Is the Sacker–Sell type spectrum equal to the contractible set?

For linear differential systems, the Sacker–Sell spectrum (dichotomy spectrum) and the contractible set are the same. However, we claim that this is not true for the linear difference equations. A counterexample is given. For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales. In fact, by a counterexample, we show that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [J. Comput. Appl. Math., 2002]. Furthermore, we find that there is no bijection between them. In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum. To counter this mismatch, we propose a new notion called generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum. Our approach is based on roughness theory and Perron's transformation. In this paper, a new method for roughness theory on time scales is provided. Moreover, we provide a time-scaled version of the Perron's transformation. However, the standard argument is invalid for Perron's transformation. Thus, some novel techniques should be employed to deal with this problem. Finally, an example is given to verify the theoretical results.


History
The well-known notion of exponential dichotomy introduced by Perron [35] extends the concept of hyperbolicity from autonomous linear systems to nonautonomous linear systems and plays a crucial role in the study of the dynamical behaviour of nonautonomous dynamical systems, such as stable and unstable invariant manifolds as well as linearization theory.Since the concept was proposed by Perron, exponential dichotomy together with its variants and extensions has been extensively studied [3-8, 11, 13, 14, 17, 19, 22, 23, 25, 28-30, 38].
Based on the study of exponential dichotomy, the well-known Sacker-Sell spectrum was introduced by Sacker and Sell [37] for skew-product flows on vector bundles with compact base and it plays an important role in describing the diagonal term.In fact, Bylov [10] introduced the concept of almost reducibility and proved that any linear system can be almost reducibility to diagonal system.Later, Lin [24] improved the Bylov's work and proved that the contractible set is equal to the Sacker-Sell spectrum of the linear system.More specifically, Lin proved that the diagonal terms are contained in the Sacker-Sell spectrum and this spectrum is the minimal compact set where the diagonal coefficients belong to.The main ideas of Lin's work are based on the roughness theory of classical exponential dichotomy and Perron's transformation [35] by which a linear system can be reduced to an upper triangular system.Later, Catañeda and Huerta [12] considered Lin's work in a nonuniform framework.Catañeda and Robledo [15] extended Lin's work to difference systems.We mention that the contractible set is a powerful tool to study the linearization with unbounded perturbations (see [16,21]).
Recently, Pötzsche [32,33] introduced the concept of the exponential dichotomy on time scales.The theory of time scale or measure chain can be traced back to Hilger [20], which allows a unified treatment of continuous systems, discrete systems and hybrid systems.With such a framework, many properties and applications of exponential dichotomies on time scales can be studied in a certain range.For instance, Aulbach and Pötzsche [2] studied the reducibility of linear dynamic equation on measure chains.Pötzsche [34], Xia et al. [40] studied the linearization of dynamic equations on measure chains and time scales, respectively.Siegmund [39] considered the exponential dichotomy which is a specialized version of the one studied in [32,33] and introduced a new notion of spectrum for this specific exponential dichotomy.Another important property of exponential dichotomy is its roughness under perturbations.Roughness of exponential dichotomy can be traced back to Massera and Schäffer [27] and then it has been widely studied for continuous or discrete systems [1,18,19,25,31,36] and the systems on general time scales [41][42][43].

Motivations and novelties
Motivated by the works of Lin [24], Huerta [21], Castañeda and Robledo [16], and Siegmund [39], in this paper, we consider the relationship between the contractible set and the dichotomy spectrum studied in [39] on time scales.We show, by a counterexample, that the contractible set may not only be different from the dichotomy spectrum established by Siegmund [39], but also there is no bijection between them.In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum, which contradicts the results in [15].Thus, the counterexample also shows that the result of [15] is questionable.This is contrary to the expectation that the spectrum should be equal to the contractible set.To counter this mismatch in expectation, we propose a new definition of generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum.In particular, if T = R, the generalized contractible set is the contractible set in [24] and the dichotomy spectrum is the Sacker-Sell spectrum.In other words, our result is consistent with the one studied in [24] when dynamical system is reduced to the continuous case.
Our approach is based on roughness theorem and Perron's transformation.In this paper, a new simple method for roughness theory on time scales is provided which is different from the method of using Lyapunov function or generalized Gröwall inequality [41][42][43].The advantages of this method are that the range of disturbance is determined and the coefficient matrix function can be unbounded.The main steps of this new method are listed as follows.
(i) Firstly, we prove that the roughness theorem holds for the system which admits exponential dichotomy with the projection I or O; (ii) Secondly, we show that the unperturbed system x Δ = A(t)x is kinematically similar to a diagonal block system y Δ = diag(A 1 (t), A 2 (t))y, where its corresponding subsystem y Δ 1 = A 1 (t)y 1 (resp.y Δ 2 = A 2 (t)y 2 ) admits exponential dichotomy with the projection I (resp.O); (iii) Lastly, we construct a Lyapunov transformation x = R(t)y by which the perturbed system x Δ = (A(t) + B(t))x can be transformed into the system However, the standard techniques to construct the Lyapunov transformation x = R(t)y for the continuous cases are not valid for the dynamic equations on time scales.In fact, it is more difficult to construct the matrix-valued function R(t) than that the continuous case.Because the relation of kinematical similarity is more complex than that in the continuous case and it is difficult to deal with the term R(σ(t)) occurred in the relation, where σ is the forward jump operator.To see how to overcome the difficulty, one can refer to (4.13) and (4.14).
Furthermore, in this paper, we provide a time-scaled version of the Perron's transformation.However, on time scales, the difficulty mentioned above still exists in the discussion of Perron's transformation.The standard arguments for Perron's transformation on R are not valid for the systems on time scales.In fact, if we use the standard arguments, the transformation x = U (t)y transforms system x Δ = A(t)x into y Δ = B(t)y, then it leads us to an inequality that , where σ is the forward jump operator and A(t) is bounded.In particular, for the continuous case T = R, B(t) + B T (t) A(t) + A T (t) .However, in order to prove the boundedness of B(t) on time scales, we have to overcome the troublesome term U T (σ(t))U (t).Therefore, we employ some novel techniques to deal with this problem (see theorem 4.1).Finally, we include an example to illustrate the effectiveness of our main result.

Organization of the paper
The rest of this paper is organized as follows.In §2, we introduce some notations and definitions.Section 3 gives a counter example to state that the contractible set may not be equal to dichotomy spectrum and introduces the new definitions of Δ-contractibility and generalized contractible set.In § 4, the main results of this paper and an example are provided.

Preliminaries
For completeness, we briefly introduce some basic terminology and notations of the calculus on time scales.More details can be found in the books [9,20].A time scale T is a nonempty closed subset of R. Throughout this paper, we always assume that a time scale T is unbounded to the right and left (two-sided time).The closed interval on time scales is denoted by [•, •] T .The forward jump operator σ : A function Λ : J → R n×n with these properties is called Lyapunov transformation function and the transformation x = Λy is called Lyapunov transformation.It is known that the corresponding linear change of variables x = Λ(t)y transforms (2.2) into (2.3).
Remark 2.1.Kinematical similarity defines an equivalence relation on the set of all linear homogeneous dynamic equation in R n .Moreover, the regressivity is preserved under kinematic similarity on any time scale [2].
An invariant projector of (2.2) is defined to be a function P : T → R n×n of projections P (t), t ∈ T such that Definition 2.2 [39].For γ ∈ R we shall say that (2.2) admits an exponential dichotomy with growth rate γ (γ-ED) if there exists an invariant projector P : T → R n×n and constants K 1 and α > 0 such that for t, s ∈ T, the dichotomy estimates hold, where I is the identity matrix.
Obviously, one can see that if system (2.2) admits γ-ED and kinematical similar to system (2.3), then system (2.3) also admits γ-ED.
Definition 2.3.The dichotomy spectrum of system (2.2) is the set

A counterexample and the new definition of contractibility
From the results in [24], we know that for linear differential systems, the Sacker-Sell spectrum and the contractible set are the same.However, we claim that this is questionable for the linear difference equations.A counterexample is given.In what follows, we show, by a counterexample, that the contractible set may not be equal to dichotomy spectrum for the difference equations.
Counterexample 1: Consider the 1-dimensional discrete system (seen as time scale T = Z) where A straightforward calculation leads to where Then for any γ > 1, there exists a constant α satisfying γ − 1 > α > 0, such that |Φ a (t, s)| = e sgn(t)•t−sgn(s)•s e (γ−α)(t−s) , for t ∈ [s, +∞) T , which implies system (3.1)admits γ-ED if γ > 1.In a similar way, we can prove that system (3.1)admits γ-ED if γ < −1.For any γ ∈ [−1, 1], it can be easily verified that there are no K 1, α > 0, such that Therefore, we have Σ(a) = [−1, 1].On the other hand, since (3.1) is a diagonal system, we see that the contractible set of system (3.1) is {e − 1, e −1 − 1}.Therefore, we conclude, in this example, that the contractible set is not equal to the dichotomy spectrum.Furthermore, there is no bijection between the dichotomy spectrum and the contractible set of this system.
On the other hand, system (3.1) can be written as  3) is the subset of {e −1 , e}.This contradicts to the result (theorem 2.4, [15], saying, the Sacker-Sell spectrum of system (3.3) is the contractible set).Therefore, their assertion is questionable.
Remark 3.1.Note that proposition 5 in [15] plays an important role in proving theorem 2.4 in [15].We now show that there is a fatal error in the proof of proposition 5 in [15].For the sake of clarity, we recall proposition 5 in [15] and its proof: "proposition 5 in [15]: If the linear system satisfies (P1)-(P2) and can be contracted to a compact set E ⊂ (0, +∞), then Σ(A) ⊆ E.

Proof of proposition 5 in [15]:
Let us choose λ / ∈ E and notice that the compactness of E allows to define α = inf x∈E |λ − x| > 0. By using definition 1.4, we have that the system x(n Since C i (n) ∈ E for any n ∈ Z and i = 1, • • • , d, without loss of generality, we can assume that We illustrate this point by way of contradiction.Suppose that there exists However, for discrete systems, the assumption (3.4)  for n 0 and C i (n) > λ for n > 0. Therefore, the assumption (3.4) is false.This is the fatal error in the proof in [15].Now we consider the 1-dimensional system on time scale T = hZ, where h > 0, h = e e−1 and a(t) is defined by (3.2).Then the evolution operator of (3.5) is given by It can be easily verified that for any γ > λ Similarly, we have that for any For any γ ∈ [λ 2 , λ 1 ] and α > 0, we have e (γ−α)h < e γh e λ1h = 1 + he − h and e (γ+α)h > e λ2h = |1 + h e −1 − h|.
In fact, the counterexample shows that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [39].Furthermore, we find that there is no bijection between them.
To counter this mismatch in expectation, we propose a new notion of contractible set, named by generalized contractible set.For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales.Suppose that S is a subset of R. Let (S) denote the minimal closed interval which contains S. The notion of Δ-contractibility is defined as follows.
2) is kinematically similar to the system where ξ μ is defined by ( is almost reducible to a diagonal system In [15], the authors give a concept of contractibility.In their paper, system (3.6) is contracted to the compact subset E ⊆ (0, +∞) if it is almost reducible to a diagonal system In our paper, we study the relation between c i (k) and the dichotomy spectrum established by Siegmund [39] (Sacker-Sell type spectrum).However, the authors of [15] consider the Sacker-Sell spectrum.

Main results
The main purpose of this paper is to prove that the generalized contractible set is equal to dichotomy spectrum.Our approach is based on roughness theorem and Perron's transformation.This section is divided into three subsections.In § 4.1, we provide a time-scaled version of Perron's transformation.In § 4.2, a new simple method for roughness theory on time scales is provided which is different from the method of using Lyapunov function or generalized Gröwall inequality [41][42][43].
In § 4.3, we prove that the generalized contractible set is exactly the dichotomy spectrum and we provide an example to illustrate the effectiveness of our result.

Perron's transformation
Since X(t) is rd-continuously differentiable, it is easily seen that U (t) and Y (t) are also.The change of variables x = U (t)y replaces the equation (2.2) by where B = (U σ ) T AU − (U σ ) T U Δ .Then we have and From (4.1) and (4.2), we have We introduce the norm defined by A := sup 3), we have We claim that A T is bounded.In fact, let A ∞ = |a ij | and we obtain which implies that μ(t m ) → 0 as m → +∞.Let This inequality can be found in the page 88 of [19].Since B is upper triangular, we have is bounded, let m → +∞ and we have the right side of the above inequality unbounded, which leads to a contradiction.Therefore, B(t) is bounded.

Roughness
Lemma 4.2.Let X(t) be a fundamental matrix of system (2.2).Then system (2.2) admits γ-ED if and only if there exist a projection matrix Remark 4.3.Obviously, system (2.2) has an evolution operator Φ A (t, s) = X(t)X −1 (s).In the proof of sufficiency we construct the invariant projector P (t) = X(t)QX −1 (t) and for the necessary condition we let Q = X −1 (τ )P (τ )X(τ ).The proof of lemma 4.2 is simple and we omit it.
which has the following properties: (b) there exists a constant K 1 such that the estimates then system also admits γ-ED with the projection where Proof.We only prove the case Q = O and the case Q = I can be proved in a similar way.Let Φ(t, s) and Ψ(t, s) denote the evolution operators of systems The other assertion can be proved in a similar way.
Remark 4.7.If T = R, then μ * = 0, δ 1 = δ 2 = α/K, which are consistent with lemma 4.6 in [25].Proof.Let k be the rank of Q.By lemma 4.5, system (2.2) is kinematically similar to the block diagonal system (4.5) by a Lyapunov transformation x = J(t)y, where for all t ∈ T, and there exists a constant K 0 1 such that the estimates Φ B1 (t, s) K 0 e (γ−α)(t−s) , t s, Note that which implies that system (4.6) is kinematically similar to the system and where H := sup t∈T H(t) .It can be seen that X is a Banach space with the norm , where I k is the identity matrix of order k.Consider a matrix function H ∈ X , and the mapping T defined by Thus we have Let us define the map ϕ : R → R + : e γs − 1 s .
It can be verified that γ ϕ(γ) e γμ * −1 and then for any H 1 , H 2 ∈ X 0 , we have Thus, (4.12) Note that D J J −1 B .It follows from (4.9), (4.10), (4.11) and (4.12) that there exists a constant δ > 0, such that B < δ implies that Therefore, T is a contraction mapping.Note that X 0 is a closed subspace of X , then X 0 is a Banach space with the norm • .By the contraction mapping principle, there exists a unique fixed point H ∈ X 0 such that Then we obtain It follows from R(t)y = 0 that R(t) is invertible for any t ∈ T. Then we obtain i.e., R −1 (t)y 2 y .Therefore, R −1 (t) 2. Note that which implies that system (4.8) is kinematically similar to the system Hence, system (4.6) is kinematically similar to system (4.15).It follows from lemma 4.6 that system (4.15)admits γ-ED when B is sufficiently small and the rank of projection is k.Then system (4.6) also admits γ-ED and the rank of projection is k.

Generalized contractibility
Lemma 4.9 Theorem 11, [39].The dichotomy spectrum Σ(A) of system (2.2) is the disjoint union of k closed intervals where 0 k n, namely, and system (2.2) is kinematically similar to the system 2) admits γ-ED.It follows from lemma 4.5 that system (2.2) is kinematically similar to where the first equation of (4.17) admits γ-ED with the invariant projector I and the second equation of (4.17) admits γ-ED with the invariant projector O.By lemma 4.9, we have Σ( The proof is completed by repeating the above steps. where Φ A (t, t 0 ) is the evolution operator of system (2.2).
Lemma 4.12.The following statements are true.Proof.(a).For any λ ∈ [c, d], system (2.2) admits λ-ED, i.e., there exist constants α λ > 0, K λ 1 and a projection Q λ such that where Φ A (t, s) is the evolution operator of system (2.2).Obviously, system (2. which implies that Φ A (t, s) e M (t−s) for t ∈ [s, +∞) T .Thus, system (2.2) admits M -ED with the projector I.Note that [λ, M ] ⊆ R − Σ(A).From the statement (a) in this lemma, we have system (2.2) that admits γ-ED with the projector I.The other assertion can be proved in a similar way.
It is clear that B(t) < δ.Therefore, z Δ = [D(t) + B(t)]z admits λ-ED.Then the system x Δ = C(t)x also admits λ-ED, which implies Σ(C) On the other hand, let λ / ∈ Σ(C) = [a, b].If λ > b, by lemma 4.12, the system x Δ = C(t)x admits λ-ED with the projector I. Then there exist constants K λ 1, It can be easily verified that Without loss of generalization, we assume that Therefore, we have Since sup t∈T B(t) δ and δ can be sufficiently small, by roughness theorem, we get system (4.18)admits λ-ED.Therefore, system (2.2) also admits λ-ED, which implies λ / ∈ Σ(A).The proof is completed.
Theorem 4.15.Assume that A(t) is bounded and the generalized contractible set of system (2.2) is denoted by F .Then F = Σ(A).
Proof.The proof is divided into several parts.Firstly, we prove that F ⊆ Σ(A).
Part 1: System (2.2) is kinematically similar to an upper triangular system.

Suppose Σ(
On the other hand, the function U (t 1 , t 0 )V (t, t 1 ) is convergent to zero as t → +∞.Then, there exists t 2 > t 1 such that Combined with the first inequality of (4.24) and (4.25), we have Therefore, it is clear that system (2.2) is Δ-contracted to the set (a i − δ, b i + δ) for any δ > 0. Since the constant δ can be sufficiently small and G δ → Σ(A) as δ → 0, we have F ⊆ Σ(A), where F is the generalized contractible set of (2.2).
On the other hand, by the definition of generalized contractible set and lemma 4.14, we have Σ(A) ⊆ F .In conclusion, we have F = Σ(A).The proof is completed.We know that the dichotomy spectrum of system (3.1) is [−1, 1], which supports our result.
function is said to be rdcontinuous if it is continuous at right-dense points in T and its left-sided limits exist at left-dense points in T. The set of rd-continuous functions is denoted by C rd .The graininess function μ is defined by μ(t) := σ(t) − t.A function p : T → R is regressive if 1 + μ(t)p(t) = 0 holds for all t ∈ T κ .The set of all regressive and rd-continuous functions is denoted by R. If p ∈ R, we define the cylinder operator ξ μ : R → C rd and the exponential function e p : T → R by ://doi.org/10.1017/prm.2023.10Published online by Cambridge University Press https x admits no γ-ED}.
(ii) if system (2.2) is contracted to F 1 , then F ⊆ F 1 .https://doi.org/10.1017/prm.2023.10Published online by Cambridge University Press 2.1) and Im(c i ) denotes the range of the function c i (t).Now we use the notion of almost reducibility to explain Δ-contractibility.If there exist sets Δ = C(t)x if for any δ > 0, system (2.2) is kinematically similar to x Δ = (C(t) + B(t))x with B δ.
Theorem 4.1 Perron's transformation.If A(t) is bounded, then system (2.2) is kinematically similar to the system x Δ = B(t)x, where B(t) is an upper triangular bounded matrix function and B ∈ R.Proof.Let X(t) be a fundamental matrix of system (2.2).By QR decomposition, we obtain a real orthogonal matrix U (t) (i.e., U (t)U (t) T = U (t) T U (t) = I holds for all t ∈ T) and a real upper triangular matrix Y (t) such that [26]a 4.11 Theorem 5,[26].Suppose that A(t) is bounded on T and let L = sup https://doi.org/10.1017/prm.2023.10Published online by Cambridge University Press t∈T A(t) .Then one has Δ i = c ii x i admits λ-ED for any i ∈ {1, 2, • • • , n}.