Nonuniform dichotomy spectrum and reducibility for nonautonomous difference equations

1 Introduction

Let A_k ∈ ℝ^N × N, k ∈ ℤ, be a sequence of invertible matrices. In this paper, we consider the following nonautonomous linear difference equations

where x_k ∈ ℝ^N, k ∈ ℤ. Let Φ:ℤ  ×  ℤ → ℝ^N × N, (k, l)↦Φ(k, l), denote the evolution operator of (1.1), i.e.,

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 ∈ ℝ^N.

An invariant projector of (1.1) is defined to be a function P:ℤ → ℝ^N × N of projections P_k, k ∈ ℤ, such that for each P_k the following property holds

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

and

where Q_l = Id − P_l 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

and

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

admits a nonuniform exponential dichotomy, but does not admit an exponential dichotomy. In fact, we have

with Pl=1000 . Therefore (1.4) holds with

Analogous arguments applied to the second equation yield the estimate (1.5). Moreover, when both k and l are even, we obtain the equality

which means that the nonuniform part ε^l = e^2al 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.

2 Nonuniform dichotomy spectrum

Consider the weighted system

where γ ∈ ℝ⁺ = (0, ∞). One can easily see that

is its evolution operator. If for some γ ∈ ℝ⁺, (2.1) admits a nonuniform exponential dichotomy with projector P_k and constants K ≥ 1, 0 < α < 1 and ε ≥ 1, then P_k is also invariant for (1.1), that is

and the dichotomy estimates of (2.1) are equivalent to

and

Let us define for γ ∈ ρ_NED(A)

and

where ε is the constant in (2.2)-(2.3). Note that 𝒮_γ and 𝒰_γ depend on ε=ε(γ) for each γ ∈ ρ_NED(A). If in addition γ ∈ ρ_ED(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 𝒮_γ and 𝒰_γ are invariant vector bundles of (1.1), here we say that a nonempty set W ⊂ Z×RN is an invariant vector bundle of (1.1) if (a) it is invariant, i.e., (l,ξ)∈W⇒(k,Φ(k,l)ξ)∈W for all k ∈ ℤ; and (b) for every l ∈ ℤ the fiber W(l)={ξ∈RN:(l,ξ)∈W} is a linear subspace of ℝ^N.

The next lemma gives the relationship between 𝒮_γ, 𝒰_γ 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.

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.

Proof. Recall that the resolvent set ρ_NED(A) is open and therefore Σ_NED(A) is the disjoint union of closed intervals. Next we will show that Σ_NED(A) consists of at most N intervals. Indeed, if Σ_NED(A) contains N + 1 components, then one can choose a collection of points ζ₁, …, ζ_N in ρ_NED(A) such that ζ₁<··· < ζ_N and each of the intervals (0, ζ₁), (ζ₁, ζ₂), …, (ζ_{N − 1}, ζ_N), (ζ_N, ∞) has nonempty intersection with the spectrum Σ_NED(A). Now alternative II of Lemma 2.5 implies

and therefore either dimSζ1=0ordimSζN=N or both. Without loss of generality, dimSζN=N , i.e., SζN=Z×RN . Assume that

admits a nonuniform exponential dichotomy with invariant projector P ≡ Id, then

also admits for every ζ>ζ_N a nonuniform exponential dichotomy with the same projector. We conclude (ζ_N, ∞) ⊂ ρ_NED(A), which is a contradiction. This proves the alternatives for Σ_NED(A).

Due to Lemma 2.5, the sets 𝒲₀, …, 𝒲_n+1 are invariant vector bundles. To prove now that dim 𝒲₁ ≥ 1, …, dim 𝒲_n ≥ 1 for n ≥ 1, let us assume that dim 𝒲₁ = 0, i.e., Uγ0∩Sγ1=Z×{0} . If (0, b₁] is a spectral interval this implies that Sγ1=Z×{0} . Then the projector of the nonuniform exponential dichotomy of

is 0 and then we get the contradiction (0, γ₁) ⊂ ρ_NED(A). If [a₁, b₁] is a spectral interval then [γ₀, γ₁]∩Σ_NED(A)≠∅ and by alternative II of Lemma 2.5 we get a contradiction. Therefore dim 𝒲₁ ≥ 1 and analogously dim𝒲_n ≥ 1. Furthermore for n ≥ 3 and i = 2, …, n  −  1 one has [γ_{i − 1}, γ_i]∩Σ_NED(A)≠∅ and again alternative II of Lemma 2.5 yields dim𝒲_i ≥ 1.

For i < j we have Wi ⊂ Sγi and Wj ⊂ Uγj−1 ⊂ Uγi and with Lemma 2.5 this gives Wi∩Wj ⊂ Sγi∩Uγi=Z×{0} , so Wi∩Wj=Z×{0} for i≠j.

To show that W0⊕⋯⊕Wn+1=Z×RN , recall the monotonicity relations Sγ0 ⊂ ⋯ ⊂ Sγn , Uγ0⊃⋯⊃Uγn , and the identity Sγ⊕Uγ=Z×RN for γ ∈ ℝ⁺. Therefore Z×RN=W0+Uγ0 . Now we have

Doing the same for Uγ1 , we get

and mathematical induction yields Z×RN=W0+⋯+Wn+1 . To finish the proof, let γ˜0,…,γ˜n∈ρNED(A) be given with the properties (2.4), (2.5) and (2.6). Then alternative I of Lemma 2.5 implies

and therefore the invariant vector bundles 𝒲₀, …, 𝒲_n+1 are independent of the choice of γ₀, …, γ_n in (2.4), (2.5) and (2.6).

Proof. Assume that (2.7) holds. Let γ > a and 0<α:=aγ<1 , then estimate (2.7) implies

Therefore (1.1) admits a nonuniform exponential dichotomy with invariant projector P = Id. It follows that γ ∈ ρ_NED(A) and also similarly for 0<γ<1a , hence ΣNED(A) ⊂ [1a,a] .

Proof. Necessity. It is easy to know that Σ_NED(A) is bounded from Lemma 2.8. Now we prove that Σ_NED(A)≠∅. It is easy to verify that Sγ=imP=Z×RN and Uγ=kerP=Z×{0} for γ>a. Set

Clearly, γ∗∈[1a,a] . In addition, we have γ* ∈ Σ_NED(A). Otherwise, by using Lemma 2.3, there exists a neighborhood J of γ* such that J ⊂ ρNED(A) and for any γ<γ* of J we have Sγ=Sγ∗=Z×RN , which is a contradiction to the definition of γ*. So Σ_NED(A)≠∅, which means that

Let γ₀ ∈ (0, a₁), γ_i ∈ (b_i, a_i+1) for i = 1, …, n  −  1 and γ_n ∈ (b_n, ∞). Clearly, from the proof of Lemma 2.8, we have Uγn=Z×{0} for γ_n>b_n and Sγ0=Z×{0} for 0<γ₀<a₁. Therefore, W0=Wn+1=Z×{0} , and then it follows from Theorem 2.6 that W1⊕⋯⊕Wn=Z×RN .

Sufficiency. Let γ₀ ∈ (0, a₁), γ_i ∈ (b_i, a_i+1) for i = 1, …, n  −  1 and γ_n ∈ (b_n, ∞). Clearly, from the proof of Lemma 2.8, we have Sγn=Z×RN for γ_n>b_n and Uγ0=Z×RN for 0 <γ₀<a₁. This means that the invariant projectors associated to γ₀ and γ_n are P_l = 0 and P_l = Id respectively.

Hence, there exist K₀ > 1, 0<α₀ < 1 and ε₀ > 1 such that

and there exist K_n > 1, 0 < α_n < 1 and ε_n > 1 such that

Now taking K = max{K₀, K_n}, ε = max{ε₀, ε_n} and a=max{a0γ0,anγn} , then we have

which shows that the evolution operator of (1.1) is nonuniformly exponentially bounded.

From Proposition 1, we know Σ_NED(A) ⊂ Σ_ED(A). Finally in this section, we present an example to illustrate that Σ_NED(A)≠Σ_ED(A) can occur.

Proof. The evolution operator of (2.8) is given by

For each γ ∈ ℝ⁺ the evolution operator of

is given by

For any γ ∈ (e^{( − ω+5a)}, +∞), it follows from (2.10) that

which implies that the equation (2.9) admits a nonuniform exponential dichotomy with invariant projector P = Id, by setting

Thus,

For any γ˜∈(0,e−ω−5a) , it follows from (2.10) that

which implies that (2.9) admits a nonuniform exponential dichotomy with P = 0, by taking

Thus,

It follows from (2.12) and (2.14) that

which implies that

Next we show that

To do this, we first prove that γ₁ = e ^{− ω+5a} ∈ Σ_NED(A). The evolution operator of the system

is given as

It is easy to see that there do not exist K, α>0 and ε>0 such that

or

Therefore γ₁ = e ^{− ω+5a} ∈ Σ_NED(A). In a similar manner, we can prove γ₂ = e ^{− ω − 5a} ∈ Σ_NED(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}] ⊂ Σ_NED(A) and therefore [e ^{− ω − 5a}, e ^{− ω+5a}] = Σ_NED(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 Σ_ED(A) = ℝ⁺.

3 Reducibility

In 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].

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]

We remark that in the setting of an exponential dichotomy the function k↦Sk−1 in Lemma 3.2 is bounded. However, in the setting of a nonuniform exponential dichotomy, Sk−1 can be unbounded, because ∣Φ(k, k)P_k∣ ≤ Kε^k for k = 0. In fact, we can see that

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).

Before stating the main results of this section, we prove several preliminary lemmas.

Proof. Let n∈Z be arbitrary but fixed. Note that the rank of the projector P_n is independent of n∈Z (see [9, page 1100]), then there exists a nondegenerate matrix T ∈ ℝ^N × N such that

with N1=dimimP˜ and N2=dimkerP˜ . Define

Then

On the other hand, we have

It follows from (3.6) and (3.7) that (1.4)-(1.5) can be rewritten in the form (3.4)-(3.5).

Proof. Suppose that S_k is weakly non-degenerate, which means that there exists M = M(ε) > 0 such that ∣S_k∣ ≤ Mε^∣k∣ and ∥Sk−1∥≤Mε|k| and such that Ak∼wBk . Let X_k = S_kY_k. It is easy to see that Y_k 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

A similar argument shows that

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.