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Nonuniform dichotomy spectrum and reducibility for nonautonomous difference equations

Nonuniform dichotomy spectrum and reducibility for nonautonomous difference equations 1IntroductionLet Ak ∈ ℝN × N, k ∈ ℤ, be a sequence of invertible matrices. In this paper, we consider the following nonautonomous linear difference equations(1.1)xk+1=Akxk,$$x_{k+1}=A_{k}x_{k},$$where xk ∈ ℝN, k ∈ ℤ. Let Φ:ℤ  ×  ℤ → ℝN × N, (k, l)↦Φ(k, l), denote the evolution operator of (1.1), i.e.,Φ ( k , l ) = A k − 1 ⋯ A l , f o r k > l , I d , f o r k = l , A k − 1 ⋯ A l − 1 − 1 , f o r k < l . $$\Phi(k,l)=\begin{cases}A_{k-1}\cdots A_{l}, &{\rm for}\quad k>l,\\{\rm Id}, &{\rm for}\quad k=l,\\A^{-1}_{k}\cdots A^{-1}_{l-1}, &{\rm for}\quad k<l.\end{cases}$$Obviously, Φ(k, m)Φ(m, l) = Φ(k, l), k, m, l ∈ ℤ, and Φ(·, l)ξ solves the initial value problem (1.1), x(l) = ξ, for l ∈ ℤ, ξ ∈ ℝN.An invariant projector of (1.1) is defined to be a function P:ℤ → ℝN × N of projections Pk, k ∈ ℤ, such that for each Pk the following property holdsPk+1Ak=AkPk,k∈Z.$$P_{k+1}A_{k}=A_{k}P_{k},\quad k\in \mathbb{Z}.$$We say that (1.1) admits an exponential dichotomy if there exist an invariant projector P and constants 0 < α < 1, K ≥ 1 such that(1.2)∥Φ(k,l)Pl∥≤Kαk−l,k≥l,$$\|\Phi(k,l)P_{l}\| \leq K\alpha^{k-l}, \quad k\geq l,$$and(1.3)∥Φ(k,l)Ql∥≤K(1α)k−l,k≤l,$$\|\Phi(k,l)Q_l\|\leq K(\tfrac{1}{\alpha})^{k-l},\quadk\leq l,$$where Ql = Id − Pl is the complementary projection.The notion of exponential dichotomy was introduced by Perron in [25] and has attracted a lot of interest during the last few decades because it plays an important role in the study of hyperbolic dynamical behavior of differential equations and difference equations. For example, see [1, 19, 28] and the references therein. We also refer to the books [10, 16, 22] for details and further references related to exponential dichotomies. On the other hand, during the last decade, inspired both by the classical notion of exponential dichotomy and by the notion of nonuniformly hyperbolic trajectory introduced by Pesin (see [5]), Barreira and Valls have introduced the notion of nonuniform exponential dichotomies and have developed the corresponding theory in a systematic way (see [6] and the references therein). As explained by Barreira and Valls, in comparison to the notion of exponential dichotomies, nonuniform exponential dichotomy is a useful and weaker notion.We say that (1.1) admits a nonuniform exponential dichotomy if there exist an invariant projector P and constants 0 < α < 1, K = 1, ε ≥ 1 such that αε2 < 1 and(1.4)∥Φ(k,l)Pl∥≤Kαk−lε|l|,k≥l,$$\|\Phi(k,l)P_{l}\| \leq K\alpha^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and(1.5)∥Φ(k,l)Ql∥≤K(1α)k−lε|l|,k≤l.$$\|\Phi(k,l)Q_l\|\leqK(\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|},\quad k\leq l.$$When ε = 1, (1.4)-(1.5) become (1.2)-(1.3), and therefore a nonuniform exponential dichotomy becomes an exponential dichotomy. Moreover, [5, Theorem 1.4.2]bp02 indicates that the condition αε2 < 1 is reasonable, which means that the nonuniform part is small.For example, given ω>5a > 0, then the linear equation(1.6)uk+1=e−ω+ak(−1)k−a(k−1)(−1)(k−1)uk,vk+1=eω−ak(−1)k+a(k−1)(−1)(k−1)vk$$u_{k+1}=e^{-\omega+ak(-1)^{k}-a(k-1)(-1)^{(k-1)}}u_{k},\,\,\,v_{k+1}=e^{\omega-ak(-1)^{k}+a(k-1)(-1)^{(k-1)}}v_{k}$$admits a nonuniform exponential dichotomy, but does not admit an exponential dichotomy. In fact, we haveΦ(k,l)Pl=e−ω(k−l−1)−a(k−l−1)(−1)k−1−al(−1)(k−1)+al(−1)l000$$\Phi(k,l)P_{l}=\left(\begin{array}{lll} e^{-\omega(k-l-1)-a(k-l-1)(-1)^{k-1}-al(-1)^{(k-1)}+al(-1)^{l}} & 0 \\ 0 & 0 \end{array}\right)$$with Pl=1000$ P_{l}=\left(\begin{array}{lll} 1 & 0 \\ 0 & 0 \end{array} \right) $. Therefore (1.4) holds withK=eω−a>1,α=e(−ω+a)∈(0,1),ε=e2a>1.$$K=e^{\omega-a}>1,\quad \alpha=e^{(-\omega+a)}\in(0,1),\quad \varepsilon=e^{2a}>1.$$Analogous arguments applied to the second equation yield the estimate (1.5). Moreover, when both k and l are even, we obtain the equality∥Φ(k,l)Pl∥=Kαk−lε|l|,k≥l,$$\|\Phi(k,l)P_{l}\| = K\alpha^{k-l}\varepsilon^{|l|}, \quad k\geql,$$which means that the nonuniform part εl = e2al cannot be removed.Among the different topics on classical exponential dichotomies, the dichotomy spectrum (also called dynamical spectrum, or Sacker-Sell spectrum) is very important and many results have been obtained. As far as we know, dynamical spectrum defined with exponential dichotomies was first introduced by Sacker and Sell in [29] in which they studied the linear skew product flows with compact base. For more results on dichotomy spectrum, we refer the reader to [2, 3, 13, 18, 23, 26, 27, 29, 31, 32] and the references therein. The definition and investigation for finite-time hyperbolicity have also been studied in [9, 17]. The dynamical spectral theorem has some important applications. For example, based on the dichotomy spectral theorem, normal forms for nonautonomous systems were established in [20,21,33]. However, all the results mentioned above were established in the setting of classical exponential dichotomies. In this paper, we establish the corresponding spectral theory for difference equations (1.1) with a nonuniform exponential dichotomy. To the best of our knowledge the nonuniform dichotomy spectral theory for linear differential equations was first studied in [14] and [35]. We refer the reader to [7, 8, 11, 12, 34] for further results on nonuniform dichotomy spectrum.This paper is organized as follows. In Section 2 we propose a definition of spectrum based on nonuniform exponential dichotomies, which is called nonuniform dichotomy spectrum. Such a spectrum can be seen as a generalization of Sacker-Sell spectrum. We prove a nonuniform dichotomy spectral theorem. In Section 3, as an application of the spectral theorem, we prove a reducibility result for (1.1). Recall that system (1.1) is called reducible if it is kinematically similar to a block diagonal system with blocks of dimension less than N.2Nonuniform dichotomy spectrumConsider the weighted system(2.1)xk+1=1γAkxk,$$x_{k+1} = \tfrac{1}{\gamma}A_{k}x_{k},$$where γ ∈ ℝ+ = (0, ∞). One can easily see thatΦγ(k,l):=(1γ)k−lΦ(k,l)$$\Phi_\gamma(k,l) := (\tfrac{1}{\gamma})^{k-l}\Phi(k,l)$$is its evolution operator. If for some γ ∈ ℝ+, (2.1) admits a nonuniform exponential dichotomy with projector Pk and constants K ≥ 1, 0 < α < 1 and ε ≥ 1, then Pk is also invariant for (1.1), that isPk+1Ak=AkPk,k∈Z,$$P_{k+1}A_{k}=A_{k}P_{k},\quad k\in \mathbb{Z},$$and the dichotomy estimates of (2.1) are equivalent to(2.2)Φ(k,l)Pl∥≤K(γα)k−lε|l|,k≥l,$$\Phi(k,l)P_{l}\| \leq K(\gamma\alpha)^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and(2.3)∥Φ(k,l)Ql∥≤K(γ1α)k−lε|l|,k≤l.$$\|\Phi(k,l)Q_l\|\leq K(\gamma\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|},\quad k\leq l.$$Definition 2.1The nonuniform dichotomy spectrum of (1.1) is the setΣNED(A)={γ∈R+:2.1admitsnononuniformexponentialdichotomy},$$\Sigma_{NED}(A) = \{\gamma \in\mathbb{R}^+:\, {2.1} \,admits\, no\, nonuniform\, exponential \,dichotomy \},$$and the resolvent set ρNED(A) = ℝ+∖ΣNED(A) is its complement. The dichotomy spectrum of (1.1) is the setΣED(A)={γ∈R+:2.1admitsnoexponentialdichotomy},$$\Sigma_{ED}(A) = \{ \gamma \in\mathbb{R}^+:\, {2.1} \,{ admits\, no\, exponential\, dichotomy} \},$$and ρED(A) = ℝ+∖ΣED(A).Proposition 1ΣNED(A) ⊂ ΣED(A).Proof. For each γ ∈ ρED(A), the weighted system (2.1) admits an exponential dichotomy. Consequently, the weighted system (2.1) admits a nonuniform exponential dichotomy. Thus, γ ∈ ρNED(A), which implies that ρED(A) ⊂ ρNED(A), and therefore ΣNED(A) ⊂ ΣED(A). □Let us define for γ ∈ ρNED(A)Sγ:=(l,ξ)∈Z×RN:supk≥l∥Φ(k,l)ξ∥γ−kε−|l|<∞,$$\mathcal{S}_\gamma : \,= \,\left\{(l,\xi) \in \mathbb{Z} \times \mathbb{R}^N:\left(\sup\limits_{k\geq l}\|\Phi(k,l)\xi\|\gamma^{-k}\right)\varepsilon^{-|l|}<\infty\right\},$$andUγ:=(l,ξ)∈Z×RN:supk≤l∥Φ(k,l)ξ∥γ−kε−|l|<∞,$$\mathcal{U}_\gamma: \,= \,\left\{(l,\xi) \in \mathbb{Z} \times \mathbb{R}^N:\left(\sup\limits_{k\leq l}\|\Phi(k,l)\xi\|\gamma^{-k}\right)\varepsilon^{-|l|}<\infty\right\},$$where ε is the constant in (2.2)-(2.3). Note that 𝒮γ and 𝒰γ depend on ε=ε(γ)$ \varepsilon=\varepsilon(\gamma) $for each γ ∈ ρNED(A). If in addition γ ∈ ρED(A) then it is shown below that these sets do not depend on ε, more precisely, εcan be set to equal 1. One may readily verify that 𝒮γ and 𝒰γ are invariant vector bundles of (1.1), here we say that a nonempty set W ⊂ Z×RN$ \mathcal{W} \subset \mathbb{Z} \times \mathbb{R}^N $is an invariant vector bundle of (1.1) if (a) it is invariant, i.e., (l,ξ)∈W⇒(k,Φ(k,l)ξ)∈W$ (l,\xi) \in \mathcal{W} \;\Rightarrow\; (k,\Phi(k,l)\xi) \in \mathcal{W} $for all k ∈ ℤ; and (b) for every l ∈ ℤ the fiber W(l)={ξ∈RN:(l,ξ)∈W}$ \mathcal{W}(l) = \{\xi \in \mathbb{R}^N \,:\, (l,\xi) \in \mathcal{W} \} $is a linear subspace of ℝN.The next lemma gives the relationship between 𝒮γ, 𝒰γ and the projector P. In [16, Chapter 2]cop it is proved in the setting of exponential dichotomies that the invariant projector is unique. The proof for the invariant projectors for (1.1) and (2.1) in the setting of nonuniform exponential dichotomies is almost identical.Lemma 2.2Assume that (2.1) admits a nonuniform exponential dichotomy with invariant projector P for γ ∈ ℝ+. ThenSγ=imP,Uγ=kerP and Sγ⊕Uγ=Z×RN.$$\mathcal{S}_{\gamma}=\operatorname{im} P, \quad \mathcal{U}_{\gamma}=\operatorname{ker} P \quad \text { and } \quad \mathcal{S}_{\gamma} \oplus \mathcal{U}_{\gamma}=\mathbb{Z} \times \mathbb{R}^{N}.$$Proof. We show only Sγ=imP$\mathcal{S}_\gamma = \text{im}\, P$. The fact 𝒰γ = ker P is analog and the fact Sγ⊕Uγ=Z×RN$\mathcal{S}_{\gamma} \oplus \mathcal{U}_{\gamma}=\mathbb{Z} \times \mathbb{R}^{N}$is clear.First we show Sγ ⊂ imP$\mathcal{S}_{\gamma} \subset \operatorname{im} P$. Let l ∈ ℤ and ξ ∈ 𝒮γ(l). Then there exists a positive constant C such that∥Φ(k,l)ξ∥≤Cγkε|l|,k≥l.$$\|\Phi(k,l)\xi\|\leq C\gamma^{k}\varepsilon^{|l|},\quad k\geq l.$$We write ξ = ξ1+ξ2 with ξ1 ∈ im Pl and ξ2 ∈ ker Pl. We show that ξ2 = 0. By invariance of P we have for k ∈ ℤ the identityξ2=Φγ(l,k)Φγ(k,l)Qlξ=Φγ(l,k)QkΦγ(k,l)ξ.$$\xi_2=\Phi_\gamma(l,k)\Phi_\gamma(k,l)Q_l\xi=\Phi_\gamma(l,k)Q_k\Phi_\gamma(k,l)\xi.$$Using the fact that (2.1) admits a nonuniform exponential dichotomy, it follows that∥Φγ(l,k)Qk∥≤K(1α)l−kε|k|.$$\|\Phi_\gamma(l,k)Q_k\|\leq K(\tfrac{1}{\alpha})^{l-k}\varepsilon^{|k|}.$$Thusξ2≤K1αl−kε|k|Φγ(k,l)ξ=K(αε)k−lεl−k+|k|1γk−l∥Φ(k,l)ξ∥≤CK(αε)k−lεl+|l|−k+|k|1γk−lγk,k≥l,$$\begin{aligned}\left\|\xi_{2}\right\| & \leq K\left(\frac{1}{\alpha}\right)^{l-k} \varepsilon^{|k|}\left\|\Phi_{\gamma}(k, l) \xi\right\| \\ &=K(\alpha \varepsilon)^{k-l} \varepsilon^{l-k+|k|}\left(\frac{1}{\gamma}\right)^{k-l}\|\Phi(k, l) \xi\| \\ & \leq C K(\alpha \varepsilon)^{k-l} \varepsilon^{l+|l|-k+|k|}\left(\frac{1}{\gamma}\right)^{k-l} \gamma^{k}, \quad k \geq l, \end{aligned}$$which implies that when k ≥ l and k > 0, we have∥ξ2∥≤CK(αε)k−lε2|l|γl,$$\|\xi_2\|\leq CK(\alpha\varepsilon)^{k-l}\varepsilon^{2|l|}\gamma^l,$$and therefore ξ2 = 0 by letting k → ∞, since αε<1.Next we show imP ⊂ Sγ$ \operatorname{im}P \subset \mathcal{S}_{\gamma} $. Let l ∈ ℤ and ξ ∈ im Pl, i.e., Plξ = ξ. Using the fact that α<1, the nonuniform exponential dichotomy estimate implies∥Φγ(k,l)ξ∥≤Kαk−lε|l|∥ξ∥≤Kε|l|∥ξ∥,k≥l,$$\|\Phi_\gamma(k,l)\xi\|\leq K\alpha^{k-l}\varepsilon^{|l|}\|\xi\|\leq K\varepsilon^{|l|}\|\xi\|,\quad k\geq l,$$which yields∥Φ(k,l)ξ∥≤Kγk−lε|l|∥ξ∥,$$\|\Phi(k,l)\xi\|\leq K\gamma^{k-l}\varepsilon^{|l|}\|\xi\|,$$and thus ξ ∈ 𝒮γ(l).Lemma 2.3The resolvent set is open, i.e., for every γ ∈ ρNED(A), there exists a β = β(γ) ∈ (0, 1) with (βγ,1βγ) ⊂ ρNED(A)$ (\beta\gamma, \frac{1}{\beta}\gamma) \subset \rho_{NED}(A) $. Furthermore,Sζ=Sγanduζ=uγforζ∈βγ,1βγ.$$\mathcal{S}_{\zeta}=\mathcal{S}_{\gamma} \quad \, { and }\, \quad u_{\zeta}=u_{\gamma} \, { for }\, \zeta \in\left(\beta \gamma, \frac{1}{\beta} \gamma\right).$$Proof. Let γ ∈ ρNED(A). Then (2.1) admits a nonuniform exponential dichotomy, i.e., the estimates (2.2)-(2.3) hold with an invariant projector P and constants K ≥ 0, 0<α <1 and ε ≥ 1. For β:=α∈(0,1)$ \beta:= \sqrt{\alpha}\in(0,1) $and ζ∈(βγ,1βγ)$ \zeta \in (\beta\gamma, \frac{1}{\beta}\gamma) $we haveΦζ(k,l)=(γζ)k−lΦγ(k,l).$$\Phi_\zeta(k,l) =(\tfrac{\gamma}{\zeta})^{k-l}\Phi_\gamma(k,l).$$Note that P is also an invariant projector for the equationxk+1=1ζAkxk.$$x_{k+1} = \tfrac{1}{\zeta}A_{k}x_{k}.$$Moreover, we have the estimates∥Φζ(k,l)Pl∥≤K(γζα)k−lε|l|≤Kβk−lε|l|,k≥l,$$\|\Phi_\zeta(k,l)P_{l}\| \leqK(\tfrac{\gamma}{\zeta}\alpha)^{k-l}\varepsilon^{|l|} \leq K\beta^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and∥Φζ(k,l)Ql∥≤K(γζ1α)k−lε|l|≤K(1β)k−lε|l|,k≤l.$$\|\Phi_\zeta(k,l)Q_{l}\|\leqK(\tfrac{\gamma}{\zeta}\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|} \leqK(\tfrac{1}{\beta})^{k-l}\varepsilon^{|l|},\quad k\leq l.$$Hence ζ ∈ ρNED(A). Therefore, ρNED(A) is an open set. Using Lemma 2.2, we know that 𝒮ζ = 𝒮γ and 𝒰ζ = 𝒰γ.Corollary 2.4ΣNED(A) is a closed set.Using the facts proved above, we can obtain the following result, whose proof is similar as [4, Lemma 2.2], and therefore we omit the proof here.Lemma 2.5Let γ1, γ2 ∈ ρNED(A) with γ1 < γ2. Then F=Uγ1∩Sγ2$ \mathcal{F}=\mathcal{U}_{\gamma_{1}} \cap \mathcal{S}_{\gamma_{2}} $is an invariant vector bundle which satisfies exactly one of the following two alternatives and the statements given in each alternative are equivalent:Alternative IAlternative II(A)F=Z×{0}.$(A) \mathcal{F}=\mathbb{Z} \times\{0\}.$A′F≠Z×{0}.$\left(\mathrm{A}^{\prime}\right) \mathcal{F} \neq \mathbb{Z} \times\{0\}.$(B)γ1,γ2 ⊂ ρNED(A).$(B) \left[\gamma_{1}, \gamma_{2}\right] \subset \rho_{N E D}(A).$(B')Thereisaζ∈γ1,γ2∩ΣNED(A).$\text{(B')}\, There\, is\, a \,\zeta \in\left(\gamma_{1}, \gamma_{2}\right) \cap \Sigma_{N E D}(A).$(C)Sγ1=Sγ2andUγ1=Uγ2.$\text{(C)}\, \,\mathcal{S}_{\gamma_{1}}=\mathcal{S}_{\gamma_{2}}\, and\,\,\mathcal{U}_{\gamma_{1}}=\mathcal{U}_{\gamma_{2}}.$C′dimSγ1<dimSγ2.$\left(\text{C}^{\prime}\right) \operatorname{dim} \mathcal{S}_{\gamma_{1}}<\operatorname{dim} \mathcal{S}_{\gamma_{2}}.$(D)Sγ=Sγ2anduγ=Uγ2forγ∈γ1,γ2.$(\mathrm{D}) \mathcal{S}_{\gamma}=\mathcal{S}_{\gamma_{2}}\, and \,u_{\gamma}=\mathcal{U}_{\gamma_{2}}\,for \,\gamma \in\left[\gamma_{1}, \gamma_{2}\right].$D′dimCγ1>dimUγ2.$\left(\mathrm{D}^{\prime}\right) \operatorname{dim} \mathcal{C}_{\gamma_{1}}>\operatorname{dim} \mathcal{U}_{\gamma_{2}}.$Now we are in a position to state and prove the nonuniform dichotomy spectral theorem which will be essential to prove the result on reducibility in Section 3. The proof follows the idea and technique of the classical dichotomy spectrum proposed in [30], we present the details for the reader’s convenience.Theorem 2.6The nonuniform dichotomy spectrum ∑NED(A)$\sum_{N E D}(A)$ of (1.1) is the disjoint union of n closed intervals (called spectral intervals) where 0≤n≤N,i.e.,ΣNED(A)=∅orΣNED(A)=R+$0 \leq n \leq N\,, i.e., \,\Sigma_{N E D}(A)=\emptyset\, or \,\Sigma_{N E D}(A)=\mathbb{R}^{+}$ or one of the four casesΣNED(A)=a1,b1or0,b1∪a2,b2∪⋯∪an−1,bn−1∪an,bnoran,∞$$\Sigma_{N E D}(A)=\left\{\begin{array}{c}{\left[a_{1}, b_{1}\right]} \\ \, { or }\, \\ \left(0, b_{1}\right]\end{array}\right\} \cup\left[a_{2}, b_{2}\right] \cup \cdots \cup\left[a_{n-1}, b_{n-1}\right] \cup\left\{\begin{array}{c}{\left[a_{n}, b_{n}\right]} \\ or \\ {\left[a_{n}, \infty\right)}\end{array}\right\}$$where 0 < a1 ≤ b1 < a2 ≤ b2 < ··· < an ≤ bn. Choose a(2.4)γ0∈ρNED(A)with0,γ0 ⊂ ρNED(A)ifpossible,$$\gamma_{0} \in \rho_{N E D}(A) \, { with }\,\left(0, \gamma_{0}\right) \subset \rho_{N E D}(A) \, { if\, possible },$$otherwise define Uγ0:=Z×RN,Sγ0:=Z×{0}$\mathcal{U}_{\gamma_{0}}:=\mathbb{Z} \times \mathbb{R}^{N}, \mathcal{S}_{\gamma_{0}}:=\mathbb{Z} \times\{0\}$. Choose a(2.5)γn∈ρNED(A)withγn,+∞ ⊂ ρNED(A)ifpossible,$$\gamma_{n} \in \rho_{N E D}(A)\, with \,\left(\gamma_{n},+\infty\right) \subset \rho_{N E D}(A)\, if \,possible,$$otherwise define Uγn:=Z×{0},Sγ0:=Z×RN$\mathcal{U}_{\gamma_{n}}:=\mathbb{Z} \times\{0\}, \mathcal{S}_{\gamma_{0}}:=\mathbb{Z} \times \mathbb{R}^{N}$. Then the setsW0=Sγ0 and Wn+1=Sγn$$\mathcal{W}_{0}=\mathcal{S}_{\gamma_{0}} \quad \text { and } \quad \mathcal{W}_{n+1}=\mathcal{S}_{\gamma_{n}}$$are invariant vector bundles of (1.1). For n = 2, choose γi ∈ ρNED(A) with(2.6)b i < γ i < a i + 1 f o r i = 1 , … , n − 1 , $$b_{i}<\gamma_{i}<a_{i+1} \, { for }\, \quad i=1, \ldots, n-1,$$then for every i = 1, …, n  −  1 the intersectionWi=Uγi−1∩Sγi$$\mathcal{W}_i=\mathcal{U}_{\gamma_{i-1}}\cap \mathcal{S}_{\gamma_{i}}$$is an invariant vector bundle of (1.1) with dim 𝒲i ≥ 1. The invariant vector bundles 𝒲i, i = 0, …, n + 1, are called spectral bundles and they are independent of the choice of γ0, …, γn in (2.4), (2.5) and (2.6). MoreoverW0⊕⋯⊕Wn+1=Z×RN$$\mathcal{W}_0\oplus\cdots\oplus\mathcal{W}_{n+1}=\mathbb{Z} \times \mathbb{R}^{N}$$is a direct sum, i.e., Wi∩Wj=Z×{0}$ \mathcal{W}_{i} \cap \mathcal{W}_{j}=\mathbb{Z} \times\{0\} $for i≠j and W0+⋯+Wn+1=Z×RN$\mathcal{W}_{0}+\cdots+\mathcal{W}_{n+1}=\mathbb{Z} \times \mathbb{R}^{N}$.Proof. Recall that the resolvent set ρNED(A) is open and therefore ΣNED(A) is the disjoint union of closed intervals. Next we will show that ΣNED(A) consists of at most N intervals. Indeed, if ΣNED(A) contains N + 1 components, then one can choose a collection of points ζ1, …, ζN in ρNED(A) such that ζ1<··· < ζN and each of the intervals (0, ζ1), (ζ1, ζ2), …, (ζN − 1, ζN), (ζN, ∞) has nonempty intersection with the spectrum ΣNED(A). Now alternative II of Lemma 2.5 implies0≤dimsζ1<⋯<dimSζN≤N$$0 \leq \operatorname{dim} s_{\zeta_{1}}<\cdots<\operatorname{dim} \mathcal{S}_{\zeta_{N}} \leq N$$and therefore either dimSζ1=0ordimSζN=N$ \operatorname{dim} \mathcal{S}_{\zeta_{1}}=0\, \text{or} \,\operatorname{dim} \mathcal{S}_{\zeta_{N}}=N $or both. Without loss of generality, dimSζN=N$ \operatorname{dim} S_{\zeta_{N}}=N $, i.e., SζN=Z×RN$\mathcal{S}_{\zeta_N}=\mathbb{Z} \times\mathbb{R}^{N}$. Assume thatxk+1=1ζNAkxk$$x_{k+1} =\tfrac{1}{\zeta_N}A_{k}x_{k}$$admits a nonuniform exponential dichotomy with invariant projector P ≡ Id, thenxk+1=1ζAkxk$$x_{k+1}=\tfrac{1}{\zeta}A_{k}x_{k}$$also admits for every ζ>ζN a nonuniform exponential dichotomy with the same projector. We conclude (ζN, ∞) ⊂ ρNED(A), which is a contradiction. This proves the alternatives for ΣNED(A).Due to Lemma 2.5, the sets 𝒲0, …, 𝒲n+1 are invariant vector bundles. To prove now that dim 𝒲1 ≥ 1, …, dim 𝒲n ≥ 1 for n ≥ 1, let us assume that dim 𝒲1 = 0, i.e., Uγ0∩Sγ1=Z×{0}$ \mathcal{U}_{\gamma_{0}} \cap \mathcal{S}_{\gamma_{1}}=\mathbb{Z} \times\{0\} $. If (0, b1] is a spectral interval this implies that Sγ1=Z×{0}$ \mathcal{S}_{\gamma_{1}}=\mathbb{Z} \times\{0\} $. Then the projector of the nonuniform exponential dichotomy ofxk+1=1γ1Akxk$$x_{k+1} =\tfrac{1}{\gamma_1}A_{k}x_{k}$$is 0 and then we get the contradiction (0, γ1) ⊂ ρNED(A). If [a1, b1] is a spectral interval then [γ0, γ1]∩ΣNED(A)≠∅ and by alternative II of Lemma 2.5 we get a contradiction. Therefore dim 𝒲1 ≥ 1 and analogously dim𝒲n ≥ 1. Furthermore for n ≥ 3 and i = 2, …, n  −  1 one has [γi − 1, γi]∩ΣNED(A)≠∅ and again alternative II of Lemma 2.5 yields dim𝒲i ≥ 1.For i < j we have Wi ⊂ Sγi$ \mathcal{W}_{i}\subset \mathcal{S}_{\gamma_i} $and Wj ⊂ Uγj−1 ⊂ Uγi$ \mathcal{W}_{j}\subset \mathcal{U}_{\gamma_{j-1}} \subset \mathcal{U}_{\gamma_i} $and with Lemma 2.5 this gives Wi∩Wj ⊂ Sγi∩Uγi=Z×{0}$ \mathcal{W}_{i} \cap \mathcal{W}_{j}\subset\mathcal{S}_{\gamma_i}\cap \mathcal{U}_{\gamma_i}=\mathbb{Z} \times \{0\} $, so Wi∩Wj=Z×{0}$ \mathcal{W}_{i} \cap\mathcal{W}_{j} =\mathbb{Z} \times \{0\} $for i≠j.To show that W0⊕⋯⊕Wn+1=Z×RN$ \mathcal{W}_0\oplus\cdots\oplus\mathcal{W}_{n+1}=\mathbb{Z} \times \mathbb{R}^{N} $, recall the monotonicity relations Sγ0 ⊂ ⋯ ⊂ Sγn$ \mathcal{S}_{\gamma_0}\subset \cdots\subset \mathcal{S}_{\gamma_n} $, Uγ0⊃⋯⊃Uγn$ \mathcal{U}_{\gamma_0}\supset\cdots\supset\mathcal{U}_{\gamma_n} $, and the identity Sγ⊕Uγ=Z×RN$ \mathcal{S}_\gamma \oplus \mathcal{U}_\gamma = \mathbb{Z} \times \mathbb{R}^N $for γ ∈ ℝ+. Therefore Z×RN=W0+Uγ0$ \mathbb{Z} \times \mathbb{R}^{N}=\mathcal{W}_0 +\mathcal{U}_{\gamma_0} $. Now we haveZ × R N = W 0 + U γ 0 ∩ [ S γ 1 + U γ 1 ] = W 0 + [ U γ 0 ∩ S γ 1 ] + U γ 1 = W 0 + W 1 + U γ 1 . $$\mathbb{Z} \times \mathbb{R}^{N} = \mathcal{W}_0 + \mathcal{U}_{\gamma_0} \cap [\mathcal{S}_{\gamma_1}+\mathcal{U}_{\gamma_1}]\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= \mathcal{W}_0 + [\mathcal{U}_{\gamma_0} \cap \mathcal{S}_{\gamma_1}]+\mathcal{U}_{\gamma_1}\\\,\,\,\,= \mathcal{W}_0 + \mathcal{W}_1+\mathcal{U}_{\gamma_1}.$$Doing the same for Uγ1$ \mathcal{U}_{\gamma_1} $, we getZ × R N = W 0 + W 1 + U γ 1 ∩ [ S γ 2 + U γ 2 ] = W 0 + W 1 + [ U γ 1 ∩ S γ 2 ] + U γ 2 = W 0 + W 1 + W 2 + U γ 2 , $$\mathbb{Z} \times \mathbb{R}^{N} = \mathcal{W}_0 + \mathcal{W}_1 + \mathcal{U}_{\gamma_1} \cap [\mathcal{S}_{\gamma_2}+\mathcal{U}_{\gamma_2}]\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= \mathcal{W}_0 + \mathcal{W}_1 + [\mathcal{U}_{\gamma_1} \cap \mathcal{S}_{\gamma_2}]+\mathcal{U}_{\gamma_2}\\\,\,\,\,= \mathcal{W}_0 + \mathcal{W}_1 + \mathcal{W}_2+\mathcal{U}_{\gamma_2},$$and mathematical induction yields Z×RN=W0+⋯+Wn+1$ \mathbb{Z} \times \mathbb{R}^{N}=\mathcal{W}_0+\cdots+\mathcal{W}_{n+1} $. To finish the proof, let γ˜0,…,γ˜n∈ρNED(A)$ \tilde{\gamma}_0,\ldots,\tilde{\gamma}_n \in\rho_{NED}(A) $be given with the properties (2.4), (2.5) and (2.6). Then alternative I of Lemma 2.5 impliesSγi=Sγ˜iandUγi=Uγ˜ifori=0,…,n$$\mathcal{S}_{\gamma_i}=\mathcal{S}_{\tilde{\gamma}_i} \quad {\rm and} \quad \mathcal{U}_{\gamma_i}=\mathcal{U}_{\tilde{\gamma}_i} \quad {\rm for} \quad i=0,\ldots,n$$and therefore the invariant vector bundles 𝒲0, …, 𝒲n+1 are independent of the choice of γ0, …, γn in (2.4), (2.5) and (2.6).Definition 2.7We say that the evolution operator of (1.1) is nonuniformly exponentially bounded if there exist K > 0, ε ≥ 1 and a ≥ 1 with(2.7)∥Φ(k,l)∥≤Ka|k−l|ε|l|,k,l∈Z.$$\begin{equation}\|\Phi(k,l)\|\leqKa^{|k-l|}\varepsilon^{|l|},\quad k,l\in\mathbb{Z}.\end{equation}$$Lemma 2.8Assume the evolution operator of (1.1) is nonuniformly exponentially bounded. Then ΣNED(A) is a bounded closed set and ΣNED(A) ⊂ [1a,a]$ \Sigma_{NED}(A)\subset [\frac{1}{a},a] $.Proof. Assume that (2.7) holds. Let γ > a and 0<α:=aγ<1$ 0<\alpha := \frac{a}{\gamma}<1 $, then estimate (2.7) implies∥Φγ(k,l)∥≤Kαk−lε|l|,k≥l.$$\|\Phi_{\gamma}(k,l)\|\leq K\alpha^{k-l}\varepsilon^{|l|}, \quad k\geq l.$$Therefore (1.1) admits a nonuniform exponential dichotomy with invariant projector P = Id. It follows that γ ∈ ρNED(A) and also similarly for 0<γ<1a$ 0<\gamma < \frac{1}{a} $, hence ΣNED(A) ⊂ [1a,a]$ \Sigma_{NED}(A) \subset [\frac{1}{a},a] $.Lemma 2.9The evolution operator of (1.1) is nonuniformly exponentially bounded if and only if the nonuniform dichotomy spectrum ΣNED(A) of (1.1) is the disjoint union of n closed intervals where 0 ≤ n ≤ N, i.e.,ΣNED(A)=[a1,b1]∪[a2,b2]∪⋯∪[an−1,bn−1]∪[an,bn],$$\Sigma_{NED}(A) =[a_1,b_1]\cup [a_2,b_2] \cup \cdots \cup [a_{n-1},b_{n-1}]\cup[a_n,b_n],$$where 0 < a1 ≤ b1 < a2 ≤ b2 < ··· < an ≤ bn < ∞. In addition, W1⊕⋯⊕Wn=Z×RN$ \mathcal{W}_1\oplus\cdots\oplus\mathcal{W}_{n}=\mathbb{Z} \times \mathbb{R}^{N} $, W0=Wn+1=Z×{0}$ \mathcal{W}_0 = \mathcal{W}_{n+1}=\mathbb{Z} \times \{0\} $, where the sets 𝒲0, …, 𝒲n+1 are invariant vector bundles defined in Theorem 2.6.Proof. Necessity. It is easy to know that ΣNED(A) is bounded from Lemma 2.8. Now we prove that ΣNED(A)≠∅. It is easy to verify that Sγ=imP=Z×RN$ \mathcal{S}_\gamma = \text{im} P =\mathbb{Z} \times \mathbb{R}^{N} $and Uγ=kerP=Z×{0}$ \mathcal{U}_\gamma = \kerP=\mathbb{Z} \times \{0\} $for γ>a. Setγ ⋆ = inf γ ∈ ρ N E D ( A ) : S γ = Z × R N .$$\gamma^{\star}=\inf \left\{\gamma \in \rho_{N E D}(A): \mathcal{S}_{\gamma}=\mathbb{Z} \times \mathbb{R}^{N}\right\}.$$Clearly, γ∗∈[1a,a]$ \gamma^{*}\in [\frac{1}{a},a] $. In addition, we have γ* ∈ ΣNED(A). Otherwise, by using Lemma 2.3, there exists a neighborhood J$ \mathcal {J} $of γ* such that J ⊂ ρNED(A)$ \mathcal {J} \subset \rho_{NED}(A) $and for any γ<γ* of J$ \mathcal {J} $we have Sγ=Sγ∗=Z×RN$ \mathcal{S}_{\gamma}=\mathcal{S}_{\gamma^{*}}=\mathbb{Z} \times \mathbb{R}^{N} $, which is a contradiction to the definition of γ*. So ΣNED(A)≠∅, which means thatΣNED(A)=[a1,b1]∪[a2,b2]∪⋯∪[an−1,bn−1]∪[an,bn].$$\Sigma_{NED}(A) =[a_1,b_1]\cup [a_2,b_2] \cup \cdots \cup [a_{n-1},b_{n-1}]\cup[a_n,b_n].$$Let γ0 ∈ (0, a1), γi ∈ (bi, ai+1) for i = 1, …, n  −  1 and γn ∈ (bn, ∞). Clearly, from the proof of Lemma 2.8, we have Uγn=Z×{0}$ \mathcal{U}_{\gamma_{n}} =\mathbb{Z}\times \{0\} $for γn>bn and Sγ0=Z×{0}$ \mathcal{S}_{\gamma_{0}} =\mathbb{Z}\times \{0\} $for 0<γ0<a1. Therefore, W0=Wn+1=Z×{0}$ \mathcal{W}_0 =\mathcal{W}_{n+1}=\mathbb{Z} \times \{0\} $, and then it follows from Theorem 2.6 that W1⊕⋯⊕Wn=Z×RN$ \mathcal{W}_1\oplus\cdots\oplus\mathcal{W}_{n}=\mathbb{Z} \times \mathbb{R}^{N} $.Sufficiency. Let γ0 ∈ (0, a1), γi ∈ (bi, ai+1) for i = 1, …, n  −  1 and γn ∈ (bn, ∞). Clearly, from the proof of Lemma 2.8, we have Sγn=Z×RN$ \mathcal{S}_{\gamma_{n}} =\mathbb{Z} \times \mathbb{R}^{N} $for γn>bn and Uγ0=Z×RN$ \mathcal{U}_{\gamma_{0}} =\mathbb{Z} \times \mathbb{R}^{N} $for 0 <γ0<a1. This means that the invariant projectors associated to γ0 and γn are Pl = 0 and Pl = Id respectively.Hence, there exist K0 > 1, 0<α0 < 1 and ε0 > 1 such that∥Φγ0(k,l)∥≤K0(1α0)k−lε0|l|,k≤l,$$\|\Phi_{\gamma_{0}}(k,l)\|\leq K_{0}(\tfrac{1}{\alpha_{0}})^{k-l}\varepsilon_{0}^{|l|},\quadk\leq l,$$and there exist Kn > 1, 0 < αn < 1 and εn > 1 such that∥Φγn(k,l)∥≤Knαnk−lεn|l|,k≥l.$$\|\Phi_{\gamma_{n}}(k,l)\|\leq K_{n}\alpha_{n}^{k-l}\varepsilon_{n}^{|l|},\quadk\geq l.$$Now taking K = max{K0, Kn}, ε = max{ε0, εn} and a=max{a0γ0,anγn}$ a=\max\{\frac{a_{0}}{\gamma_{0}}, a_{n}\gamma_{n}\} $, then we have∥Φ(k,l)∥≤Kα|k−l|ε|l|,for k,l∈Z,$$\|\Phi(k,l)\|\leqK\alpha^{|k-l|}\varepsilon^{|l|},\quad {\rm for ~}k, l\in \mathbb{Z},$$which shows that the evolution operator of (1.1) is nonuniformly exponentially bounded.From Proposition 1, we know ΣNED(A) ⊂ ΣED(A). Finally in this section, we present an example to illustrate that ΣNED(A)≠ΣED(A) can occur.Example 2.10Given ω>5a > 0. Consider the scalar equation(2.8)uk+1=Akuk$$u_{k+1}=A_{k}u_{k}$$withAk=e−ω+ak(−1)k−a(k−1)(−1)(k−1).$$A_{k}=e^{-\omega+ak(-1)^{k}-a(k-1)(-1)^{(k-1)}}.$$Then ΣNED(A) = [e − ω − 5a, e − ω+5a] and ΣED(A) = ℝ+.Proof. The evolution operator of (2.8) is given byΦ ( k , l ) = e − ω ( k − l − 1 ) − a ( k − l − 1 ) ( − 1 ) k − 1 − a l ( − 1 ) ( k − 1 ) + a l ( − 1 ) l . $$\Phi(k,l)=e^{-\omega(k-l-1)-a(k-l-1)(-1)^{k-1}-al(-1)^{(k-1)}+al(-1)^{l}}.$$For each γ ∈ ℝ+ the evolution operator of(2.9)ukuk+1=1γAkuk$${uk}u_{k+1}=\tfrac{1}{\gamma}A_{k}u_{k}$$is given by(2.10)Φ γ ( k , l ) = ( 1 γ ) ( k − l ) e − ω ( k − l − 1 ) − a ( k − l − 1 ) ( − 1 ) k − 1 − a l ( − 1 ) ( k − 1 ) + a l ( − 1 ) l . $$\begin{equation}\Phi_{\gamma}(k,l)=(\tfrac{1}{\gamma})^{(k-l)}e^{-\omega(k-l-1)-a(k-l-1)(-1)^{k-1}-al(-1)^{(k-1)}+al(-1)^{l}}.\end{equation}$$For any γ ∈ (e( − ω+5a), +∞), it follows from (2.10) that(2.11)Φ γ ( k , l ) ≤ e ω − a e − ω + a γ k − l e 2 a | l | , k ≥ l , $$\left|\Phi_{\gamma}(k, l)\right| \leq e^{\omega-a}\left(\frac{e^{-\omega+a}}{\gamma}\right)^{k-l} e^{2 a|l|}, k \geq l,$$which implies that the equation (2.9) admits a nonuniform exponential dichotomy with invariant projector P = Id, by settingK=eω−a,  α=e−ω+aγ<1,  ε=e2a>0.$$K=e^{\omega-a},~~\alpha=\frac{e^{-\omega+a}}{\gamma}<1,~~\varepsilon=e^{2a}>0.$$Thus,(2.12)(e−ω+5a,+∞) ⊂ ρNED(A).$$\begin{equation}(e^{-\omega+5a},+\infty)\subset \rho_{NED}(A).\end{equation}$$For any γ˜∈(0,e−ω−5a)$ \widetilde{\gamma}\in (0, e^{-\omega-5a}) $, it follows from (2.10) that(2.13)Φγ(k,l)≤eω+ae−ω−aγk−le2a|l|,k≤l,$$\left|\Phi_{\gamma}(k, l)\right| \leq e^{\omega+a}\left(\frac{e^{-\omega-a}}{\gamma}\right)^{k-l} e^{2 a|l|}, k \leq l,$$which implies that (2.9) admits a nonuniform exponential dichotomy with P = 0, by takingK=eω+a,  α=γe−ω−a<1,  ε=e2a>0.$$K=e^{\omega+a},~~\alpha=\frac{\gamma}{e^{-\omega-a}}<1,~~\varepsilon=e^{2a}>0.$$Thus,(2.14)(0,e−ω−5a) ⊂ ρNED(A).$$\begin{equation}(0, e^{-\omega-5a})\subset \rho_{NED}(A).\end{equation}$$It follows from (2.12) and (2.14) that(0,e−ω−5a)∪(e−ω+5a,+∞) ⊂ ρNED(A),$$(0, e^{-\omega-5a}) \cup (e^{-\omega+5a},+\infty)\subset\rho_{NED}(A),$$which implies thatΣNED(A) ⊂ [e−ω−5a,e−ω+5a].$$\Sigma_{NED}(A) \subset[e^{-\omega-5a},e^{-\omega+5a}].$$Next we show that[e−ω−5a,e−ω+5a] ⊂ ΣNED(A).$$[e^{-\omega-5a},e^{-\omega+5a}]\subset\Sigma_{NED}(A).$$To do this, we first prove that γ1 = e − ω+5a ∈ ΣNED(A). The evolution operator of the systemuk+1=1γ1Akuk$$u_{k+1}=\tfrac{1}{\gamma_{1}}A_{k}u_{k}$$is given asΦ γ 1 ( k , l ) = e ω − a e − a ( k − l − 1 ) ( 1 + ( − 1 ) k − 1 ) − a l ( − 1 ) ( k − 1 ) + a l ( − 1 ) l . $$\Phi_{\gamma_{1}}(k,l)=e^{\omega-a}e^{-a(k-l-1)(1+(-1)^{k-1})-al(-1)^{(k-1)}+al(-1)^{l}}.$$It is easy to see that there do not exist K, α>0 and ε>0 such that∥Φγ1(k,l)∥≤Kαk−lε|l|,    fork≥l,$$\|\Phi_{\gamma_{1}}(k,l)\| \leqK\alpha^{k-l}\varepsilon^{|l|}, \ \ \ \ \mbox{for} \,\,\, k\geq l,$$or∥Φγ1(k,l)∥≤K(1α)k−lε|l|,  fork≤l.$$\|\Phi_{\gamma_{1}}(k,l)\|\leqK(\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|},\ \ \mbox{for} \,\,\,k\leq l.$$Therefore γ1 = e − ω+5a ∈ ΣNED(A). In a similar manner, we can prove γ2 = e − ω − 5a ∈ ΣNED(A). We can see from Theorem 2.6 that (2.8) has at most one nonuniform dichotomy spectral interval, which means that [e − ω − 5a, e − ω+5a] ⊂ ΣNED(A) and therefore [e − ω − 5a, e − ω+5a] = ΣNED(A).On the other hand, using a similar argument as in equations (1.6), we know that the nonuniform part ε∣l∣ cannot be removed in the estimates (2.11) and (2.13). Therefore, (2.8) does not admit an exponential dichotomy, which means that ΣED(A) = ℝ+.3ReducibilityIn this section we utilize Theorem 2.6 to show a reducibility result. For the reducibility results in the setting of an exponential dichotomy, we refer the reader to [15, 24, 32] and the references therein. First we recall the definition of kinematic similarity given in Coppel [16] and Aulbach et al. [2].Definition 3.1Equation (1.1) is said to be kinematically similar to another equation(3.1)yk+1=Bkyk$$y_{k+1}=B_{k}y_{k}$$with k ∈ ℤ, if there exists an invertible matrix Sk with ∣Sk∣ ≤ M and ∥Sk−1∥≤M$ \|S_{k}^{-1}\|\leq M $(M > 0), which satisfies the difference equationSk+1Bk=AkSk.$$S_{k+1}B_{k}=A_{k}S_{k}.$$The change of variables xk = Skyk then transforms (1.1) into (3.1).The next lemma is important to establish the reducibility results and its proof follows along the lines of the proof of Siegmund [32]. See also Coppel [16] and Aulbach et al. [2]Lemma 3.2[16, Chapter 5] Let P be an orthogonal projection (PT = P) and let X be an invertible matrix function. Then there exists an invertible matrix function S:Z→RN×N$ S:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N} $such thatSkPSk−1=XkPXk−1,SkQSk−1=XkQXk−1,$$S_{k}PS^{-1}_{k}=X_{k}PX^{-1}_{k},\,\,\,\,\,\,\, \,\,\,\,\,\,\,S_{k}QS^{-1}_{k}=X_{k}QX^{-1}_{k},$$and∥Sk∥≤2,$$\begin{equation*}\|S_{k}\|{\leq}\sqrt{2},\end{equation*}$$∥Sk−1∥≤[∥XkPXk−1∥2+∥Xk(I−P)Xk−1∥2]12,$$\begin{equation*}\|S^{-1}_{k}\|{\leq}\big[\|X_{k}PX^{-1}_{k}\|^{2}+\|X_{k}(I-P)X^{-1}_{k}\|^{2}\big]^{\frac{1}{2}},\end{equation*}$$where k ∈ ℤ and Q = Id − P. DefineR˜:Z→RN×N,k↦PXkTXkP+[Id−P]XkTXk[Id−P].$$\widetilde{R} :\mathbb{Z}\rightarrow \mathbb{R}^{N \timesN},\;k \mapsto PX_{k}^T X_{k} P+ [{\rm Id} - P] X_{k}^T X_{k} [{\rm Id} -P].$$Then the mapping is a positive definite, symmetric matrix for every k∈Z$ k \in \mathbb{Z} $. Moreover, there is a unique functionR:Z→RN×N$$R : \mathbb{Z}\rightarrow \mathbb{R}^{N \times N}$$of positive definite symmetric matrices Rk, k∈Z$ k \in \mathbb{Z} $, withRk2=R˜k,PRk=RkP.$$R_{k}^2 = \widetilde{R}_{k},\quadPR_{k} = R_{k} P\;.$$We remark that in the setting of an exponential dichotomy the function k↦Sk−1$ k \mapsto S^{-1}_{k} $in Lemma 3.2 is bounded. However, in the setting of a nonuniform exponential dichotomy, Sk−1$ S^{-1}_{k} $can be unbounded, because ∣Φ(k, k)Pk∣ ≤ Kεk for k = 0. In fact, we can see that(3.2)∥Sk−1∥≤2Kε|k|.$$\|S_k^{-1}\|\leq \sqrt{2}K\varepsilon^{|k|}.$$To overcome the difficulty, we introduce a new version of non-degeneracy, so-called weak non-degeneracy and define the concept of weak kinematical similarity, which is a very natural notion if we note the fact (3.2).Definition 3.3S:Z→RN×N$S:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N}$is called weakly non-degenerate if there exists an M = M(ε)>0 such that∥Sk∥≤Mε|k|and∥Sk−1∥≤Mε|k|,forallk∈Z,$$\|S_{k}\|\leq M\varepsilon^{|k|}\,\,\,\,\,{and}\,\,\,\, \|S^{-1}_{k}\| \leq M\varepsilon^{|k|},\,\,\,\,\,{for\, all}\,\,\,\, k\in \mathbb{Z},$$where ε is the same constant in (1.4)-(1.5).Definition 3.4If there exists a weakly non-degenerate matrix Sk such thatSk+1Bk=AkSk,$$S_{k+1}B_{k}=A_{k}S_{k},$$then equation (1.1) is weakly kinematically similar to equation (3.1). For short, we denote (1.1) ∼w$ \overset{w}{\sim} $(3.1) or Ak∼wBk$ A_{k} \overset{w}{\sim} B_{k} $.We analogously denote kinematical similarity by (1.1) ∼ (3.1) or Ak ∼ Bk.Definition 3.5Equation (1.1) is called reducible, if it is weakly kinematically similar to equation (3.1) whose coefficient matrix Bk has the block diagonal form(3.3)Bk100Bk2,$$\begin{equation}\left(\begin{array}{lll}B^1_{k} & 0\\0 & B^2_{k}\end{array}\right),\end{equation}$$where Bk1$ B^1_{k} $and Bk2$ B^2_{k} $are matrices of smaller size than Bk.Before stating the main results of this section, we prove several preliminary lemmas.Lemma 3.6Assume that (1.1) admits a nonuniform exponential dichotomy. Then the projector of equation (1.1) can be chosen as P˜=IN1000N2$ \tilde{P}= \left(\begin{array}{lll} I_{N_1} & 0 \\ 0 & 0_{N_2} \end{array} \right) $with N1=dimimP˜$ N_1 = {\rm dim\, im} \tilde{P} $and N2=dimkerP˜$ N_2 = {\rm dim\, ker} \tilde{P} $, and the fundamental matrix Xk can be chosen suitably such that the estimates (1.4)-(1.5) can be rewritten as(3.4)∥XkP˜Xl−1∥≤Kαk−lε|l|,k≥l,$$\|X_{k}\tilde{P}X^{-1}_{l}\|\leqK\alpha^{k-l}\varepsilon^{|l|},\quad k\geq l,$$and(3.5)∥XkQ˜Xl−1∥≤K(1α)k−lε|l|,k≤l,$$\|X_{k}\tilde{Q}X^{-1}_{l}\|\leqK(\frac{1}{\alpha})^{k-l}\varepsilon^{|l|},\quad k\leq l,$$where Q˜=Id−P˜$ \tilde{Q}={\rm Id}-\tilde{P} $.Proof. Let n∈Z$ n\in \mathbb{Z} $be arbitrary but fixed. Note that the rank of the projector Pn is independent of n∈Z$ n\in \mathbb{Z} $(see [9, page 1100]), then there exists a nondegenerate matrix T ∈ ℝN × N such thatP˜:=IN1000N2=TPnT−1$$\tilde{P}:=\left(\begin{array}{lll} I_{N_1} & 0\\0 & 0_{N_2}\end{array}\right)=TP_{n}T^{-1}$$with N1=dimimP˜$ N_1 = {\rm dim\, im} \tilde{P} $and N2=dimkerP˜$ N_2 = {\rm dim\, ker} \tilde{P} $. DefineXk:=Φ(k,n)T−1fork∈ZandP˜:=IN1000N2=TPnT−1.$$X_{k} := \Phi(k,n)T^{-1} \quad \text{for} k \in \mathbb{Z}\quad \text{and} \quad\tilde{P}:= \left(\begin{array}{lll}I_{N_1} & 0\\0 & 0_{N_2}\end{array}\right)= T P_{n} T^{-1}.$$Then(3.6)∥XkP˜Xl−1∥=∥Φ(k,n)T−1P˜TΦ−1(l,n)∥=∥Φ(k,n)PnΦ−1(l,n)∥.$$\begin{equation}\|X_{k}\tilde{P}X^{-1}_{l}\|=\|\Phi(k,n)T^{-1} \tilde{P}T\Phi^{-1}(l,n)\|=\|\Phi(k,n)P_{n}\Phi^{-1}(l,n)\|.\end{equation}$$On the other hand, we have(3.7)∥ Φ ( k , l ) P l ∥ = ∥ Φ ( k , n ) Φ ( n , l ) P l ∥ = ∥ Φ ( k , n ) P n Φ ( n , l ) ∥ = ∥ Φ ( k , n ) P n Φ − 1 ( l , n ) ∥ . $$\|\Phi(k,l)P_{l}\| = \|\Phi(k,n)\Phi(n,l)P_{l}\|\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \|\Phi(k,n)P_{n}\Phi(n,l)\|\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \|\Phi(k,n)P_{n}\Phi^{-1}(l,n)\|.$$It follows from (3.6) and (3.7) that (1.4)-(1.5) can be rewritten in the form (3.4)-(3.5).Lemma 3.7Assume that (1.1) admits a nonuniform exponential dichotomy with the form of estimates (3.4)-(3.5) and rank(P˜)=N1,(0<N1<N)$ rank(\tilde{P})=N_{1}, (0 < N_{1}< N) $, and there exists a weakly non-degenerate transformation Sk such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $. Then system (3.1) also admits a nonuniform exponential dichotomy, and the projector has the same rank.Proof. Suppose that Sk is weakly non-degenerate, which means that there exists M = M(ε) > 0 such that ∣Sk∣ ≤ Mε∣k∣ and ∥Sk−1∥≤Mε|k|$ \|S^{-1}_{k}\| \leq M\varepsilon^{|k|} $and such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $. Let Xk = SkYk. It is easy to see that Yk is the fundamental matrix of system (3.1). To prove that system (3.1) admits a nonuniform exponential dichotomy, we first consider the case k ≥ l and obtain(3.8)∥YkP˜Yl−1∥=∥Sk−1XkP˜Xl−1Sl∥≤∥Sk−1∥⋅∥XkP˜Xl−1∥⋅∥Sl∥≤KM2ε|k|αk−lε2|l|=KM2(αε2)k−lε2l−2k+|k|+2|l|≤KM2(αε2)k−lεl−k+3|l|≤KM2(αε2)k−lε3|l|,k≥l.$$\begin{equation}\begin{array}{lll}\|Y_{k}\tilde{P}Y^{-1}_{l}\|&=&\|S^{-1}_{k}X_{k}\tilde{P}X^{-1}_{l}S_{l}\|\\&\leq&\|S^{-1}_{k}\|\cdot\|X_{k}\tilde{P}X^{-1}_{l}\|\cdot\|S_{l}\|\\&\leq&K M^{2}\varepsilon^{|k|} \alpha^{k-l}\varepsilon^{2|l|}\\&=& KM^2(\alpha\varepsilon^2)^{k-l}\varepsilon^{2l-2k+|k|+2|l|}\\&\leq&K M^{2}(\alpha\varepsilon^2)^{k-l}\varepsilon^{l-k+3|l|}\\&\leq & K M^{2}(\alpha\varepsilon^2)^{k-l}\varepsilon^{3|l|},\,\,k\geq l.\end{array}\end{equation}$$A similar argument shows that(3.9)∥YkQ˜Yl−1∥≤KM2(1αε2)k−lε3|l|,k≤l.$$\begin{equation}\|Y_{k}\tilde{Q}Y^{-1}_{l}\|\leq KM^{2}(\tfrac{1}{\alpha\varepsilon^2})^{k-l}\varepsilon^{3|l|},\,\,k\leq l.\end{equation}$$Form (3.8) and (3.9), it is easy to see that system (3.1) admits a nonuniform exponential dichotomy. Clearly, the rank of the projector is N1.Lemma 3.8Assume that the systems (1.1) and (3.1) are weakly kinematically similar via Sk. If for a constant γ ∈ ℝ+ the system (2.1) admits a nonuniform exponential dichotomy with constants K > 0, 0 < α < 1, ε ≥ 1 and invariant projector P, then the system(3.10)yk+1=1γBkyk$$y_{k+1}=\tfrac{1}{\gamma}B_{k}y_{k}$$also admits a nonuniform exponential dichotomy.Proof. Obviously, P is also an invariant projector for (1.1). The dichotomy estimates are equivalent to∥XkPXl−1∥≤Kαk−lε|l|,k≥l,$$\|X_{k}P X^{-1}_{l}\| \leq K\alpha^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and∥XkPXl−1∥≤K(1α)k−lε|l|,k≤l.$$\|X_{k}P X^{-1}_{l}\|\leqK(\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|},\quad k\leq l.$$Using Lemma 3.7, it is easy to see that∥YkPYl−1∥≤Kγ′(εα)k−lε|l|,k≥l,$$\|Y_{k}P Y^{-1}_{l}\| \leqK'_\gamma(\varepsilon\alpha)^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and∥YkPYl−1∥≤Kγ′(1εα)k−lε|l|,k≤l,$$\|Y_{k}P Y^{-1}_{l}\|\leqK'_\gamma(\tfrac{1}{\varepsilon\alpha})^{k-l}\varepsilon^{|l|},\quad k\leql,$$for some constant K′γ ≥ 1. Therefore, (3.10) admits a nonuniform exponential dichotomy.The following result follows directly from Lemma 3.8.Corollary 3.9Assume that there exists a weakly non-degenerate transformation Sk such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $. Then ΣNED(A) = ΣNED(B), i.e.,ΣNED(A)=a1,b1 or 0,b1∪a2,b2∪⋯∪an−1,bn−1∪an,bn or an,∞=ΣNED(B).$$\begin{equation}\Sigma_{N E D}(A)=\left\{\begin{array}{c}{\left[a_{1}, b_{1}\right]} \\ \text { or } \\ \left(0, b_{1}\right]\end{array}\right\} \cup\left[a_{2}, b_{2}\right] \cup \cdots \cup\left[a_{n-1}, b_{n-1}\right] \cup\left\{\begin{array}{c}{\left[\begin{array}{c}\left.a_{n}, b_{n}\right] \\ \text { or } \\ {\left[a_{n}, \infty\right)}\end{array}\right\}}\end{array}\right\}=\Sigma_{N E D}(B). \end{equation}$$The following theorem states that if (1.1) admits a nonuniform exponential dichotomy, then there exists a weakly non-degenerate transformation such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $and Bk has the block form (3.3), i.e., system (1.1) is reducible.Theorem 3.10Assume that (1.1) admits a nonuniform exponential dichotomy (not necessary ) of the form (3.4)-(3.5) with the invariant projector satisfying Pk≠0, Id. Then (1.1) is reducible. More precisely, it is weakly kinematically similar to a decoupled system(3.11)xk+1=Bk100Bk2xk$$\begin{equation}x_{k+1} = \begin{pmatrix} B^1_{k} & 0 \\ 0 & B^2_{k} \end{pmatrix} x_{k} \end{equation}$$for some matrix functionsB1:Z→RN1×N1andB2:Z→RN2×N2$$B^1 : \mathbb{Z} \rightarrow \mathbb{R}^{N_1 \times N_1}\quad \text{and} \quad B^2 : \mathbb{Z} \rightarrow \mathbb{R}^{N_2\times N_2}$$where N 1 := dim ⁡ im P ~ $ N_1 := \dim \text{im}\, \tilde{P} $and N2:=dimkerP˜$ N_2 := \dim \ker \tilde{P} $.Proof. Since equation (1.1) admits a nonuniform exponential dichotomy of the form (1.4)-(1.5) with the invariant projector satisfying Pk≠0, Id, by Lemma 3.6, one can choose a suitable fundamental matrix Xk and the projector P˜=IN1000$ \tilde{P}= \left(\begin{array}{lll} I_{N_1} & 0 \\ 0 & 0 \end{array}\right) $, (0 < N1<N) such that the estimates (3.4)-(3.5) hold. By Lemma 3.2 and the estimates (3.4)-(3.5), there exists an M = M(ε)>0 large enough such thatSk≤2≤Mε|k|,Sk−1≤XkP˜Xk−12+Xk(Id−P˜)Xk−1212≤2Kε|k|.$$\begin{equation}\begin{array}{c}\left\|S_{k}\right\| \leq \sqrt{2} \leq M \varepsilon^{|k|}, \\ \left\|S_{k}^{-1}\right\| \leq\left[\left\|X_{k} \tilde{P} X_{k}^{-1}\right\|^{2}+\left\|X_{k}(\mathrm{Id}-\tilde{P}) X_{k}^{-1}\right\|^{2}\right]^{\frac{1}{2}} \leq \sqrt{2} K \varepsilon^{|k|}.\end{array} \end{equation}$$Thus, S is weakly non-degenerate. SettingBk=Rk+1Rk−1,$$B_{k}=R_{k+1}R^{-1}_{k},$$where Rk is defined in Lemma 3.2 and Xk = SkRk. Obviously, Rk is the fundamental matrix of the linear systemyk+1=Bkyk.$$\begin{equation*}y_{k+1}=B_{k}y_{k}.\end{equation*}$$Now we need to show that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $and Bk has the block diagonal formBk=Bk100Bk2,fork∈Z.$$\begin{equation*}B_{k}= \left(\begin{array}{lll} B^1_{k} & 0\\0 & B^{2}_{k}\end{array}\right),\,\,\,\,\mbox{for} \,\,\,\, k\in \mathbb{Z}.\end{equation*}$$First, we show that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $. In fact,Sk+1Bk=Xk+1Rk+1−1Bk=AkXkRk−1Bk−1Bk=AkSk,$$\begin{equation}\begin{aligned} S_{k+1} B_{k} &=X_{k+1} R_{k+1}^{-1} B_{k} \\ &=A_{k} X_{k} R_{k}^{-1} B_{k}^{-1} B_{k} \\ &=A_{k} S_{k}, \end{aligned} \end{equation}$$which implies that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $.Now we show that system (1.1) is weakly kinematically similar to (3.11). By Lemma 3.2, Rk+1 and Rk−1$ R_{k}^{-1} $commute with the matrix P˜$ \tilde{P} $for every k∈Z$ k \in \mathbb{Z} $. It follows that(3.12)P˜Bk=BkP˜$$\begin{equation}\tilde{P} B_{k}=B_{k} \tilde{P}\end{equation}$$for all k∈Z$ k \in \mathbb{Z} $. Now we decompose B k : Z → R N × N $B_{k}: \mathbb{Z} \rightarrow \mathbb{R}^{N \times N}$into four functionsBk1:Z→RN1×N1,Bk2:Z→RN2×N2,Bk3:Z→RN1×N2,Bk4:Z→RN2×N1,$$\begin{equation}\begin{array}{ll}B_{k}^{1}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{1} \times N_{1}}, & B_{k}^{2}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{2} \times N_{2}}, \\ B_{k}^{3}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{1} \times N_{2}}, & B_{k}^{4}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{2} \times N_{1}},\end{array} \end{equation}$$withBk=Bk1Bk3Bk4Bk2,k∈Z.$$B_{k}= \left(\begin{array}{lll} B^1_{k} & B^3_{k}\\B^4_{k} & B^{2}_{k}\end{array}\right), \quad \,\,\,\, k\in \mathbb{Z}.$$Identity (3.12) implies thatBk1Bk300=Bk10Bk40,k∈Z.$$\left(\begin{array}{lll} B^1_{k} & B^3_{k}\\0 & 0\end{array}\right)=\left(\begin{array}{lll} B^1_{k} & 0\\B^4_{k} & 0\end{array} \right), \quad \,\,\,\, k\in \mathbb{Z}.$$Therefore Bk3≡0$ B^3_{k}\equiv 0 $and Bk4≡0$ B^4_{k}\equiv 0 $. Thus Bk has the block formBk=Bk100Bk2,k∈Z.$$B_{k}= \left(\begin{array}{lll} B^1_{k} & 0\\0 & B^{2}_{k}\end{array} \right), \quad \,\,\,\, k\in \mathbb{Z}.$$Now the proof is finished.From Theorem 3.10, we know that if (1.1) admits a nonuniform exponential dichotomy, then there exists a weakly non-degenerate transformation Sk such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $and Bk has two blocks of the form (3.3). Now we are in a position to prove the reducibility result.Theorem 3.11Assume that (1.1) admits a nonuniform exponential dichotomy. Due to Theorem 2.6, the dichotomy spectrum is either empty or the disjoint union of n closed spectral intervals J1, ⋅, Jn with 1 ≤ N ≤ N, i.e.,ΣNED(A)=∅(n=0) or ΣNED(A)=J1∪⋯∪Jn.$$\begin{equation}\Sigma_{N E D}(A)=\emptyset \quad(n=0) \quad \text { or } \quad \Sigma_{N E D}(A)=\mathcal{J}_{1} \cup \cdots \cup \mathcal{J}_{n}. \end{equation}$$Then there exists a weakly kinematic similarity action S : Z → R N × N $S: \mathbb{Z} \rightarrow \mathbb{R}^{N \times N}$between (1.1) and a block diagonal systemxk+1=Bk0⋱Bkn+1xk$$\begin{equation}x_{k+1}=\left(\begin{array}{lll}B_{k}^{0} & & \\ & \ddots & \\ & & B_{k}^{n+1}\end{array}\right) x_{k} \end{equation}$$with B i : Z → R N i × N i $B^{i}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{i} \times N_{i}}$, Ni = dim𝒲i, andΣNEDB0=∅,ΣNEDB1=J1,…,ΣNEDBn=Jn,ΣNEDBn+1=∅.$$\begin{equation}\Sigma_{N E D}\left(B^{0}\right)=\emptyset, \Sigma_{N E D}\left(B^{1}\right)=\mathcal{J}_{1}, \ldots, \Sigma_{N E D}\left(B^{n}\right)=\mathcal{J}_{n}, \Sigma_{N E D}\left(B^{n+1}\right)=\emptyset. \end{equation}$$Proof. If for any γ ∈ ℝ+, system (2.1) admits a nonuniform exponential dichotomy, then ΣNED(A) = ∅. Conversely, for any γ ∈ ℝ+, system (2.1) does not admit a nonuniform exponential dichotomy, then ΣNED(A) = ℝ+. Now, we prove the theorem for the nontrivial case (ΣNED(A)≠∅ and ΣNED(A)≠ℝ+).Recall that the resolvent set ρNED(A) is open and therefore the dichotomy spectrum ΣNED(A) is the disjoint union of closed intervals. Using Theorem 2.6, we can assumeJ1=a1,b1 or 0,b1,J2=a2,b2,…,Jn−1=an−1,bn−1,Jn=an,bn or an,∞$$\begin{equation}\mathcal{J}_{1}=\left\{\begin{array}{c}{\left[a_{1}, b_{1}\right]} \\ \text { or } \\ \left(0, b_{1}\right]\end{array}\right\}, \mathcal{J}_{2}=\left[a_{2}, b_{2}\right], \ldots, \mathcal{J}_{n-1}=\left[a_{n-1}, b_{n-1}\right], \mathcal{J}_{n}=\left\{\begin{array}{c}{\left[\begin{array}{c}\left.a_{n}, b_{n}\right] \\ \text { or } \\ {\left[a_{n}, \infty\right)}\end{array}\right\}}\end{array}\right\} \end{equation}$$with 0 < a1 ≤ b1 < a2 ≤ b2 < … < an ≤ bn.If J1 = [a1, b1] is a spectral interval, then (0, γ0) ⊂ ρNED(A) and W 0 = S γ 0 $\mathcal{W}_{0}=\mathcal{S}_{\gamma_{0}}$for some γ0 < a1 due to Theorem 2.6, which implies thatxk+1=1γ0Akxk$$x_{k+1}=\tfrac{1}{\gamma_0}A_{k}x_{k}$$admits a nonuniform exponential dichotomy, let its invariant projector be denoted by P˜0$ \tilde{P}_0 $. By Theorem 3.10 and Corollary 3.9, there exists a weakly non-degenerate transformation xk=Sk0xk(0)$ x_k=S_k^{0}x_k^{(0)} $with ∥Sk0∥≤M0ε|k|$ \|S_{k}^{0}\|\leq M_0\varepsilon^{|k|} $and ∥(Sk0)−1∥≤M0ε|k|$ \|(S_{k}^{0})^{-1}\| \leq M_0\varepsilon^{|k|} $for some positive constant M0 = M0(ε) and such that Ak∼wAk0$ A_{k}\overset{w}{\sim} A^{0}_{k} $and Ak0$ A^{0}_{k} $has two blocks of the form Ak0=Bk000Bk0,∗$ A^{0}_{k}=\left(\begin{array}{lll} B^{0}_{k} & 0 \\ 0 & B^{0,*}_{k} \end{array}\right) $with dim ⁡ B k 0 = d i m i m ⁡ P ~ 0 = dim S γ 0 = dim W 0 =: N 0 $\operatorname{dim} B_{k}^{0}=\operatorname{dim\,im} \tilde{P}_{0}=\operatorname{dim} \mathcal{S}_{\gamma_{0}}=\operatorname{dim} \mathcal{W}_{0}=: N_{0}$due to Theorem 3.10, Lemma 2.5 and Theorem 2.6. If J1 = (0, b1] is a spectral interval, a block Bk0$ B_{k}^{0} $is omitted.Now we consider the following systemxk+1(0)=Ak0xk(0)=Bk000Bk0,∗xk(0).$$x_{k+1}^{(0)}=A^{0}_{k}x_{k}^{(0)}=\left(\begin{array}{lll}B^{0}_{k} & 0\\0 & B^{0,*}_{k}\end{array}\right)x_{k}^{(0)}.$$By using Lemma 2.5, we take γ1 ∈ (b1, a2). In view of (b1,a2) ⊂ ρNED(Bk0,∗)$ (b_{1},a_{2})\subset \rho_{NED}(B_k^{0,*}) $, γ1∈ρNED(Bk0,∗)$ \gamma_1\in \rho_{NED}(B_k^{0,*}) $, which implies thatxk+1(0)=1γ1Bk000Bk0,∗xk(0)$$x_{k+1}^{(0)}=\frac{1}{\gamma_1}\left(\begin{array}{lll}B^{0}_{k} & 0\\0 & B^{0,*}_{k}\end{array}\right)x_{k}^{(0)}$$admits a nonuniform exponential dichotomy. Its invariant projector P˜1$ \tilde{P}_1 $satisfies P˜1≠0,I$ \tilde{P}_1\neq 0,\,I $. Similarly, by Theorem 3.10 and Corollary 3.9, there exists a weakly non-degenerate transformationxk(0)=Sk1xk(1)=IN000S˜k1xk(1)$$x_k^{(0)}=S_k^{1}x_k^{(1)}=\left(\begin{array}{lll}I_{N_{0}} & 0\\0 & \tilde{S}^{1}_{k}\end{array}\right)x_k^{(1)}$$with ∥S˜k1∥≤M1ε|k|$ \|\tilde{S}^{1}_{k}\|\leq M_1\varepsilon^{|k|} $and ∥(S˜k1)−1∥≤M1ε|k|$ \|(\tilde{S}^{1}_{k})^{-1}\| \leq M_1\varepsilon^{|k|} $for some positive constant M1 = M1(ε) and such that Bk0,∗∼wB˜k0,∗$ B^{0,*}_{k}\overset{w}{\sim} \tilde{B}^{0,*}_{k} $and B˜k0,∗$ \tilde{B}^{0,*}_{k} $has two blocks of the form B˜k0,∗=Bk100Bk1,∗$ \tilde{B}^{0,*}_{k}=\left(\begin{array}{lll} B^{1}_{k} & 0 \\ 0 & B^{1,*}_{k} \end{array} \right) $with dim ⁡ B k 1 = dim ⁡ im P ~ 1 = dim S γ 1 ≥ dim ⁡ ( U γ 0 ∩ S γ 1 ) = dim W 1 =: N 1 $\dim B_{k}^{1}=\dim \text{im} \,\tilde{P}_1=\dim\mathcal{S}_{\gamma_1}\geq\dim(\mathcal{U}_{\gamma_0}\cap \mathcal{S}_{\gamma_1})=\dim\mathcal{W}_{1}=:N_{1}$due to Theorem 3.10, Lemma 2.5 and Theorem 2.6. In addition, using Theorem 3.10 and Corollary 3.9, we haveΣNEDBk1=a1,b1 or 0,b1,ΣNEDBk1,⋆=a2,b2∪⋯∪an−1,bn−1∪an,bn or an,∞.$$\begin{equation}\Sigma_{N E D}\left(B_{k}^{1}\right)=\left\{\begin{array}{c}{\left[a_{1}, b_{1}\right]} \\ \text { or } \\ \left(0, b_{1}\right]\end{array}\right\}, \quad \Sigma_{N E D}\left(B_{k}^{1, \star}\right)=\left[a_{2}, b_{2}\right] \cup \cdots \cup\left[a_{n-1}, b_{n-1}\right] \cup\left\{\begin{array}{c}{\left[a_{n}, b_{n}\right]} \\ \text { or } \\ {\left[a_{n}, \infty\right)}\end{array}\right\}. \end{equation}$$Now we can construct a weakly non-degenerate transformation xk=S˜kxk(1)$ x_k=\tilde{S}_kx_k^{(1)} $with S˜k=Sk0Sk1=Sk0IN000S˜k1$ \tilde{S}_{k}=S^{0}_{k}S^{1}_{k}=S^{0}_{k}\left(\begin{array}{lll} I_{N_{0}} & 0 \\ 0 & \tilde{S}^{1}_{k} \end{array}\right) $, where ∥S˜k∥≤M0M1ε2|k|$ \| \tilde{S}_{k}\|\leq M_0 M_1\varepsilon^{2|k|} $and ∥S˜k−1∥≤M0M1ε2|k|$ \| \tilde{S}_{k}^{-1}\| \leq M_0 M_1\varepsilon^{2|k|} $. Then Ak∼wAk1$ A_{k}\overset{w}{\sim} A^{1}_{k} $and Ak1$ A^{1}_{k} $has three blocks of the formAk1=Bk0Bk1Bk1,⋆.$$\begin{equation}A_{k}^{1}=\left(\begin{array}{ccc}B_{k}^{0} & & \\ & B_{k}^{1} & \\ & & B_{k}^{1, \star}\end{array}\right). \end{equation}$$Applying similar procedures to γ2 ∈ (b2, a3), γ3 ∈ (b3, a4), …, we can construct a weakly non-degenerate transformation xk=Skxk(n+1)$ x_k=S_kx_k^{(n+1)} $withSk=Sk0IN000S˜k1IN0+N100S˜k2⋯IN0+…+Nn−100S˜kn$$S_{k}=S^{0}_{k}\left(\begin{array}{lll}I_{N_{0}} & 0\\0 & \tilde{S}^{1}_{k}\end{array}\right)\left(\begin{array}{lll}I_{N_{0}+N_{1}} & 0\\0 & \tilde{S}^{2}_{k}\end{array}\right)\cdots \left(\begin{array}{lll}I_{N_{0}+\ldots+N_{n-1}} & 0\\0 & \tilde{S}^{n}_{k}\end{array}\right)$$such that ∣ Sk∣ ≤ Mε εn∣k∣ and ∥Sk−1∥≤Mεεn|k|$ \| S_{k}^{-1}\| \leq M_{\varepsilon} \varepsilon^{n|k|} $with Mε = M0  ×  ···  ×  Mn. Now we can proveAk∼wAkn:=Bk=Bk0⋱Bkn+1$$\begin{equation}A_{k} \stackrel{w}{\sim} A_{k}^{n}:=B_{k}=\left(\begin{array}{lll}B_{k}^{0} & & \\ & \ddots & \\ & & B_{k}^{n+1}\end{array}\right) \end{equation}$$with locally integrable functions B i : Z → R N i × N i $B^{i}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{i} \times N_{i}}$andΣNEDB0=∅,ΣNEDB1=J1,…,ΣNEDBn=Jn,ΣNEDBn+1=∅.$$\begin{equation}\Sigma_{N E D}\left(B^{0}\right)=\emptyset, \Sigma_{N E D}\left(B^{1}\right)=\mathcal{J}_{1}, \ldots, \Sigma_{N E D}\left(B^{n}\right)=\mathcal{J}_{n}, \Sigma_{N E D}\left(B^{n+1}\right)=\emptyset. \end{equation}$$Finally, we show that Ni = dim𝒲i. From the claim above, we note that dim ⁡ B k 0 = dim W 0 ,   dim ⁡ B k 1 ≥ dim W 1 , … , dim ⁡ B k n ≥ dim W n ,   dim ⁡ B k n + 1 = dim W n + 1 $ \dim B^{0}_k=\dim \mathcal{W}_0, ~\dim B^{1}_k\geq \dim \mathcal{W}_1,\ldots, \dim B^{n}_k\geq \dim \mathcal{W}_n, ~\dim B^{n+1}_k=\dim \mathcal{W}_{n+1} $and using Theorem 2.6 this yields dim𝒲0 +···+dim𝒲n+1 = N, so dim ⁡ B k i = dim W i $ \dim B^{i}_k=\dim \mathcal{W}_i $for i = 0, …, n + 1. Now the proof is finished. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Nonlinear Analysis de Gruyter

Nonuniform dichotomy spectrum and reducibility for nonautonomous difference equations

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de Gruyter
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© 2021 Jifeng Chu et al., published by De Gruyter
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2191-9496
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2191-950X
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10.1515/anona-2020-0198
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Abstract

1IntroductionLet Ak ∈ ℝN × N, k ∈ ℤ, be a sequence of invertible matrices. In this paper, we consider the following nonautonomous linear difference equations(1.1)xk+1=Akxk,$$x_{k+1}=A_{k}x_{k},$$where xk ∈ ℝN, k ∈ ℤ. Let Φ:ℤ  ×  ℤ → ℝN × N, (k, l)↦Φ(k, l), denote the evolution operator of (1.1), i.e.,Φ ( k , l ) = A k − 1 ⋯ A l , f o r k > l , I d , f o r k = l , A k − 1 ⋯ A l − 1 − 1 , f o r k < l . $$\Phi(k,l)=\begin{cases}A_{k-1}\cdots A_{l}, &{\rm for}\quad k>l,\\{\rm Id}, &{\rm for}\quad k=l,\\A^{-1}_{k}\cdots A^{-1}_{l-1}, &{\rm for}\quad k<l.\end{cases}$$Obviously, Φ(k, m)Φ(m, l) = Φ(k, l), k, m, l ∈ ℤ, and Φ(·, l)ξ solves the initial value problem (1.1), x(l) = ξ, for l ∈ ℤ, ξ ∈ ℝN.An invariant projector of (1.1) is defined to be a function P:ℤ → ℝN × N of projections Pk, k ∈ ℤ, such that for each Pk the following property holdsPk+1Ak=AkPk,k∈Z.$$P_{k+1}A_{k}=A_{k}P_{k},\quad k\in \mathbb{Z}.$$We say that (1.1) admits an exponential dichotomy if there exist an invariant projector P and constants 0 < α < 1, K ≥ 1 such that(1.2)∥Φ(k,l)Pl∥≤Kαk−l,k≥l,$$\|\Phi(k,l)P_{l}\| \leq K\alpha^{k-l}, \quad k\geq l,$$and(1.3)∥Φ(k,l)Ql∥≤K(1α)k−l,k≤l,$$\|\Phi(k,l)Q_l\|\leq K(\tfrac{1}{\alpha})^{k-l},\quadk\leq l,$$where Ql = Id − Pl is the complementary projection.The notion of exponential dichotomy was introduced by Perron in [25] and has attracted a lot of interest during the last few decades because it plays an important role in the study of hyperbolic dynamical behavior of differential equations and difference equations. For example, see [1, 19, 28] and the references therein. We also refer to the books [10, 16, 22] for details and further references related to exponential dichotomies. On the other hand, during the last decade, inspired both by the classical notion of exponential dichotomy and by the notion of nonuniformly hyperbolic trajectory introduced by Pesin (see [5]), Barreira and Valls have introduced the notion of nonuniform exponential dichotomies and have developed the corresponding theory in a systematic way (see [6] and the references therein). As explained by Barreira and Valls, in comparison to the notion of exponential dichotomies, nonuniform exponential dichotomy is a useful and weaker notion.We say that (1.1) admits a nonuniform exponential dichotomy if there exist an invariant projector P and constants 0 < α < 1, K = 1, ε ≥ 1 such that αε2 < 1 and(1.4)∥Φ(k,l)Pl∥≤Kαk−lε|l|,k≥l,$$\|\Phi(k,l)P_{l}\| \leq K\alpha^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and(1.5)∥Φ(k,l)Ql∥≤K(1α)k−lε|l|,k≤l.$$\|\Phi(k,l)Q_l\|\leqK(\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|},\quad k\leq l.$$When ε = 1, (1.4)-(1.5) become (1.2)-(1.3), and therefore a nonuniform exponential dichotomy becomes an exponential dichotomy. Moreover, [5, Theorem 1.4.2]bp02 indicates that the condition αε2 < 1 is reasonable, which means that the nonuniform part is small.For example, given ω>5a > 0, then the linear equation(1.6)uk+1=e−ω+ak(−1)k−a(k−1)(−1)(k−1)uk,vk+1=eω−ak(−1)k+a(k−1)(−1)(k−1)vk$$u_{k+1}=e^{-\omega+ak(-1)^{k}-a(k-1)(-1)^{(k-1)}}u_{k},\,\,\,v_{k+1}=e^{\omega-ak(-1)^{k}+a(k-1)(-1)^{(k-1)}}v_{k}$$admits a nonuniform exponential dichotomy, but does not admit an exponential dichotomy. In fact, we haveΦ(k,l)Pl=e−ω(k−l−1)−a(k−l−1)(−1)k−1−al(−1)(k−1)+al(−1)l000$$\Phi(k,l)P_{l}=\left(\begin{array}{lll} e^{-\omega(k-l-1)-a(k-l-1)(-1)^{k-1}-al(-1)^{(k-1)}+al(-1)^{l}} & 0 \\ 0 & 0 \end{array}\right)$$with Pl=1000$ P_{l}=\left(\begin{array}{lll} 1 & 0 \\ 0 & 0 \end{array} \right) $. Therefore (1.4) holds withK=eω−a>1,α=e(−ω+a)∈(0,1),ε=e2a>1.$$K=e^{\omega-a}>1,\quad \alpha=e^{(-\omega+a)}\in(0,1),\quad \varepsilon=e^{2a}>1.$$Analogous arguments applied to the second equation yield the estimate (1.5). Moreover, when both k and l are even, we obtain the equality∥Φ(k,l)Pl∥=Kαk−lε|l|,k≥l,$$\|\Phi(k,l)P_{l}\| = K\alpha^{k-l}\varepsilon^{|l|}, \quad k\geql,$$which means that the nonuniform part εl = e2al cannot be removed.Among the different topics on classical exponential dichotomies, the dichotomy spectrum (also called dynamical spectrum, or Sacker-Sell spectrum) is very important and many results have been obtained. As far as we know, dynamical spectrum defined with exponential dichotomies was first introduced by Sacker and Sell in [29] in which they studied the linear skew product flows with compact base. For more results on dichotomy spectrum, we refer the reader to [2, 3, 13, 18, 23, 26, 27, 29, 31, 32] and the references therein. The definition and investigation for finite-time hyperbolicity have also been studied in [9, 17]. The dynamical spectral theorem has some important applications. For example, based on the dichotomy spectral theorem, normal forms for nonautonomous systems were established in [20,21,33]. However, all the results mentioned above were established in the setting of classical exponential dichotomies. In this paper, we establish the corresponding spectral theory for difference equations (1.1) with a nonuniform exponential dichotomy. To the best of our knowledge the nonuniform dichotomy spectral theory for linear differential equations was first studied in [14] and [35]. We refer the reader to [7, 8, 11, 12, 34] for further results on nonuniform dichotomy spectrum.This paper is organized as follows. In Section 2 we propose a definition of spectrum based on nonuniform exponential dichotomies, which is called nonuniform dichotomy spectrum. Such a spectrum can be seen as a generalization of Sacker-Sell spectrum. We prove a nonuniform dichotomy spectral theorem. In Section 3, as an application of the spectral theorem, we prove a reducibility result for (1.1). Recall that system (1.1) is called reducible if it is kinematically similar to a block diagonal system with blocks of dimension less than N.2Nonuniform dichotomy spectrumConsider the weighted system(2.1)xk+1=1γAkxk,$$x_{k+1} = \tfrac{1}{\gamma}A_{k}x_{k},$$where γ ∈ ℝ+ = (0, ∞). One can easily see thatΦγ(k,l):=(1γ)k−lΦ(k,l)$$\Phi_\gamma(k,l) := (\tfrac{1}{\gamma})^{k-l}\Phi(k,l)$$is its evolution operator. If for some γ ∈ ℝ+, (2.1) admits a nonuniform exponential dichotomy with projector Pk and constants K ≥ 1, 0 < α < 1 and ε ≥ 1, then Pk is also invariant for (1.1), that isPk+1Ak=AkPk,k∈Z,$$P_{k+1}A_{k}=A_{k}P_{k},\quad k\in \mathbb{Z},$$and the dichotomy estimates of (2.1) are equivalent to(2.2)Φ(k,l)Pl∥≤K(γα)k−lε|l|,k≥l,$$\Phi(k,l)P_{l}\| \leq K(\gamma\alpha)^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and(2.3)∥Φ(k,l)Ql∥≤K(γ1α)k−lε|l|,k≤l.$$\|\Phi(k,l)Q_l\|\leq K(\gamma\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|},\quad k\leq l.$$Definition 2.1The nonuniform dichotomy spectrum of (1.1) is the setΣNED(A)={γ∈R+:2.1admitsnononuniformexponentialdichotomy},$$\Sigma_{NED}(A) = \{\gamma \in\mathbb{R}^+:\, {2.1} \,admits\, no\, nonuniform\, exponential \,dichotomy \},$$and the resolvent set ρNED(A) = ℝ+∖ΣNED(A) is its complement. The dichotomy spectrum of (1.1) is the setΣED(A)={γ∈R+:2.1admitsnoexponentialdichotomy},$$\Sigma_{ED}(A) = \{ \gamma \in\mathbb{R}^+:\, {2.1} \,{ admits\, no\, exponential\, dichotomy} \},$$and ρED(A) = ℝ+∖ΣED(A).Proposition 1ΣNED(A) ⊂ ΣED(A).Proof. For each γ ∈ ρED(A), the weighted system (2.1) admits an exponential dichotomy. Consequently, the weighted system (2.1) admits a nonuniform exponential dichotomy. Thus, γ ∈ ρNED(A), which implies that ρED(A) ⊂ ρNED(A), and therefore ΣNED(A) ⊂ ΣED(A). □Let us define for γ ∈ ρNED(A)Sγ:=(l,ξ)∈Z×RN:supk≥l∥Φ(k,l)ξ∥γ−kε−|l|<∞,$$\mathcal{S}_\gamma : \,= \,\left\{(l,\xi) \in \mathbb{Z} \times \mathbb{R}^N:\left(\sup\limits_{k\geq l}\|\Phi(k,l)\xi\|\gamma^{-k}\right)\varepsilon^{-|l|}<\infty\right\},$$andUγ:=(l,ξ)∈Z×RN:supk≤l∥Φ(k,l)ξ∥γ−kε−|l|<∞,$$\mathcal{U}_\gamma: \,= \,\left\{(l,\xi) \in \mathbb{Z} \times \mathbb{R}^N:\left(\sup\limits_{k\leq l}\|\Phi(k,l)\xi\|\gamma^{-k}\right)\varepsilon^{-|l|}<\infty\right\},$$where ε is the constant in (2.2)-(2.3). Note that 𝒮γ and 𝒰γ depend on ε=ε(γ)$ \varepsilon=\varepsilon(\gamma) $for each γ ∈ ρNED(A). If in addition γ ∈ ρED(A) then it is shown below that these sets do not depend on ε, more precisely, εcan be set to equal 1. One may readily verify that 𝒮γ and 𝒰γ are invariant vector bundles of (1.1), here we say that a nonempty set W ⊂ Z×RN$ \mathcal{W} \subset \mathbb{Z} \times \mathbb{R}^N $is an invariant vector bundle of (1.1) if (a) it is invariant, i.e., (l,ξ)∈W⇒(k,Φ(k,l)ξ)∈W$ (l,\xi) \in \mathcal{W} \;\Rightarrow\; (k,\Phi(k,l)\xi) \in \mathcal{W} $for all k ∈ ℤ; and (b) for every l ∈ ℤ the fiber W(l)={ξ∈RN:(l,ξ)∈W}$ \mathcal{W}(l) = \{\xi \in \mathbb{R}^N \,:\, (l,\xi) \in \mathcal{W} \} $is a linear subspace of ℝN.The next lemma gives the relationship between 𝒮γ, 𝒰γ and the projector P. In [16, Chapter 2]cop it is proved in the setting of exponential dichotomies that the invariant projector is unique. The proof for the invariant projectors for (1.1) and (2.1) in the setting of nonuniform exponential dichotomies is almost identical.Lemma 2.2Assume that (2.1) admits a nonuniform exponential dichotomy with invariant projector P for γ ∈ ℝ+. ThenSγ=imP,Uγ=kerP and Sγ⊕Uγ=Z×RN.$$\mathcal{S}_{\gamma}=\operatorname{im} P, \quad \mathcal{U}_{\gamma}=\operatorname{ker} P \quad \text { and } \quad \mathcal{S}_{\gamma} \oplus \mathcal{U}_{\gamma}=\mathbb{Z} \times \mathbb{R}^{N}.$$Proof. We show only Sγ=imP$\mathcal{S}_\gamma = \text{im}\, P$. The fact 𝒰γ = ker P is analog and the fact Sγ⊕Uγ=Z×RN$\mathcal{S}_{\gamma} \oplus \mathcal{U}_{\gamma}=\mathbb{Z} \times \mathbb{R}^{N}$is clear.First we show Sγ ⊂ imP$\mathcal{S}_{\gamma} \subset \operatorname{im} P$. Let l ∈ ℤ and ξ ∈ 𝒮γ(l). Then there exists a positive constant C such that∥Φ(k,l)ξ∥≤Cγkε|l|,k≥l.$$\|\Phi(k,l)\xi\|\leq C\gamma^{k}\varepsilon^{|l|},\quad k\geq l.$$We write ξ = ξ1+ξ2 with ξ1 ∈ im Pl and ξ2 ∈ ker Pl. We show that ξ2 = 0. By invariance of P we have for k ∈ ℤ the identityξ2=Φγ(l,k)Φγ(k,l)Qlξ=Φγ(l,k)QkΦγ(k,l)ξ.$$\xi_2=\Phi_\gamma(l,k)\Phi_\gamma(k,l)Q_l\xi=\Phi_\gamma(l,k)Q_k\Phi_\gamma(k,l)\xi.$$Using the fact that (2.1) admits a nonuniform exponential dichotomy, it follows that∥Φγ(l,k)Qk∥≤K(1α)l−kε|k|.$$\|\Phi_\gamma(l,k)Q_k\|\leq K(\tfrac{1}{\alpha})^{l-k}\varepsilon^{|k|}.$$Thusξ2≤K1αl−kε|k|Φγ(k,l)ξ=K(αε)k−lεl−k+|k|1γk−l∥Φ(k,l)ξ∥≤CK(αε)k−lεl+|l|−k+|k|1γk−lγk,k≥l,$$\begin{aligned}\left\|\xi_{2}\right\| & \leq K\left(\frac{1}{\alpha}\right)^{l-k} \varepsilon^{|k|}\left\|\Phi_{\gamma}(k, l) \xi\right\| \\ &=K(\alpha \varepsilon)^{k-l} \varepsilon^{l-k+|k|}\left(\frac{1}{\gamma}\right)^{k-l}\|\Phi(k, l) \xi\| \\ & \leq C K(\alpha \varepsilon)^{k-l} \varepsilon^{l+|l|-k+|k|}\left(\frac{1}{\gamma}\right)^{k-l} \gamma^{k}, \quad k \geq l, \end{aligned}$$which implies that when k ≥ l and k > 0, we have∥ξ2∥≤CK(αε)k−lε2|l|γl,$$\|\xi_2\|\leq CK(\alpha\varepsilon)^{k-l}\varepsilon^{2|l|}\gamma^l,$$and therefore ξ2 = 0 by letting k → ∞, since αε<1.Next we show imP ⊂ Sγ$ \operatorname{im}P \subset \mathcal{S}_{\gamma} $. Let l ∈ ℤ and ξ ∈ im Pl, i.e., Plξ = ξ. Using the fact that α<1, the nonuniform exponential dichotomy estimate implies∥Φγ(k,l)ξ∥≤Kαk−lε|l|∥ξ∥≤Kε|l|∥ξ∥,k≥l,$$\|\Phi_\gamma(k,l)\xi\|\leq K\alpha^{k-l}\varepsilon^{|l|}\|\xi\|\leq K\varepsilon^{|l|}\|\xi\|,\quad k\geq l,$$which yields∥Φ(k,l)ξ∥≤Kγk−lε|l|∥ξ∥,$$\|\Phi(k,l)\xi\|\leq K\gamma^{k-l}\varepsilon^{|l|}\|\xi\|,$$and thus ξ ∈ 𝒮γ(l).Lemma 2.3The resolvent set is open, i.e., for every γ ∈ ρNED(A), there exists a β = β(γ) ∈ (0, 1) with (βγ,1βγ) ⊂ ρNED(A)$ (\beta\gamma, \frac{1}{\beta}\gamma) \subset \rho_{NED}(A) $. Furthermore,Sζ=Sγanduζ=uγforζ∈βγ,1βγ.$$\mathcal{S}_{\zeta}=\mathcal{S}_{\gamma} \quad \, { and }\, \quad u_{\zeta}=u_{\gamma} \, { for }\, \zeta \in\left(\beta \gamma, \frac{1}{\beta} \gamma\right).$$Proof. Let γ ∈ ρNED(A). Then (2.1) admits a nonuniform exponential dichotomy, i.e., the estimates (2.2)-(2.3) hold with an invariant projector P and constants K ≥ 0, 0<α <1 and ε ≥ 1. For β:=α∈(0,1)$ \beta:= \sqrt{\alpha}\in(0,1) $and ζ∈(βγ,1βγ)$ \zeta \in (\beta\gamma, \frac{1}{\beta}\gamma) $we haveΦζ(k,l)=(γζ)k−lΦγ(k,l).$$\Phi_\zeta(k,l) =(\tfrac{\gamma}{\zeta})^{k-l}\Phi_\gamma(k,l).$$Note that P is also an invariant projector for the equationxk+1=1ζAkxk.$$x_{k+1} = \tfrac{1}{\zeta}A_{k}x_{k}.$$Moreover, we have the estimates∥Φζ(k,l)Pl∥≤K(γζα)k−lε|l|≤Kβk−lε|l|,k≥l,$$\|\Phi_\zeta(k,l)P_{l}\| \leqK(\tfrac{\gamma}{\zeta}\alpha)^{k-l}\varepsilon^{|l|} \leq K\beta^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and∥Φζ(k,l)Ql∥≤K(γζ1α)k−lε|l|≤K(1β)k−lε|l|,k≤l.$$\|\Phi_\zeta(k,l)Q_{l}\|\leqK(\tfrac{\gamma}{\zeta}\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|} \leqK(\tfrac{1}{\beta})^{k-l}\varepsilon^{|l|},\quad k\leq l.$$Hence ζ ∈ ρNED(A). Therefore, ρNED(A) is an open set. Using Lemma 2.2, we know that 𝒮ζ = 𝒮γ and 𝒰ζ = 𝒰γ.Corollary 2.4ΣNED(A) is a closed set.Using the facts proved above, we can obtain the following result, whose proof is similar as [4, Lemma 2.2], and therefore we omit the proof here.Lemma 2.5Let γ1, γ2 ∈ ρNED(A) with γ1 < γ2. Then F=Uγ1∩Sγ2$ \mathcal{F}=\mathcal{U}_{\gamma_{1}} \cap \mathcal{S}_{\gamma_{2}} $is an invariant vector bundle which satisfies exactly one of the following two alternatives and the statements given in each alternative are equivalent:Alternative IAlternative II(A)F=Z×{0}.$(A) \mathcal{F}=\mathbb{Z} \times\{0\}.$A′F≠Z×{0}.$\left(\mathrm{A}^{\prime}\right) \mathcal{F} \neq \mathbb{Z} \times\{0\}.$(B)γ1,γ2 ⊂ ρNED(A).$(B) \left[\gamma_{1}, \gamma_{2}\right] \subset \rho_{N E D}(A).$(B')Thereisaζ∈γ1,γ2∩ΣNED(A).$\text{(B')}\, There\, is\, a \,\zeta \in\left(\gamma_{1}, \gamma_{2}\right) \cap \Sigma_{N E D}(A).$(C)Sγ1=Sγ2andUγ1=Uγ2.$\text{(C)}\, \,\mathcal{S}_{\gamma_{1}}=\mathcal{S}_{\gamma_{2}}\, and\,\,\mathcal{U}_{\gamma_{1}}=\mathcal{U}_{\gamma_{2}}.$C′dimSγ1<dimSγ2.$\left(\text{C}^{\prime}\right) \operatorname{dim} \mathcal{S}_{\gamma_{1}}<\operatorname{dim} \mathcal{S}_{\gamma_{2}}.$(D)Sγ=Sγ2anduγ=Uγ2forγ∈γ1,γ2.$(\mathrm{D}) \mathcal{S}_{\gamma}=\mathcal{S}_{\gamma_{2}}\, and \,u_{\gamma}=\mathcal{U}_{\gamma_{2}}\,for \,\gamma \in\left[\gamma_{1}, \gamma_{2}\right].$D′dimCγ1>dimUγ2.$\left(\mathrm{D}^{\prime}\right) \operatorname{dim} \mathcal{C}_{\gamma_{1}}>\operatorname{dim} \mathcal{U}_{\gamma_{2}}.$Now we are in a position to state and prove the nonuniform dichotomy spectral theorem which will be essential to prove the result on reducibility in Section 3. The proof follows the idea and technique of the classical dichotomy spectrum proposed in [30], we present the details for the reader’s convenience.Theorem 2.6The nonuniform dichotomy spectrum ∑NED(A)$\sum_{N E D}(A)$ of (1.1) is the disjoint union of n closed intervals (called spectral intervals) where 0≤n≤N,i.e.,ΣNED(A)=∅orΣNED(A)=R+$0 \leq n \leq N\,, i.e., \,\Sigma_{N E D}(A)=\emptyset\, or \,\Sigma_{N E D}(A)=\mathbb{R}^{+}$ or one of the four casesΣNED(A)=a1,b1or0,b1∪a2,b2∪⋯∪an−1,bn−1∪an,bnoran,∞$$\Sigma_{N E D}(A)=\left\{\begin{array}{c}{\left[a_{1}, b_{1}\right]} \\ \, { or }\, \\ \left(0, b_{1}\right]\end{array}\right\} \cup\left[a_{2}, b_{2}\right] \cup \cdots \cup\left[a_{n-1}, b_{n-1}\right] \cup\left\{\begin{array}{c}{\left[a_{n}, b_{n}\right]} \\ or \\ {\left[a_{n}, \infty\right)}\end{array}\right\}$$where 0 < a1 ≤ b1 < a2 ≤ b2 < ··· < an ≤ bn. Choose a(2.4)γ0∈ρNED(A)with0,γ0 ⊂ ρNED(A)ifpossible,$$\gamma_{0} \in \rho_{N E D}(A) \, { with }\,\left(0, \gamma_{0}\right) \subset \rho_{N E D}(A) \, { if\, possible },$$otherwise define Uγ0:=Z×RN,Sγ0:=Z×{0}$\mathcal{U}_{\gamma_{0}}:=\mathbb{Z} \times \mathbb{R}^{N}, \mathcal{S}_{\gamma_{0}}:=\mathbb{Z} \times\{0\}$. Choose a(2.5)γn∈ρNED(A)withγn,+∞ ⊂ ρNED(A)ifpossible,$$\gamma_{n} \in \rho_{N E D}(A)\, with \,\left(\gamma_{n},+\infty\right) \subset \rho_{N E D}(A)\, if \,possible,$$otherwise define Uγn:=Z×{0},Sγ0:=Z×RN$\mathcal{U}_{\gamma_{n}}:=\mathbb{Z} \times\{0\}, \mathcal{S}_{\gamma_{0}}:=\mathbb{Z} \times \mathbb{R}^{N}$. Then the setsW0=Sγ0 and Wn+1=Sγn$$\mathcal{W}_{0}=\mathcal{S}_{\gamma_{0}} \quad \text { and } \quad \mathcal{W}_{n+1}=\mathcal{S}_{\gamma_{n}}$$are invariant vector bundles of (1.1). For n = 2, choose γi ∈ ρNED(A) with(2.6)b i < γ i < a i + 1 f o r i = 1 , … , n − 1 , $$b_{i}<\gamma_{i}<a_{i+1} \, { for }\, \quad i=1, \ldots, n-1,$$then for every i = 1, …, n  −  1 the intersectionWi=Uγi−1∩Sγi$$\mathcal{W}_i=\mathcal{U}_{\gamma_{i-1}}\cap \mathcal{S}_{\gamma_{i}}$$is an invariant vector bundle of (1.1) with dim 𝒲i ≥ 1. The invariant vector bundles 𝒲i, i = 0, …, n + 1, are called spectral bundles and they are independent of the choice of γ0, …, γn in (2.4), (2.5) and (2.6). MoreoverW0⊕⋯⊕Wn+1=Z×RN$$\mathcal{W}_0\oplus\cdots\oplus\mathcal{W}_{n+1}=\mathbb{Z} \times \mathbb{R}^{N}$$is a direct sum, i.e., Wi∩Wj=Z×{0}$ \mathcal{W}_{i} \cap \mathcal{W}_{j}=\mathbb{Z} \times\{0\} $for i≠j and W0+⋯+Wn+1=Z×RN$\mathcal{W}_{0}+\cdots+\mathcal{W}_{n+1}=\mathbb{Z} \times \mathbb{R}^{N}$.Proof. Recall that the resolvent set ρNED(A) is open and therefore ΣNED(A) is the disjoint union of closed intervals. Next we will show that ΣNED(A) consists of at most N intervals. Indeed, if ΣNED(A) contains N + 1 components, then one can choose a collection of points ζ1, …, ζN in ρNED(A) such that ζ1<··· < ζN and each of the intervals (0, ζ1), (ζ1, ζ2), …, (ζN − 1, ζN), (ζN, ∞) has nonempty intersection with the spectrum ΣNED(A). Now alternative II of Lemma 2.5 implies0≤dimsζ1<⋯<dimSζN≤N$$0 \leq \operatorname{dim} s_{\zeta_{1}}<\cdots<\operatorname{dim} \mathcal{S}_{\zeta_{N}} \leq N$$and therefore either dimSζ1=0ordimSζN=N$ \operatorname{dim} \mathcal{S}_{\zeta_{1}}=0\, \text{or} \,\operatorname{dim} \mathcal{S}_{\zeta_{N}}=N $or both. Without loss of generality, dimSζN=N$ \operatorname{dim} S_{\zeta_{N}}=N $, i.e., SζN=Z×RN$\mathcal{S}_{\zeta_N}=\mathbb{Z} \times\mathbb{R}^{N}$. Assume thatxk+1=1ζNAkxk$$x_{k+1} =\tfrac{1}{\zeta_N}A_{k}x_{k}$$admits a nonuniform exponential dichotomy with invariant projector P ≡ Id, thenxk+1=1ζAkxk$$x_{k+1}=\tfrac{1}{\zeta}A_{k}x_{k}$$also admits for every ζ>ζN a nonuniform exponential dichotomy with the same projector. We conclude (ζN, ∞) ⊂ ρNED(A), which is a contradiction. This proves the alternatives for ΣNED(A).Due to Lemma 2.5, the sets 𝒲0, …, 𝒲n+1 are invariant vector bundles. To prove now that dim 𝒲1 ≥ 1, …, dim 𝒲n ≥ 1 for n ≥ 1, let us assume that dim 𝒲1 = 0, i.e., Uγ0∩Sγ1=Z×{0}$ \mathcal{U}_{\gamma_{0}} \cap \mathcal{S}_{\gamma_{1}}=\mathbb{Z} \times\{0\} $. If (0, b1] is a spectral interval this implies that Sγ1=Z×{0}$ \mathcal{S}_{\gamma_{1}}=\mathbb{Z} \times\{0\} $. Then the projector of the nonuniform exponential dichotomy ofxk+1=1γ1Akxk$$x_{k+1} =\tfrac{1}{\gamma_1}A_{k}x_{k}$$is 0 and then we get the contradiction (0, γ1) ⊂ ρNED(A). If [a1, b1] is a spectral interval then [γ0, γ1]∩ΣNED(A)≠∅ and by alternative II of Lemma 2.5 we get a contradiction. Therefore dim 𝒲1 ≥ 1 and analogously dim𝒲n ≥ 1. Furthermore for n ≥ 3 and i = 2, …, n  −  1 one has [γi − 1, γi]∩ΣNED(A)≠∅ and again alternative II of Lemma 2.5 yields dim𝒲i ≥ 1.For i < j we have Wi ⊂ Sγi$ \mathcal{W}_{i}\subset \mathcal{S}_{\gamma_i} $and Wj ⊂ Uγj−1 ⊂ Uγi$ \mathcal{W}_{j}\subset \mathcal{U}_{\gamma_{j-1}} \subset \mathcal{U}_{\gamma_i} $and with Lemma 2.5 this gives Wi∩Wj ⊂ Sγi∩Uγi=Z×{0}$ \mathcal{W}_{i} \cap \mathcal{W}_{j}\subset\mathcal{S}_{\gamma_i}\cap \mathcal{U}_{\gamma_i}=\mathbb{Z} \times \{0\} $, so Wi∩Wj=Z×{0}$ \mathcal{W}_{i} \cap\mathcal{W}_{j} =\mathbb{Z} \times \{0\} $for i≠j.To show that W0⊕⋯⊕Wn+1=Z×RN$ \mathcal{W}_0\oplus\cdots\oplus\mathcal{W}_{n+1}=\mathbb{Z} \times \mathbb{R}^{N} $, recall the monotonicity relations Sγ0 ⊂ ⋯ ⊂ Sγn$ \mathcal{S}_{\gamma_0}\subset \cdots\subset \mathcal{S}_{\gamma_n} $, Uγ0⊃⋯⊃Uγn$ \mathcal{U}_{\gamma_0}\supset\cdots\supset\mathcal{U}_{\gamma_n} $, and the identity Sγ⊕Uγ=Z×RN$ \mathcal{S}_\gamma \oplus \mathcal{U}_\gamma = \mathbb{Z} \times \mathbb{R}^N $for γ ∈ ℝ+. Therefore Z×RN=W0+Uγ0$ \mathbb{Z} \times \mathbb{R}^{N}=\mathcal{W}_0 +\mathcal{U}_{\gamma_0} $. Now we haveZ × R N = W 0 + U γ 0 ∩ [ S γ 1 + U γ 1 ] = W 0 + [ U γ 0 ∩ S γ 1 ] + U γ 1 = W 0 + W 1 + U γ 1 . $$\mathbb{Z} \times \mathbb{R}^{N} = \mathcal{W}_0 + \mathcal{U}_{\gamma_0} \cap [\mathcal{S}_{\gamma_1}+\mathcal{U}_{\gamma_1}]\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= \mathcal{W}_0 + [\mathcal{U}_{\gamma_0} \cap \mathcal{S}_{\gamma_1}]+\mathcal{U}_{\gamma_1}\\\,\,\,\,= \mathcal{W}_0 + \mathcal{W}_1+\mathcal{U}_{\gamma_1}.$$Doing the same for Uγ1$ \mathcal{U}_{\gamma_1} $, we getZ × R N = W 0 + W 1 + U γ 1 ∩ [ S γ 2 + U γ 2 ] = W 0 + W 1 + [ U γ 1 ∩ S γ 2 ] + U γ 2 = W 0 + W 1 + W 2 + U γ 2 , $$\mathbb{Z} \times \mathbb{R}^{N} = \mathcal{W}_0 + \mathcal{W}_1 + \mathcal{U}_{\gamma_1} \cap [\mathcal{S}_{\gamma_2}+\mathcal{U}_{\gamma_2}]\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= \mathcal{W}_0 + \mathcal{W}_1 + [\mathcal{U}_{\gamma_1} \cap \mathcal{S}_{\gamma_2}]+\mathcal{U}_{\gamma_2}\\\,\,\,\,= \mathcal{W}_0 + \mathcal{W}_1 + \mathcal{W}_2+\mathcal{U}_{\gamma_2},$$and mathematical induction yields Z×RN=W0+⋯+Wn+1$ \mathbb{Z} \times \mathbb{R}^{N}=\mathcal{W}_0+\cdots+\mathcal{W}_{n+1} $. To finish the proof, let γ˜0,…,γ˜n∈ρNED(A)$ \tilde{\gamma}_0,\ldots,\tilde{\gamma}_n \in\rho_{NED}(A) $be given with the properties (2.4), (2.5) and (2.6). Then alternative I of Lemma 2.5 impliesSγi=Sγ˜iandUγi=Uγ˜ifori=0,…,n$$\mathcal{S}_{\gamma_i}=\mathcal{S}_{\tilde{\gamma}_i} \quad {\rm and} \quad \mathcal{U}_{\gamma_i}=\mathcal{U}_{\tilde{\gamma}_i} \quad {\rm for} \quad i=0,\ldots,n$$and therefore the invariant vector bundles 𝒲0, …, 𝒲n+1 are independent of the choice of γ0, …, γn in (2.4), (2.5) and (2.6).Definition 2.7We say that the evolution operator of (1.1) is nonuniformly exponentially bounded if there exist K > 0, ε ≥ 1 and a ≥ 1 with(2.7)∥Φ(k,l)∥≤Ka|k−l|ε|l|,k,l∈Z.$$\begin{equation}\|\Phi(k,l)\|\leqKa^{|k-l|}\varepsilon^{|l|},\quad k,l\in\mathbb{Z}.\end{equation}$$Lemma 2.8Assume the evolution operator of (1.1) is nonuniformly exponentially bounded. Then ΣNED(A) is a bounded closed set and ΣNED(A) ⊂ [1a,a]$ \Sigma_{NED}(A)\subset [\frac{1}{a},a] $.Proof. Assume that (2.7) holds. Let γ > a and 0<α:=aγ<1$ 0<\alpha := \frac{a}{\gamma}<1 $, then estimate (2.7) implies∥Φγ(k,l)∥≤Kαk−lε|l|,k≥l.$$\|\Phi_{\gamma}(k,l)\|\leq K\alpha^{k-l}\varepsilon^{|l|}, \quad k\geq l.$$Therefore (1.1) admits a nonuniform exponential dichotomy with invariant projector P = Id. It follows that γ ∈ ρNED(A) and also similarly for 0<γ<1a$ 0<\gamma < \frac{1}{a} $, hence ΣNED(A) ⊂ [1a,a]$ \Sigma_{NED}(A) \subset [\frac{1}{a},a] $.Lemma 2.9The evolution operator of (1.1) is nonuniformly exponentially bounded if and only if the nonuniform dichotomy spectrum ΣNED(A) of (1.1) is the disjoint union of n closed intervals where 0 ≤ n ≤ N, i.e.,ΣNED(A)=[a1,b1]∪[a2,b2]∪⋯∪[an−1,bn−1]∪[an,bn],$$\Sigma_{NED}(A) =[a_1,b_1]\cup [a_2,b_2] \cup \cdots \cup [a_{n-1},b_{n-1}]\cup[a_n,b_n],$$where 0 < a1 ≤ b1 < a2 ≤ b2 < ··· < an ≤ bn < ∞. In addition, W1⊕⋯⊕Wn=Z×RN$ \mathcal{W}_1\oplus\cdots\oplus\mathcal{W}_{n}=\mathbb{Z} \times \mathbb{R}^{N} $, W0=Wn+1=Z×{0}$ \mathcal{W}_0 = \mathcal{W}_{n+1}=\mathbb{Z} \times \{0\} $, where the sets 𝒲0, …, 𝒲n+1 are invariant vector bundles defined in Theorem 2.6.Proof. Necessity. It is easy to know that ΣNED(A) is bounded from Lemma 2.8. Now we prove that ΣNED(A)≠∅. It is easy to verify that Sγ=imP=Z×RN$ \mathcal{S}_\gamma = \text{im} P =\mathbb{Z} \times \mathbb{R}^{N} $and Uγ=kerP=Z×{0}$ \mathcal{U}_\gamma = \kerP=\mathbb{Z} \times \{0\} $for γ>a. Setγ ⋆ = inf γ ∈ ρ N E D ( A ) : S γ = Z × R N .$$\gamma^{\star}=\inf \left\{\gamma \in \rho_{N E D}(A): \mathcal{S}_{\gamma}=\mathbb{Z} \times \mathbb{R}^{N}\right\}.$$Clearly, γ∗∈[1a,a]$ \gamma^{*}\in [\frac{1}{a},a] $. In addition, we have γ* ∈ ΣNED(A). Otherwise, by using Lemma 2.3, there exists a neighborhood J$ \mathcal {J} $of γ* such that J ⊂ ρNED(A)$ \mathcal {J} \subset \rho_{NED}(A) $and for any γ<γ* of J$ \mathcal {J} $we have Sγ=Sγ∗=Z×RN$ \mathcal{S}_{\gamma}=\mathcal{S}_{\gamma^{*}}=\mathbb{Z} \times \mathbb{R}^{N} $, which is a contradiction to the definition of γ*. So ΣNED(A)≠∅, which means thatΣNED(A)=[a1,b1]∪[a2,b2]∪⋯∪[an−1,bn−1]∪[an,bn].$$\Sigma_{NED}(A) =[a_1,b_1]\cup [a_2,b_2] \cup \cdots \cup [a_{n-1},b_{n-1}]\cup[a_n,b_n].$$Let γ0 ∈ (0, a1), γi ∈ (bi, ai+1) for i = 1, …, n  −  1 and γn ∈ (bn, ∞). Clearly, from the proof of Lemma 2.8, we have Uγn=Z×{0}$ \mathcal{U}_{\gamma_{n}} =\mathbb{Z}\times \{0\} $for γn>bn and Sγ0=Z×{0}$ \mathcal{S}_{\gamma_{0}} =\mathbb{Z}\times \{0\} $for 0<γ0<a1. Therefore, W0=Wn+1=Z×{0}$ \mathcal{W}_0 =\mathcal{W}_{n+1}=\mathbb{Z} \times \{0\} $, and then it follows from Theorem 2.6 that W1⊕⋯⊕Wn=Z×RN$ \mathcal{W}_1\oplus\cdots\oplus\mathcal{W}_{n}=\mathbb{Z} \times \mathbb{R}^{N} $.Sufficiency. Let γ0 ∈ (0, a1), γi ∈ (bi, ai+1) for i = 1, …, n  −  1 and γn ∈ (bn, ∞). Clearly, from the proof of Lemma 2.8, we have Sγn=Z×RN$ \mathcal{S}_{\gamma_{n}} =\mathbb{Z} \times \mathbb{R}^{N} $for γn>bn and Uγ0=Z×RN$ \mathcal{U}_{\gamma_{0}} =\mathbb{Z} \times \mathbb{R}^{N} $for 0 <γ0<a1. This means that the invariant projectors associated to γ0 and γn are Pl = 0 and Pl = Id respectively.Hence, there exist K0 > 1, 0<α0 < 1 and ε0 > 1 such that∥Φγ0(k,l)∥≤K0(1α0)k−lε0|l|,k≤l,$$\|\Phi_{\gamma_{0}}(k,l)\|\leq K_{0}(\tfrac{1}{\alpha_{0}})^{k-l}\varepsilon_{0}^{|l|},\quadk\leq l,$$and there exist Kn > 1, 0 < αn < 1 and εn > 1 such that∥Φγn(k,l)∥≤Knαnk−lεn|l|,k≥l.$$\|\Phi_{\gamma_{n}}(k,l)\|\leq K_{n}\alpha_{n}^{k-l}\varepsilon_{n}^{|l|},\quadk\geq l.$$Now taking K = max{K0, Kn}, ε = max{ε0, εn} and a=max{a0γ0,anγn}$ a=\max\{\frac{a_{0}}{\gamma_{0}}, a_{n}\gamma_{n}\} $, then we have∥Φ(k,l)∥≤Kα|k−l|ε|l|,for k,l∈Z,$$\|\Phi(k,l)\|\leqK\alpha^{|k-l|}\varepsilon^{|l|},\quad {\rm for ~}k, l\in \mathbb{Z},$$which shows that the evolution operator of (1.1) is nonuniformly exponentially bounded.From Proposition 1, we know ΣNED(A) ⊂ ΣED(A). Finally in this section, we present an example to illustrate that ΣNED(A)≠ΣED(A) can occur.Example 2.10Given ω>5a > 0. Consider the scalar equation(2.8)uk+1=Akuk$$u_{k+1}=A_{k}u_{k}$$withAk=e−ω+ak(−1)k−a(k−1)(−1)(k−1).$$A_{k}=e^{-\omega+ak(-1)^{k}-a(k-1)(-1)^{(k-1)}}.$$Then ΣNED(A) = [e − ω − 5a, e − ω+5a] and ΣED(A) = ℝ+.Proof. The evolution operator of (2.8) is given byΦ ( k , l ) = e − ω ( k − l − 1 ) − a ( k − l − 1 ) ( − 1 ) k − 1 − a l ( − 1 ) ( k − 1 ) + a l ( − 1 ) l . $$\Phi(k,l)=e^{-\omega(k-l-1)-a(k-l-1)(-1)^{k-1}-al(-1)^{(k-1)}+al(-1)^{l}}.$$For each γ ∈ ℝ+ the evolution operator of(2.9)ukuk+1=1γAkuk$${uk}u_{k+1}=\tfrac{1}{\gamma}A_{k}u_{k}$$is given by(2.10)Φ γ ( k , l ) = ( 1 γ ) ( k − l ) e − ω ( k − l − 1 ) − a ( k − l − 1 ) ( − 1 ) k − 1 − a l ( − 1 ) ( k − 1 ) + a l ( − 1 ) l . $$\begin{equation}\Phi_{\gamma}(k,l)=(\tfrac{1}{\gamma})^{(k-l)}e^{-\omega(k-l-1)-a(k-l-1)(-1)^{k-1}-al(-1)^{(k-1)}+al(-1)^{l}}.\end{equation}$$For any γ ∈ (e( − ω+5a), +∞), it follows from (2.10) that(2.11)Φ γ ( k , l ) ≤ e ω − a e − ω + a γ k − l e 2 a | l | , k ≥ l , $$\left|\Phi_{\gamma}(k, l)\right| \leq e^{\omega-a}\left(\frac{e^{-\omega+a}}{\gamma}\right)^{k-l} e^{2 a|l|}, k \geq l,$$which implies that the equation (2.9) admits a nonuniform exponential dichotomy with invariant projector P = Id, by settingK=eω−a,  α=e−ω+aγ<1,  ε=e2a>0.$$K=e^{\omega-a},~~\alpha=\frac{e^{-\omega+a}}{\gamma}<1,~~\varepsilon=e^{2a}>0.$$Thus,(2.12)(e−ω+5a,+∞) ⊂ ρNED(A).$$\begin{equation}(e^{-\omega+5a},+\infty)\subset \rho_{NED}(A).\end{equation}$$For any γ˜∈(0,e−ω−5a)$ \widetilde{\gamma}\in (0, e^{-\omega-5a}) $, it follows from (2.10) that(2.13)Φγ(k,l)≤eω+ae−ω−aγk−le2a|l|,k≤l,$$\left|\Phi_{\gamma}(k, l)\right| \leq e^{\omega+a}\left(\frac{e^{-\omega-a}}{\gamma}\right)^{k-l} e^{2 a|l|}, k \leq l,$$which implies that (2.9) admits a nonuniform exponential dichotomy with P = 0, by takingK=eω+a,  α=γe−ω−a<1,  ε=e2a>0.$$K=e^{\omega+a},~~\alpha=\frac{\gamma}{e^{-\omega-a}}<1,~~\varepsilon=e^{2a}>0.$$Thus,(2.14)(0,e−ω−5a) ⊂ ρNED(A).$$\begin{equation}(0, e^{-\omega-5a})\subset \rho_{NED}(A).\end{equation}$$It follows from (2.12) and (2.14) that(0,e−ω−5a)∪(e−ω+5a,+∞) ⊂ ρNED(A),$$(0, e^{-\omega-5a}) \cup (e^{-\omega+5a},+\infty)\subset\rho_{NED}(A),$$which implies thatΣNED(A) ⊂ [e−ω−5a,e−ω+5a].$$\Sigma_{NED}(A) \subset[e^{-\omega-5a},e^{-\omega+5a}].$$Next we show that[e−ω−5a,e−ω+5a] ⊂ ΣNED(A).$$[e^{-\omega-5a},e^{-\omega+5a}]\subset\Sigma_{NED}(A).$$To do this, we first prove that γ1 = e − ω+5a ∈ ΣNED(A). The evolution operator of the systemuk+1=1γ1Akuk$$u_{k+1}=\tfrac{1}{\gamma_{1}}A_{k}u_{k}$$is given asΦ γ 1 ( k , l ) = e ω − a e − a ( k − l − 1 ) ( 1 + ( − 1 ) k − 1 ) − a l ( − 1 ) ( k − 1 ) + a l ( − 1 ) l . $$\Phi_{\gamma_{1}}(k,l)=e^{\omega-a}e^{-a(k-l-1)(1+(-1)^{k-1})-al(-1)^{(k-1)}+al(-1)^{l}}.$$It is easy to see that there do not exist K, α>0 and ε>0 such that∥Φγ1(k,l)∥≤Kαk−lε|l|,    fork≥l,$$\|\Phi_{\gamma_{1}}(k,l)\| \leqK\alpha^{k-l}\varepsilon^{|l|}, \ \ \ \ \mbox{for} \,\,\, k\geq l,$$or∥Φγ1(k,l)∥≤K(1α)k−lε|l|,  fork≤l.$$\|\Phi_{\gamma_{1}}(k,l)\|\leqK(\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|},\ \ \mbox{for} \,\,\,k\leq l.$$Therefore γ1 = e − ω+5a ∈ ΣNED(A). In a similar manner, we can prove γ2 = e − ω − 5a ∈ ΣNED(A). We can see from Theorem 2.6 that (2.8) has at most one nonuniform dichotomy spectral interval, which means that [e − ω − 5a, e − ω+5a] ⊂ ΣNED(A) and therefore [e − ω − 5a, e − ω+5a] = ΣNED(A).On the other hand, using a similar argument as in equations (1.6), we know that the nonuniform part ε∣l∣ cannot be removed in the estimates (2.11) and (2.13). Therefore, (2.8) does not admit an exponential dichotomy, which means that ΣED(A) = ℝ+.3ReducibilityIn this section we utilize Theorem 2.6 to show a reducibility result. For the reducibility results in the setting of an exponential dichotomy, we refer the reader to [15, 24, 32] and the references therein. First we recall the definition of kinematic similarity given in Coppel [16] and Aulbach et al. [2].Definition 3.1Equation (1.1) is said to be kinematically similar to another equation(3.1)yk+1=Bkyk$$y_{k+1}=B_{k}y_{k}$$with k ∈ ℤ, if there exists an invertible matrix Sk with ∣Sk∣ ≤ M and ∥Sk−1∥≤M$ \|S_{k}^{-1}\|\leq M $(M > 0), which satisfies the difference equationSk+1Bk=AkSk.$$S_{k+1}B_{k}=A_{k}S_{k}.$$The change of variables xk = Skyk then transforms (1.1) into (3.1).The next lemma is important to establish the reducibility results and its proof follows along the lines of the proof of Siegmund [32]. See also Coppel [16] and Aulbach et al. [2]Lemma 3.2[16, Chapter 5] Let P be an orthogonal projection (PT = P) and let X be an invertible matrix function. Then there exists an invertible matrix function S:Z→RN×N$ S:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N} $such thatSkPSk−1=XkPXk−1,SkQSk−1=XkQXk−1,$$S_{k}PS^{-1}_{k}=X_{k}PX^{-1}_{k},\,\,\,\,\,\,\, \,\,\,\,\,\,\,S_{k}QS^{-1}_{k}=X_{k}QX^{-1}_{k},$$and∥Sk∥≤2,$$\begin{equation*}\|S_{k}\|{\leq}\sqrt{2},\end{equation*}$$∥Sk−1∥≤[∥XkPXk−1∥2+∥Xk(I−P)Xk−1∥2]12,$$\begin{equation*}\|S^{-1}_{k}\|{\leq}\big[\|X_{k}PX^{-1}_{k}\|^{2}+\|X_{k}(I-P)X^{-1}_{k}\|^{2}\big]^{\frac{1}{2}},\end{equation*}$$where k ∈ ℤ and Q = Id − P. DefineR˜:Z→RN×N,k↦PXkTXkP+[Id−P]XkTXk[Id−P].$$\widetilde{R} :\mathbb{Z}\rightarrow \mathbb{R}^{N \timesN},\;k \mapsto PX_{k}^T X_{k} P+ [{\rm Id} - P] X_{k}^T X_{k} [{\rm Id} -P].$$Then the mapping is a positive definite, symmetric matrix for every k∈Z$ k \in \mathbb{Z} $. Moreover, there is a unique functionR:Z→RN×N$$R : \mathbb{Z}\rightarrow \mathbb{R}^{N \times N}$$of positive definite symmetric matrices Rk, k∈Z$ k \in \mathbb{Z} $, withRk2=R˜k,PRk=RkP.$$R_{k}^2 = \widetilde{R}_{k},\quadPR_{k} = R_{k} P\;.$$We remark that in the setting of an exponential dichotomy the function k↦Sk−1$ k \mapsto S^{-1}_{k} $in Lemma 3.2 is bounded. However, in the setting of a nonuniform exponential dichotomy, Sk−1$ S^{-1}_{k} $can be unbounded, because ∣Φ(k, k)Pk∣ ≤ Kεk for k = 0. In fact, we can see that(3.2)∥Sk−1∥≤2Kε|k|.$$\|S_k^{-1}\|\leq \sqrt{2}K\varepsilon^{|k|}.$$To overcome the difficulty, we introduce a new version of non-degeneracy, so-called weak non-degeneracy and define the concept of weak kinematical similarity, which is a very natural notion if we note the fact (3.2).Definition 3.3S:Z→RN×N$S:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N}$is called weakly non-degenerate if there exists an M = M(ε)>0 such that∥Sk∥≤Mε|k|and∥Sk−1∥≤Mε|k|,forallk∈Z,$$\|S_{k}\|\leq M\varepsilon^{|k|}\,\,\,\,\,{and}\,\,\,\, \|S^{-1}_{k}\| \leq M\varepsilon^{|k|},\,\,\,\,\,{for\, all}\,\,\,\, k\in \mathbb{Z},$$where ε is the same constant in (1.4)-(1.5).Definition 3.4If there exists a weakly non-degenerate matrix Sk such thatSk+1Bk=AkSk,$$S_{k+1}B_{k}=A_{k}S_{k},$$then equation (1.1) is weakly kinematically similar to equation (3.1). For short, we denote (1.1) ∼w$ \overset{w}{\sim} $(3.1) or Ak∼wBk$ A_{k} \overset{w}{\sim} B_{k} $.We analogously denote kinematical similarity by (1.1) ∼ (3.1) or Ak ∼ Bk.Definition 3.5Equation (1.1) is called reducible, if it is weakly kinematically similar to equation (3.1) whose coefficient matrix Bk has the block diagonal form(3.3)Bk100Bk2,$$\begin{equation}\left(\begin{array}{lll}B^1_{k} & 0\\0 & B^2_{k}\end{array}\right),\end{equation}$$where Bk1$ B^1_{k} $and Bk2$ B^2_{k} $are matrices of smaller size than Bk.Before stating the main results of this section, we prove several preliminary lemmas.Lemma 3.6Assume that (1.1) admits a nonuniform exponential dichotomy. Then the projector of equation (1.1) can be chosen as P˜=IN1000N2$ \tilde{P}= \left(\begin{array}{lll} I_{N_1} & 0 \\ 0 & 0_{N_2} \end{array} \right) $with N1=dimimP˜$ N_1 = {\rm dim\, im} \tilde{P} $and N2=dimkerP˜$ N_2 = {\rm dim\, ker} \tilde{P} $, and the fundamental matrix Xk can be chosen suitably such that the estimates (1.4)-(1.5) can be rewritten as(3.4)∥XkP˜Xl−1∥≤Kαk−lε|l|,k≥l,$$\|X_{k}\tilde{P}X^{-1}_{l}\|\leqK\alpha^{k-l}\varepsilon^{|l|},\quad k\geq l,$$and(3.5)∥XkQ˜Xl−1∥≤K(1α)k−lε|l|,k≤l,$$\|X_{k}\tilde{Q}X^{-1}_{l}\|\leqK(\frac{1}{\alpha})^{k-l}\varepsilon^{|l|},\quad k\leq l,$$where Q˜=Id−P˜$ \tilde{Q}={\rm Id}-\tilde{P} $.Proof. Let n∈Z$ n\in \mathbb{Z} $be arbitrary but fixed. Note that the rank of the projector Pn is independent of n∈Z$ n\in \mathbb{Z} $(see [9, page 1100]), then there exists a nondegenerate matrix T ∈ ℝN × N such thatP˜:=IN1000N2=TPnT−1$$\tilde{P}:=\left(\begin{array}{lll} I_{N_1} & 0\\0 & 0_{N_2}\end{array}\right)=TP_{n}T^{-1}$$with N1=dimimP˜$ N_1 = {\rm dim\, im} \tilde{P} $and N2=dimkerP˜$ N_2 = {\rm dim\, ker} \tilde{P} $. DefineXk:=Φ(k,n)T−1fork∈ZandP˜:=IN1000N2=TPnT−1.$$X_{k} := \Phi(k,n)T^{-1} \quad \text{for} k \in \mathbb{Z}\quad \text{and} \quad\tilde{P}:= \left(\begin{array}{lll}I_{N_1} & 0\\0 & 0_{N_2}\end{array}\right)= T P_{n} T^{-1}.$$Then(3.6)∥XkP˜Xl−1∥=∥Φ(k,n)T−1P˜TΦ−1(l,n)∥=∥Φ(k,n)PnΦ−1(l,n)∥.$$\begin{equation}\|X_{k}\tilde{P}X^{-1}_{l}\|=\|\Phi(k,n)T^{-1} \tilde{P}T\Phi^{-1}(l,n)\|=\|\Phi(k,n)P_{n}\Phi^{-1}(l,n)\|.\end{equation}$$On the other hand, we have(3.7)∥ Φ ( k , l ) P l ∥ = ∥ Φ ( k , n ) Φ ( n , l ) P l ∥ = ∥ Φ ( k , n ) P n Φ ( n , l ) ∥ = ∥ Φ ( k , n ) P n Φ − 1 ( l , n ) ∥ . $$\|\Phi(k,l)P_{l}\| = \|\Phi(k,n)\Phi(n,l)P_{l}\|\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \|\Phi(k,n)P_{n}\Phi(n,l)\|\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \|\Phi(k,n)P_{n}\Phi^{-1}(l,n)\|.$$It follows from (3.6) and (3.7) that (1.4)-(1.5) can be rewritten in the form (3.4)-(3.5).Lemma 3.7Assume that (1.1) admits a nonuniform exponential dichotomy with the form of estimates (3.4)-(3.5) and rank(P˜)=N1,(0<N1<N)$ rank(\tilde{P})=N_{1}, (0 < N_{1}< N) $, and there exists a weakly non-degenerate transformation Sk such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $. Then system (3.1) also admits a nonuniform exponential dichotomy, and the projector has the same rank.Proof. Suppose that Sk is weakly non-degenerate, which means that there exists M = M(ε) > 0 such that ∣Sk∣ ≤ Mε∣k∣ and ∥Sk−1∥≤Mε|k|$ \|S^{-1}_{k}\| \leq M\varepsilon^{|k|} $and such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $. Let Xk = SkYk. It is easy to see that Yk is the fundamental matrix of system (3.1). To prove that system (3.1) admits a nonuniform exponential dichotomy, we first consider the case k ≥ l and obtain(3.8)∥YkP˜Yl−1∥=∥Sk−1XkP˜Xl−1Sl∥≤∥Sk−1∥⋅∥XkP˜Xl−1∥⋅∥Sl∥≤KM2ε|k|αk−lε2|l|=KM2(αε2)k−lε2l−2k+|k|+2|l|≤KM2(αε2)k−lεl−k+3|l|≤KM2(αε2)k−lε3|l|,k≥l.$$\begin{equation}\begin{array}{lll}\|Y_{k}\tilde{P}Y^{-1}_{l}\|&=&\|S^{-1}_{k}X_{k}\tilde{P}X^{-1}_{l}S_{l}\|\\&\leq&\|S^{-1}_{k}\|\cdot\|X_{k}\tilde{P}X^{-1}_{l}\|\cdot\|S_{l}\|\\&\leq&K M^{2}\varepsilon^{|k|} \alpha^{k-l}\varepsilon^{2|l|}\\&=& KM^2(\alpha\varepsilon^2)^{k-l}\varepsilon^{2l-2k+|k|+2|l|}\\&\leq&K M^{2}(\alpha\varepsilon^2)^{k-l}\varepsilon^{l-k+3|l|}\\&\leq & K M^{2}(\alpha\varepsilon^2)^{k-l}\varepsilon^{3|l|},\,\,k\geq l.\end{array}\end{equation}$$A similar argument shows that(3.9)∥YkQ˜Yl−1∥≤KM2(1αε2)k−lε3|l|,k≤l.$$\begin{equation}\|Y_{k}\tilde{Q}Y^{-1}_{l}\|\leq KM^{2}(\tfrac{1}{\alpha\varepsilon^2})^{k-l}\varepsilon^{3|l|},\,\,k\leq l.\end{equation}$$Form (3.8) and (3.9), it is easy to see that system (3.1) admits a nonuniform exponential dichotomy. Clearly, the rank of the projector is N1.Lemma 3.8Assume that the systems (1.1) and (3.1) are weakly kinematically similar via Sk. If for a constant γ ∈ ℝ+ the system (2.1) admits a nonuniform exponential dichotomy with constants K > 0, 0 < α < 1, ε ≥ 1 and invariant projector P, then the system(3.10)yk+1=1γBkyk$$y_{k+1}=\tfrac{1}{\gamma}B_{k}y_{k}$$also admits a nonuniform exponential dichotomy.Proof. Obviously, P is also an invariant projector for (1.1). The dichotomy estimates are equivalent to∥XkPXl−1∥≤Kαk−lε|l|,k≥l,$$\|X_{k}P X^{-1}_{l}\| \leq K\alpha^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and∥XkPXl−1∥≤K(1α)k−lε|l|,k≤l.$$\|X_{k}P X^{-1}_{l}\|\leqK(\tfrac{1}{\alpha})^{k-l}\varepsilon^{|l|},\quad k\leq l.$$Using Lemma 3.7, it is easy to see that∥YkPYl−1∥≤Kγ′(εα)k−lε|l|,k≥l,$$\|Y_{k}P Y^{-1}_{l}\| \leqK'_\gamma(\varepsilon\alpha)^{k-l}\varepsilon^{|l|}, \quad k\geq l,$$and∥YkPYl−1∥≤Kγ′(1εα)k−lε|l|,k≤l,$$\|Y_{k}P Y^{-1}_{l}\|\leqK'_\gamma(\tfrac{1}{\varepsilon\alpha})^{k-l}\varepsilon^{|l|},\quad k\leql,$$for some constant K′γ ≥ 1. Therefore, (3.10) admits a nonuniform exponential dichotomy.The following result follows directly from Lemma 3.8.Corollary 3.9Assume that there exists a weakly non-degenerate transformation Sk such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $. Then ΣNED(A) = ΣNED(B), i.e.,ΣNED(A)=a1,b1 or 0,b1∪a2,b2∪⋯∪an−1,bn−1∪an,bn or an,∞=ΣNED(B).$$\begin{equation}\Sigma_{N E D}(A)=\left\{\begin{array}{c}{\left[a_{1}, b_{1}\right]} \\ \text { or } \\ \left(0, b_{1}\right]\end{array}\right\} \cup\left[a_{2}, b_{2}\right] \cup \cdots \cup\left[a_{n-1}, b_{n-1}\right] \cup\left\{\begin{array}{c}{\left[\begin{array}{c}\left.a_{n}, b_{n}\right] \\ \text { or } \\ {\left[a_{n}, \infty\right)}\end{array}\right\}}\end{array}\right\}=\Sigma_{N E D}(B). \end{equation}$$The following theorem states that if (1.1) admits a nonuniform exponential dichotomy, then there exists a weakly non-degenerate transformation such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $and Bk has the block form (3.3), i.e., system (1.1) is reducible.Theorem 3.10Assume that (1.1) admits a nonuniform exponential dichotomy (not necessary ) of the form (3.4)-(3.5) with the invariant projector satisfying Pk≠0, Id. Then (1.1) is reducible. More precisely, it is weakly kinematically similar to a decoupled system(3.11)xk+1=Bk100Bk2xk$$\begin{equation}x_{k+1} = \begin{pmatrix} B^1_{k} & 0 \\ 0 & B^2_{k} \end{pmatrix} x_{k} \end{equation}$$for some matrix functionsB1:Z→RN1×N1andB2:Z→RN2×N2$$B^1 : \mathbb{Z} \rightarrow \mathbb{R}^{N_1 \times N_1}\quad \text{and} \quad B^2 : \mathbb{Z} \rightarrow \mathbb{R}^{N_2\times N_2}$$where N 1 := dim ⁡ im P ~ $ N_1 := \dim \text{im}\, \tilde{P} $and N2:=dimkerP˜$ N_2 := \dim \ker \tilde{P} $.Proof. Since equation (1.1) admits a nonuniform exponential dichotomy of the form (1.4)-(1.5) with the invariant projector satisfying Pk≠0, Id, by Lemma 3.6, one can choose a suitable fundamental matrix Xk and the projector P˜=IN1000$ \tilde{P}= \left(\begin{array}{lll} I_{N_1} & 0 \\ 0 & 0 \end{array}\right) $, (0 < N1<N) such that the estimates (3.4)-(3.5) hold. By Lemma 3.2 and the estimates (3.4)-(3.5), there exists an M = M(ε)>0 large enough such thatSk≤2≤Mε|k|,Sk−1≤XkP˜Xk−12+Xk(Id−P˜)Xk−1212≤2Kε|k|.$$\begin{equation}\begin{array}{c}\left\|S_{k}\right\| \leq \sqrt{2} \leq M \varepsilon^{|k|}, \\ \left\|S_{k}^{-1}\right\| \leq\left[\left\|X_{k} \tilde{P} X_{k}^{-1}\right\|^{2}+\left\|X_{k}(\mathrm{Id}-\tilde{P}) X_{k}^{-1}\right\|^{2}\right]^{\frac{1}{2}} \leq \sqrt{2} K \varepsilon^{|k|}.\end{array} \end{equation}$$Thus, S is weakly non-degenerate. SettingBk=Rk+1Rk−1,$$B_{k}=R_{k+1}R^{-1}_{k},$$where Rk is defined in Lemma 3.2 and Xk = SkRk. Obviously, Rk is the fundamental matrix of the linear systemyk+1=Bkyk.$$\begin{equation*}y_{k+1}=B_{k}y_{k}.\end{equation*}$$Now we need to show that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $and Bk has the block diagonal formBk=Bk100Bk2,fork∈Z.$$\begin{equation*}B_{k}= \left(\begin{array}{lll} B^1_{k} & 0\\0 & B^{2}_{k}\end{array}\right),\,\,\,\,\mbox{for} \,\,\,\, k\in \mathbb{Z}.\end{equation*}$$First, we show that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $. In fact,Sk+1Bk=Xk+1Rk+1−1Bk=AkXkRk−1Bk−1Bk=AkSk,$$\begin{equation}\begin{aligned} S_{k+1} B_{k} &=X_{k+1} R_{k+1}^{-1} B_{k} \\ &=A_{k} X_{k} R_{k}^{-1} B_{k}^{-1} B_{k} \\ &=A_{k} S_{k}, \end{aligned} \end{equation}$$which implies that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $.Now we show that system (1.1) is weakly kinematically similar to (3.11). By Lemma 3.2, Rk+1 and Rk−1$ R_{k}^{-1} $commute with the matrix P˜$ \tilde{P} $for every k∈Z$ k \in \mathbb{Z} $. It follows that(3.12)P˜Bk=BkP˜$$\begin{equation}\tilde{P} B_{k}=B_{k} \tilde{P}\end{equation}$$for all k∈Z$ k \in \mathbb{Z} $. Now we decompose B k : Z → R N × N $B_{k}: \mathbb{Z} \rightarrow \mathbb{R}^{N \times N}$into four functionsBk1:Z→RN1×N1,Bk2:Z→RN2×N2,Bk3:Z→RN1×N2,Bk4:Z→RN2×N1,$$\begin{equation}\begin{array}{ll}B_{k}^{1}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{1} \times N_{1}}, & B_{k}^{2}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{2} \times N_{2}}, \\ B_{k}^{3}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{1} \times N_{2}}, & B_{k}^{4}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{2} \times N_{1}},\end{array} \end{equation}$$withBk=Bk1Bk3Bk4Bk2,k∈Z.$$B_{k}= \left(\begin{array}{lll} B^1_{k} & B^3_{k}\\B^4_{k} & B^{2}_{k}\end{array}\right), \quad \,\,\,\, k\in \mathbb{Z}.$$Identity (3.12) implies thatBk1Bk300=Bk10Bk40,k∈Z.$$\left(\begin{array}{lll} B^1_{k} & B^3_{k}\\0 & 0\end{array}\right)=\left(\begin{array}{lll} B^1_{k} & 0\\B^4_{k} & 0\end{array} \right), \quad \,\,\,\, k\in \mathbb{Z}.$$Therefore Bk3≡0$ B^3_{k}\equiv 0 $and Bk4≡0$ B^4_{k}\equiv 0 $. Thus Bk has the block formBk=Bk100Bk2,k∈Z.$$B_{k}= \left(\begin{array}{lll} B^1_{k} & 0\\0 & B^{2}_{k}\end{array} \right), \quad \,\,\,\, k\in \mathbb{Z}.$$Now the proof is finished.From Theorem 3.10, we know that if (1.1) admits a nonuniform exponential dichotomy, then there exists a weakly non-degenerate transformation Sk such that Ak∼wBk$ A_{k}\overset{w}{\sim} B_{k} $and Bk has two blocks of the form (3.3). Now we are in a position to prove the reducibility result.Theorem 3.11Assume that (1.1) admits a nonuniform exponential dichotomy. Due to Theorem 2.6, the dichotomy spectrum is either empty or the disjoint union of n closed spectral intervals J1, ⋅, Jn with 1 ≤ N ≤ N, i.e.,ΣNED(A)=∅(n=0) or ΣNED(A)=J1∪⋯∪Jn.$$\begin{equation}\Sigma_{N E D}(A)=\emptyset \quad(n=0) \quad \text { or } \quad \Sigma_{N E D}(A)=\mathcal{J}_{1} \cup \cdots \cup \mathcal{J}_{n}. \end{equation}$$Then there exists a weakly kinematic similarity action S : Z → R N × N $S: \mathbb{Z} \rightarrow \mathbb{R}^{N \times N}$between (1.1) and a block diagonal systemxk+1=Bk0⋱Bkn+1xk$$\begin{equation}x_{k+1}=\left(\begin{array}{lll}B_{k}^{0} & & \\ & \ddots & \\ & & B_{k}^{n+1}\end{array}\right) x_{k} \end{equation}$$with B i : Z → R N i × N i $B^{i}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{i} \times N_{i}}$, Ni = dim𝒲i, andΣNEDB0=∅,ΣNEDB1=J1,…,ΣNEDBn=Jn,ΣNEDBn+1=∅.$$\begin{equation}\Sigma_{N E D}\left(B^{0}\right)=\emptyset, \Sigma_{N E D}\left(B^{1}\right)=\mathcal{J}_{1}, \ldots, \Sigma_{N E D}\left(B^{n}\right)=\mathcal{J}_{n}, \Sigma_{N E D}\left(B^{n+1}\right)=\emptyset. \end{equation}$$Proof. If for any γ ∈ ℝ+, system (2.1) admits a nonuniform exponential dichotomy, then ΣNED(A) = ∅. Conversely, for any γ ∈ ℝ+, system (2.1) does not admit a nonuniform exponential dichotomy, then ΣNED(A) = ℝ+. Now, we prove the theorem for the nontrivial case (ΣNED(A)≠∅ and ΣNED(A)≠ℝ+).Recall that the resolvent set ρNED(A) is open and therefore the dichotomy spectrum ΣNED(A) is the disjoint union of closed intervals. Using Theorem 2.6, we can assumeJ1=a1,b1 or 0,b1,J2=a2,b2,…,Jn−1=an−1,bn−1,Jn=an,bn or an,∞$$\begin{equation}\mathcal{J}_{1}=\left\{\begin{array}{c}{\left[a_{1}, b_{1}\right]} \\ \text { or } \\ \left(0, b_{1}\right]\end{array}\right\}, \mathcal{J}_{2}=\left[a_{2}, b_{2}\right], \ldots, \mathcal{J}_{n-1}=\left[a_{n-1}, b_{n-1}\right], \mathcal{J}_{n}=\left\{\begin{array}{c}{\left[\begin{array}{c}\left.a_{n}, b_{n}\right] \\ \text { or } \\ {\left[a_{n}, \infty\right)}\end{array}\right\}}\end{array}\right\} \end{equation}$$with 0 < a1 ≤ b1 < a2 ≤ b2 < … < an ≤ bn.If J1 = [a1, b1] is a spectral interval, then (0, γ0) ⊂ ρNED(A) and W 0 = S γ 0 $\mathcal{W}_{0}=\mathcal{S}_{\gamma_{0}}$for some γ0 < a1 due to Theorem 2.6, which implies thatxk+1=1γ0Akxk$$x_{k+1}=\tfrac{1}{\gamma_0}A_{k}x_{k}$$admits a nonuniform exponential dichotomy, let its invariant projector be denoted by P˜0$ \tilde{P}_0 $. By Theorem 3.10 and Corollary 3.9, there exists a weakly non-degenerate transformation xk=Sk0xk(0)$ x_k=S_k^{0}x_k^{(0)} $with ∥Sk0∥≤M0ε|k|$ \|S_{k}^{0}\|\leq M_0\varepsilon^{|k|} $and ∥(Sk0)−1∥≤M0ε|k|$ \|(S_{k}^{0})^{-1}\| \leq M_0\varepsilon^{|k|} $for some positive constant M0 = M0(ε) and such that Ak∼wAk0$ A_{k}\overset{w}{\sim} A^{0}_{k} $and Ak0$ A^{0}_{k} $has two blocks of the form Ak0=Bk000Bk0,∗$ A^{0}_{k}=\left(\begin{array}{lll} B^{0}_{k} & 0 \\ 0 & B^{0,*}_{k} \end{array}\right) $with dim ⁡ B k 0 = d i m i m ⁡ P ~ 0 = dim S γ 0 = dim W 0 =: N 0 $\operatorname{dim} B_{k}^{0}=\operatorname{dim\,im} \tilde{P}_{0}=\operatorname{dim} \mathcal{S}_{\gamma_{0}}=\operatorname{dim} \mathcal{W}_{0}=: N_{0}$due to Theorem 3.10, Lemma 2.5 and Theorem 2.6. If J1 = (0, b1] is a spectral interval, a block Bk0$ B_{k}^{0} $is omitted.Now we consider the following systemxk+1(0)=Ak0xk(0)=Bk000Bk0,∗xk(0).$$x_{k+1}^{(0)}=A^{0}_{k}x_{k}^{(0)}=\left(\begin{array}{lll}B^{0}_{k} & 0\\0 & B^{0,*}_{k}\end{array}\right)x_{k}^{(0)}.$$By using Lemma 2.5, we take γ1 ∈ (b1, a2). In view of (b1,a2) ⊂ ρNED(Bk0,∗)$ (b_{1},a_{2})\subset \rho_{NED}(B_k^{0,*}) $, γ1∈ρNED(Bk0,∗)$ \gamma_1\in \rho_{NED}(B_k^{0,*}) $, which implies thatxk+1(0)=1γ1Bk000Bk0,∗xk(0)$$x_{k+1}^{(0)}=\frac{1}{\gamma_1}\left(\begin{array}{lll}B^{0}_{k} & 0\\0 & B^{0,*}_{k}\end{array}\right)x_{k}^{(0)}$$admits a nonuniform exponential dichotomy. Its invariant projector P˜1$ \tilde{P}_1 $satisfies P˜1≠0,I$ \tilde{P}_1\neq 0,\,I $. Similarly, by Theorem 3.10 and Corollary 3.9, there exists a weakly non-degenerate transformationxk(0)=Sk1xk(1)=IN000S˜k1xk(1)$$x_k^{(0)}=S_k^{1}x_k^{(1)}=\left(\begin{array}{lll}I_{N_{0}} & 0\\0 & \tilde{S}^{1}_{k}\end{array}\right)x_k^{(1)}$$with ∥S˜k1∥≤M1ε|k|$ \|\tilde{S}^{1}_{k}\|\leq M_1\varepsilon^{|k|} $and ∥(S˜k1)−1∥≤M1ε|k|$ \|(\tilde{S}^{1}_{k})^{-1}\| \leq M_1\varepsilon^{|k|} $for some positive constant M1 = M1(ε) and such that Bk0,∗∼wB˜k0,∗$ B^{0,*}_{k}\overset{w}{\sim} \tilde{B}^{0,*}_{k} $and B˜k0,∗$ \tilde{B}^{0,*}_{k} $has two blocks of the form B˜k0,∗=Bk100Bk1,∗$ \tilde{B}^{0,*}_{k}=\left(\begin{array}{lll} B^{1}_{k} & 0 \\ 0 & B^{1,*}_{k} \end{array} \right) $with dim ⁡ B k 1 = dim ⁡ im P ~ 1 = dim S γ 1 ≥ dim ⁡ ( U γ 0 ∩ S γ 1 ) = dim W 1 =: N 1 $\dim B_{k}^{1}=\dim \text{im} \,\tilde{P}_1=\dim\mathcal{S}_{\gamma_1}\geq\dim(\mathcal{U}_{\gamma_0}\cap \mathcal{S}_{\gamma_1})=\dim\mathcal{W}_{1}=:N_{1}$due to Theorem 3.10, Lemma 2.5 and Theorem 2.6. In addition, using Theorem 3.10 and Corollary 3.9, we haveΣNEDBk1=a1,b1 or 0,b1,ΣNEDBk1,⋆=a2,b2∪⋯∪an−1,bn−1∪an,bn or an,∞.$$\begin{equation}\Sigma_{N E D}\left(B_{k}^{1}\right)=\left\{\begin{array}{c}{\left[a_{1}, b_{1}\right]} \\ \text { or } \\ \left(0, b_{1}\right]\end{array}\right\}, \quad \Sigma_{N E D}\left(B_{k}^{1, \star}\right)=\left[a_{2}, b_{2}\right] \cup \cdots \cup\left[a_{n-1}, b_{n-1}\right] \cup\left\{\begin{array}{c}{\left[a_{n}, b_{n}\right]} \\ \text { or } \\ {\left[a_{n}, \infty\right)}\end{array}\right\}. \end{equation}$$Now we can construct a weakly non-degenerate transformation xk=S˜kxk(1)$ x_k=\tilde{S}_kx_k^{(1)} $with S˜k=Sk0Sk1=Sk0IN000S˜k1$ \tilde{S}_{k}=S^{0}_{k}S^{1}_{k}=S^{0}_{k}\left(\begin{array}{lll} I_{N_{0}} & 0 \\ 0 & \tilde{S}^{1}_{k} \end{array}\right) $, where ∥S˜k∥≤M0M1ε2|k|$ \| \tilde{S}_{k}\|\leq M_0 M_1\varepsilon^{2|k|} $and ∥S˜k−1∥≤M0M1ε2|k|$ \| \tilde{S}_{k}^{-1}\| \leq M_0 M_1\varepsilon^{2|k|} $. Then Ak∼wAk1$ A_{k}\overset{w}{\sim} A^{1}_{k} $and Ak1$ A^{1}_{k} $has three blocks of the formAk1=Bk0Bk1Bk1,⋆.$$\begin{equation}A_{k}^{1}=\left(\begin{array}{ccc}B_{k}^{0} & & \\ & B_{k}^{1} & \\ & & B_{k}^{1, \star}\end{array}\right). \end{equation}$$Applying similar procedures to γ2 ∈ (b2, a3), γ3 ∈ (b3, a4), …, we can construct a weakly non-degenerate transformation xk=Skxk(n+1)$ x_k=S_kx_k^{(n+1)} $withSk=Sk0IN000S˜k1IN0+N100S˜k2⋯IN0+…+Nn−100S˜kn$$S_{k}=S^{0}_{k}\left(\begin{array}{lll}I_{N_{0}} & 0\\0 & \tilde{S}^{1}_{k}\end{array}\right)\left(\begin{array}{lll}I_{N_{0}+N_{1}} & 0\\0 & \tilde{S}^{2}_{k}\end{array}\right)\cdots \left(\begin{array}{lll}I_{N_{0}+\ldots+N_{n-1}} & 0\\0 & \tilde{S}^{n}_{k}\end{array}\right)$$such that ∣ Sk∣ ≤ Mε εn∣k∣ and ∥Sk−1∥≤Mεεn|k|$ \| S_{k}^{-1}\| \leq M_{\varepsilon} \varepsilon^{n|k|} $with Mε = M0  ×  ···  ×  Mn. Now we can proveAk∼wAkn:=Bk=Bk0⋱Bkn+1$$\begin{equation}A_{k} \stackrel{w}{\sim} A_{k}^{n}:=B_{k}=\left(\begin{array}{lll}B_{k}^{0} & & \\ & \ddots & \\ & & B_{k}^{n+1}\end{array}\right) \end{equation}$$with locally integrable functions B i : Z → R N i × N i $B^{i}: \mathbb{Z} \rightarrow \mathbb{R}^{N_{i} \times N_{i}}$andΣNEDB0=∅,ΣNEDB1=J1,…,ΣNEDBn=Jn,ΣNEDBn+1=∅.$$\begin{equation}\Sigma_{N E D}\left(B^{0}\right)=\emptyset, \Sigma_{N E D}\left(B^{1}\right)=\mathcal{J}_{1}, \ldots, \Sigma_{N E D}\left(B^{n}\right)=\mathcal{J}_{n}, \Sigma_{N E D}\left(B^{n+1}\right)=\emptyset. \end{equation}$$Finally, we show that Ni = dim𝒲i. From the claim above, we note that dim ⁡ B k 0 = dim W 0 ,   dim ⁡ B k 1 ≥ dim W 1 , … , dim ⁡ B k n ≥ dim W n ,   dim ⁡ B k n + 1 = dim W n + 1 $ \dim B^{0}_k=\dim \mathcal{W}_0, ~\dim B^{1}_k\geq \dim \mathcal{W}_1,\ldots, \dim B^{n}_k\geq \dim \mathcal{W}_n, ~\dim B^{n+1}_k=\dim \mathcal{W}_{n+1} $and using Theorem 2.6 this yields dim𝒲0 +···+dim𝒲n+1 = N, so dim ⁡ B k i = dim W i $ \dim B^{i}_k=\dim \mathcal{W}_i $for i = 0, …, n + 1. Now the proof is finished.

Journal

Advances in Nonlinear Analysis – de Gruyter

Published: Jan 1, 2022

Keywords: Dichotomy spectrum; nonuniform exponential dichotomy; reducibility; 37D25; 37B55

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