by using Proposition 1 and recalling that the Lipschitz constant of the system (5) is L = ||C||(1 + ||B||) + δ, for any couple ξ,ξ in the annulus Int Λ j we have that and we conclude that for any fixed k, the map ξ → G(k, ξ) is continuous with the possible exception of the sets D j , which have Lebesgue measure zero. Proof. It will be useful to consider the sets of initial conditions where E is any subset of R d . The proof will be decomposed in several steps.
Step 1. for any fixed k ∈ Z, we prove the following identity If ξ ∈ r∈Z G(k, D r ), there exists a unique η ∈ D j such that ξ = G(k, η).

2298ÁLVARO CASTAÑEDA AND GONZALO ROBLEDO
In consequence, it follows that if ξ ∈ I k , then H(k, ξ) ∈ D j for some j ∈ Z or equivalently ξ ∈ G(k, D j ).
Step 2. We will prove that N (k, η) is constant for any η ∈ G(k, Λ j ).
Step 3. We will prove that ξ → H(k, ξ) is continuous with the possible exception of the set I k .
In fact, by using again Proposition 1 with η,η ∈ G(k, Int Λ j ), we can deduce that , we can deduce that the η → H(k, η) is continuous with the possible exception of the set I k .
Step 4. We will show that I k has Lebesgue measure zero.
By using (30), we only need to verify that G(k, D j ) has Lebesgue measure zero for any j ∈ Z.
Notice that D j can be seen as the limit of the sequence of sets D n j defined by D n j = ξ ∈ R d \ {0} : |ξ| = θ j r n , where 0 < r n < r n+1 < 1 and r n → 1 .
As ξ → G(k, ξ) is Lipschitz in the interior of Λ j and m(D n j ) = 0 then the measure of G(k, D n j ) is zero. (See Theorem 3.33 from [25] and its details). Now, note that G(k, D j ) can be seen that as the limit of the sequence of sets E n+1 ⊂ E n defined by and m(E n ) = 0 follows from the fact that the sets G(k, D i j ) are pairwise disjoints.

Remark 8.
A careful inspection of the map G shows that it can be seen as the composition of a continuous function with a piecewise constant map. This fact prompted us to consider the almost topological equivalence.
By considering g(n, y(n)) = 0 and following the lines of the above proof, we can deduce the following result. Corollary 1. The linear systems (4) and (14) are almost topologically equivalent.

5.
Proof of Theorem 2.6. In order to verify that the systems (12) and (13) verify assumption (Q1), we have the following Lemma Lemma 5.1. Let us consider the system y(n + 1) = C(n){I + B(n)}y(n) + g(n, y(n)) where C(n), B(n) are the same of the system (12) are satisfied with g ∈ F 2 having Lipschitz constant L g such that then there exist 0 < K(L g ) < 1 such that the solutions Y (n, n 0 , ξ) of (31) verifies | Y (n, n 0 , ξ) | 2 ≤ K(L g ) n−m | Y (m, n 0 , ξ) | 2 for any n > m.
Now, by using the Cauchy-Schwarz inequality combined of properties of g it is easy to deduce that g . We can see that 0 < K(L g ) < 1 when δ verifies (32) and we can prove by an inductive approach that w(n) ≤ K(L g ) n−m w(m) for any n > m, and the results follows.
Remark 9. An interesting feature of the above result is that the uniform asymptotic stability of (12) is preserved for any F 2 -perturbation whose Lipschitz constant has an upper bound which is dependent of Σ(C[I + B]) = Σ(A).
The end of proof is now clear, indeed is a trivial consequence of Theorem 2.5 combined with the fact that (almost) topological equivalence is an equivalence relation: (i) As the systems (12) and (13) satisfies (Q1)-(Q2), Theorem 2.5 says that are almost topologically equivalent. (ii) By Theorem 1 from [6], the systems (1) and (12) are δ-kinematically similar by a Lypaunov transformation F . (iii) By using the same transformation F , the system (2) and (13) are topologically equivalent. (iv) By (ii) and (iii) we can deduce that the couples (1)- (12) and (2)- (13) are topologically equivalent since the map ξ → F −1 (n, δ)ξ fulfils the properties of topological equivalence.
Lemma 6.3. The map ξ → H(n, ξ) is continuous for any fixed n ∈ Z.
Proof. By (42), we only need to prove that the map ξ → Z(n, (n, ξ)) is continuous for any fixed n. We will follow the lines of the proof of Theorem 1 from [7]. Indeed, let us recall n → Z(n, (n 0 , ξ)) is the unique bounded solution of (40), which was constructed by successive approximations in Proposition 2. That is lim j→+∞ Z j (n, (n 0 , ξ)) = Z(n, (n 0 , ξ)),
By summarizing all this Lemmas, we easily conclude that the systems (2) and (36) are topologically equivalents. As (1) and (36) are almost topologically equivalent by Theorem 2.6, the result follows.