2 Nonuniform spectrum

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Evolution families and nonuniform spectrum

Luis Barreira

^B

and Claudia Valls

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

Received 2 March 2016, appeared 11 July 2016 Communicated by Michal Feˇckan

Abstract. We give a complete description of all possible forms of the nonuniform spectrum for an evolution family on a Banach space. Moreover, for each form we provide an explicit example of a nonautonomous differential equation on l²(N)whose evolution family has that spectrum. As an application, we show that the asymptotic behavior persists under sufficiently small nonlinear perturbations, in the sense that the lower and upper Lyapunov exponents of the nonlinear dynamics are in the same connected component of the nonuniform spectrum.

Keywords: nonuniform hyperbolicity, spectrum, Lyapunov regularity.

2010 Mathematics Subject Classification: 37D99.

1 Introduction

For an evolution family on a Banach space, we give a complete description of all possible forms of the nonuniform spectrum. This notion of spectrum is inspired on the one introduced by Sacker and Sell in [12] in terms of uniform exponential dichotomies. Instead, we consider nonuniform exponential dichotomies with an arbitrarily small nonuniform part, for which the conditional stability, although exponential, need not be uniformly exponential on the initial time. We emphasize that these exponential dichotomies are very common in the context of ergodic theory—in strong contrast, the notion of uniform exponential dichotomy is much more restrictive. In particular, almost all trajectories with nonzero Lyapunov exponents of a measure-preserving flow give rise to a linear variational equation admitting a nonuniform exponential dichotomy with an arbitrarily small nonuniform part. Our results can also be considered a contribution to the theory of nonuniform hyperbolicity, which is an important tool in the study of stochastic behavior. We refer the reader to [1] for a detailed exposition of the theory, which goes back to landmark works of Oseledets [8] and particularly Pesin [9].

Given an evolution familyT(t,s)of linear operators acting on a Banach space, itsnonuni- form spectrumis the set Σ of all numbers a ∈ _Rsuch that the evolution family e⁻^a⁽^t⁻^s⁾T(t,s) does not admit a nonuniform exponential dichotomy with an arbitrarily small nonuniform part. Our main aim is to describe the structure of the nonuniform spectrum (see Theorem2.2):

BCorresponding author. Email: barreira@math.tecnico.ulisboa.pt

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Main Theorem. The nonuniform spectrumΣis either∅,R, a finite union of disjoint closed intervals (possibly unbounded), or there exists numbers

b₁ ≥a₁>b₂≥ a₂ >b₃ ≥a₃>· · · such that

Σ= I₁∪

[∞ n=2

[a_n,b_n] or Σ= I₁∪

[∞ n=2

[a_n,b_n]∪(−_∞,a_∞], where I₁ = [a₁,b₁]or I₁= [a₁,+_∞), respectively if an → −_∞or an→ a_∞.

Moreover, we describe how the nonuniform spectrum relates to certain invariant subspaces (see Theorems 2.7and 2.8). In particular, we show that each trajectory of the evolution family has lower and upper Lyapunov exponents inside the same connected component of the nonuniform spectrum. For related work we refer the reader to [3,6,13].

In addition, the asymptotic behavior persists under sufficiently small nonlinear perturbations, in the sense that the lower and upper Lyapunov exponents of the nonlinear dynamics belong to the same connected component of the nonuniform spectrum (see Theorem 2.10).

More precisely, consider a nonzero global solutionx(t)of the nonlinear equation x(t) =T(t,s)x(s) +

Z _t

s T(t,τ)f(τ,x(τ))dτ such that

a_p≤lim inf

t→+_∞

1

t logkx(t)k ≤lim sup

t→+_∞

1

t logkx(t)k ≤supΣ and

t→+lim_∞ Z _t+1

t e^δτkf(τ,x(τ))k

kx(τ)k ^dτ=0 for someδ >0. Then there existsi∈ {_{1, . . . ,}p}_{such that}

a_i ≤_{lim inf}

t→+_∞

1

t logkx(t)k ≤_{lim sup}

t→+_∞

1

t logkx(t)k ≤b_i.

A related result was established by Coppel in [4] for perturbations of a linear differential equation with constant coefficients. Corresponding results for perturbations of autonomous delay equations were established by Pituk [10,11] (for values in a finite-dimensional space) and by Matsui, Matsunaga and Murakami [7] (for values in a Banach space).

Finally, for each possible form of the nonuniform spectrumΣ we provide an explicit example of an evolution family onl²(_N)having that spectrum (see Section3).

2 Nonuniform spectrum

2.1 Preliminaries

Let B(X) be the set of all bounded linear operators acting on a Banach space X. A family T(t,s), fort,s ∈_Rwitht ≥s, of linear operators in B(X)is called anevolution familyif:

1. T(t,t) =Id fort∈_R;

2. T(t,s)T(s,τ) =T(t,τ)_fort,s,τ∈_R_witht≥ s≥_τ.

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We say that an evolution family T(t,τ) admits a nonuniform exponential dichotomy with an arbitrarily small nonuniform partor simply anonuniform dichotomyif:

1. there exist projectionsP_t: X→Xfort∈_Rwith dim KerP_t<+_∞satisfying

T(t,s)P_s =P_tT(t,s) _(2.1) fort≥ ssuch that the map

T(t,s)|KerP_s: KerP_s→KerP_t is invertible;

2. there existλ>0 and for eachε>0 a constantD= D(ε)>0 such that

kT(t,s)P_sk ≤De⁻^λ⁽^t⁻^s⁾⁺^ε^|^s^| fort≥ s (2.2) and

kT(t,s)Q_sk ≤De⁻^λ⁽^s⁻^t⁾⁺^ε^|^s^| fort ≤s, (2.3) where Q_t=Id−P_t and

T(t,s) = (T(s,t)|KerPt)⁻¹: KerPt→KerPs

fort< s.

The sets ImPt and ImQt are called, respectively, stableandunstable spacesof the nonuniform dichotomy. We note that the hypothesis that the unstable spaces are finite-dimensional already appeared for example in [5,13].

Proposition 2.1. For each t∈_{R, we have} ImP_t=

v∈ X: sup

s≥t

kT(s,t)vk< +_∞

andImQ_t consists of all vectors v∈ X for which there exists a function x: (−_∞,t] → X such that x(t) =v, x(t₁) =T(t₁,t₂)x(t₂)for t≥t₁ ≥t₂ andsup_s_≤_tkx(s)k<+_∞.

Proof. By (2.2) we have

sup

s≥t

kT(s,t)vk<+_∞ (2.4)

forv∈ImP_t. On the other hand, ifv∈ Xsatisfies (2.4), then it follows from (2.2) that sup

s≥t

kT(s,t)Qtvk<+_∞. (2.5) By (2.3), fors≥ twe have

kQtvk ≤De⁻^λ⁽^s⁻^t⁾⁺^ε^|^s^|kT(s,t)Qtvk. WheneverQ_tv6=0, takingε<λwe obtain

sup

s≥t

kT(s,t)Qtvk= +_∞, which contradicts to (2.5). Hence,Q_tv =_{0 and}v∈_ImP_t.

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Now take a vectorv ∈ImQ_t and consider the function x: (−_∞,t]→Xdefined byx(s) = T(s,t)v for s ≤ t. Then x(t₁) = T(t₁,t2)x(t2) for t ≥ t₁ ≥ t2 and it follows from (2.3) that sup_s_≤_tkx(s)k < +∞. On the other hand, there exists no v ∈ ImP_t\ {0}for which there is a functionx: (−_∞,t]→ Xas in the proposition. Indeed, it follows from (2.1) and (2.2) that

kvk=kT(t,s)P_sx(s)k ≤De⁻^λ⁽^t⁻^s⁾⁺^ε^|^s^|kx(s)k fors ≤t. Takingε<λyields that sup_s_≤_tkx(s)k= +_∞.

Thenonuniform spectrumof an evolution familyT(t,s)is the setΣof all numbersa∈_Rsuch that the evolution familyT_a(t,s) =e⁻^a⁽^t⁻^s⁾T(t,s)does not admit a nonuniform dichotomy. For eacha∈_Randt∈_{R, let}

S_a(t) =

v ∈X: sup

s≥t

e⁻^a⁽^s⁻^t⁾kT(s,t)vk <+_∞

and letUa(t)be the set of all vectorsv ∈ X for which there exists a functionx: (−_∞,t] → X such thatx(t) =v, x(t₁) =T(t₁,t₂)x(t₂)fort≥ t₁ ≥t₂and

sup

s≤t

e⁻^a⁽^s⁻^t⁾kx(s)k <+_∞. We note that ifa<b, then

S_a(t)⊂S_b(t) and U_b(t)⊂U_a(t) fort ∈R. By Proposition2.1, if a∈_R\_{Σ, then}

X=S_a(t)⊕U_a(t) fort ∈_R and the dimensions dimS_a(t)and dimU_a(t)are independent of t.

2.2 Main result

The following theorem is our main result. It describes all possible forms of the nonuniform spectrum.

Theorem 2.2. For an evolution family T(_t,_s) on a Banach space, one of the following alternatives holds:

1. Σ=_∅;

2. Σ=_R;

3. Σis a finite union of disjoint closed intervals (possibly unbounded);

4. Σ= I₁∪^S^∞_n₌₂[an,bn], where I₁ = [a₁,b₁]or I₁ = [a₁,+_∞), for some numbers

b₁≥ a₁> b₂ ≥a₂>b₃≥ a₃ >· · · (2.6) withlim_n→+_∞a_n=−_∞;

5. Σ= I₁∪^S^∞_n₌₂[a_n,b_n]∪(−_∞,a_∞], where I₁ = [a₁,b₁]or I₁ = [a₁,+_∞), for some numbers as in(2.6)with a_∞ =limn→+_∞an.

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Proof. We first establish some auxiliary results.

Lemma 2.3. The set Σ ⊂ _Ris closed and for each a ∈ _R\_Σ we have S_a(t) = S_b(t)and U_a(t) = U_b(t)for all t∈_Rand all b in some open neighborhood of a.

Proof of the lemma. Given a∈ _R\Σ, there exist projections P_t for t∈ _Rsatisfying (2.1), a constant λ>0 and for eachε>0 a constant D= D(ε)>0 such that

ke⁻^a⁽^t⁻^s⁾T(t,s)Psk ≤De⁻^λ⁽^t⁻^s⁾⁺^ε^|^s^| fort ≥sand

ke⁻^a⁽^t⁻^s⁾T(t,s)Q_sk ≤De⁻^λ⁽^s⁻^t⁾⁺^ε^|^s^| fort ≤s. Therefore, for each b∈_R,

ke⁻^b⁽^t⁻^s⁾T(t,s)Psk ≤De⁻⁽^λ⁻^a⁺^b⁾⁽^t⁻^s⁾⁺^ε^|^s^| fort ≥sand

ke⁻^b⁽^t⁻^s⁾T(t,s)Q_sk ≤De⁻⁽^λ⁺^a⁻^b⁾⁽^s⁻^t⁾⁺^ε^|^s^|

for t ≤ s. Hence, b ∈ _R\_Σ _whenever |a−b| < λ and it follows from Proposition 2.1 that S_b(t) =Sa(t)andU_b(t) =Ua(t)fort∈ _R.

Lemma 2.4. Take a₁,a₂ ∈ _R\_Σ with a₁ < a₂. Then [a₁,a₂]∩_Σ 6= _∅if and only ifdimU_a₁(t) >

dimUa₂(t).

Proof of the lemma. Assume that dimU_a₁(t) = dimU_a₂(t). Then U_a₁(t) = U_a₂(t) andS_a₁(t) = Sa₂(t)fort∈R. Hence, by Proposition2.1, there exist projections Pt fort∈_Rsatisfying (2.1), constants λ₁,λ₂ > 0 and for eachε > 0 constants D₁ = D₁(ε),D₂ = D₂(ε) > 0 such that for i=1, 2 we have

ke⁻^aⁱ⁽^t⁻^s⁾T(t,s)Psk ≤D_ie⁻^λⁱ⁽^t⁻^s⁾⁺^ε^|^s^| fort ≥s (2.7) and

ke⁻^aⁱ⁽^t⁻^s⁾T(t,s)Qsk ≤D_ie⁻^λⁱ⁽^s⁻^t⁾⁺^ε^|^s^| fort ≤s. (2.8) For each a∈[a₁,a₂], by (2.7) we obtain

ke⁻^a⁽^t⁻^s⁾T(t,s)P_sk ≤D₁e⁻^λ¹⁽^t⁻^s⁾⁺^ε^|^s^| fort≥s and similarly, by (2.8),

ke⁻^a⁽^t⁻^s⁾T(t,s)Q_sk ≤D₂e⁻^λ²⁽^s⁻^t⁾⁺^ε^|^s^| fort≤s.

Takingλ=min{λ₁,λ₂}andD=max{D₁,D₂}yields that[a₁,a₂]⊂_R\_Σ. For the converse, assume that dimUa₁(t)>dimUa2(t)and let

b=inf

a∈_R\_Σ: dimU_a(t) =dimU_a₂(t) .

Since dimU_a₁(t) > dimU_a₂(t), it follows from Lemma 2.3 that a₁ < b < a₂. Now assume that b 6∈ Σ. Then either dimU_b(t) = _dimU_a₂(t)_{or dim}U_b(t) 6= _dimU_a₂(t). In the first case, by Lemma 2.3, there exists ε > 0 such that dimU_b⁰(t) = dimUa₂(t) and b⁰ ∈ _R\_Σfor b⁰ ∈ (b−ε,b]. But this contradicts to the definition of b. In the second case, again by Lemma 2.3, there exists ε > 0 such that dimU_b⁰(t) 6= dimU_a₂(t)_andb⁰ ∈ _R\_Σfor b⁰ ∈ [b,b+ε)_{, which} again contradicts to the definition of b. Hence,b∈ _Σand[a₁,a₂]∩_Σ6=_∅.

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Lemma 2.5. For each c∈/_{Σ, the set} _Σ∩[c,+_∞)is the union of finitely many closed intervals.

Proof of the lemma. Let

d=dimUc(t) =dim KerPt,

where Pt are the projections associated to the nonuniform dichotomy of the evolution family e⁻^c⁽^t⁻^s⁾T(t,s). We assume thatΣ∩[c,+_∞)has at leastd+2 connected componentsI_i = [α_i,β_i], fori=1, . . . ,d+2, where

α₁≤ β₁ <α₂≤ β₂ <· · ·<α_d+2≤ β_d+2 ≤+_∞.

Fori=1, . . . ,d+1, takec_i ∈(β_i,α_i+1). It follows from Lemma2.4that d >dimUc₁(t)>dimUc₂(t)>· · · >dimUc_d+1(t), which is impossible.

Now we assume thatΣis not given by one of the first three alternatives in the theorem and take c₁ ∈/ Σ. By Lemma 2.5, the setΣ∩[c₁,+_∞)is the union of finitely many disjoint closed intervals, say I₁, . . . ,I_k. We note that Σ∩(−_∞,c₁) 6= ∅, since otherwise Σ = I₁∪ · · · ∪I_k, which contradicts to our assumption. Moreover, there exists c2 < c₁ such that c2 ∈/ _Σ and (c₂,c₁)∩_Σ6=∅. Otherwise,(−_∞,c₁)∩_Σ= (−_∞,a]for somea< c₁and thus,

Σ= (−_∞,a]∪I₁· · · ∪I_k,

which again contradicts to our assumption. Proceeding inductively, we obtain a decreasing sequence(cn)_n_∈_N⊂_Rsuch that

cn∈/_Σ and (cn+₁,cn)∩_Σ6=_∅

forn ∈ _N. Now either lim_n→+_∞c_n = −_∞or lim_n→+_∞c_n = a_∞ for some a_∞ ∈ _R. In the first case, it follows from Lemma2.5 thatΣis given by alternative 4. In the second case, it follows from Lemma2.5that

(a_∞,∞)∩_Σ= I₁∪

[∞ n=2

[a_n,b_n],

where I₁ = [a₁,b₁]or I₁ = [a₁,+_∞), for some sequences (an)_n_∈_N and(bn)_n_∈_N as in (2.6) with a_∞ = lim_n→+_∞a_n. Again by Lemma2.5, we have (−_∞,a_∞]⊂ _Σ and soΣis given by the last alternative.

The finite-dimensional case is simpler.

Theorem 2.6. For an evolution family on a finite-dimensional space, the nonuniform spectrum is given by one of the first three alternatives in Theorem2.2.

Proof. Assume that the ambient space has dimension d. We will show that Σ is the union of at most d+1 disjoint closed intervals. This implies that Σ is never given by the last two alternatives in Theorem2.2.

Assume that Σ has at least d+2 connected components. Then there exist numbers c₁, . . . ,c_d+1 ∈ _R\_Σ such that c_i < c_i+1 and (c_i,c_i+1)∩_Σ 6= _∅ for i = 1, . . . ,d. It follows from Lemma2.4that

d ≥dimU_c₁(t)>dimU_c₂(t)>. . .>dimU_c_d₊₁(t), which is impossible.

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2.3 Further properties

In this section we assume that Σ is neither ∅ nor R. Let (c_k)_k ⊂ _R be a finite or infinite sequence such thatc_k ∈ (b_k₊₁,a_k)for eachk, with the numbersa_k andb_k as in (2.6) and define

E_k(s) =S_c_k(s)∩U_c_k₊₁(s), k =1, 2, . . . Moreover, whenΣ∩_R⁺is bounded, takec₀ >b₁ and define

E0(s) =Sc0(s)∩Uc₁(s).

By Lemma2.4, the subspaces E_k(s)are independent of the numbersc_k.

Theorem 2.7. Assume thatΣis neither∅norR. For each k =1, 2, . . ., s∈ _Rand v∈ E_k(s)\ {0}, we have "

lim inf

t→+_∞

1

t logkT(t,s)vk, lim sup

t→+_∞

1

t logkT(t,s)vk

#

⊂[a_k+₁,b_k+₁]. WhenΣ∩_R⁺is bounded, this statement also holds for k=0.

Proof. Sincec_k ∈/ Σ, the evolution familye⁻^c^k⁽^t⁻^s⁾T(t,s)admits a nonuniform dichotomy and so there exist projections P_t fort ∈ _R satisfying (2.1), a constant λ > 0 and for each ε > 0 a constant D= D(ε)>0 such that

kT(t,s)P_sk ≤De⁽^c^k⁻^λ⁾⁽^t⁻^s⁾⁺^ε^|^s^| fort≥s (2.9) and

kT(t,s)Q_sk ≤De⁻⁽^λ⁺^c^k⁾⁽^s⁻^t⁾⁺^ε^|^s^| fort≤s,

where Qt = Id−Pt. By Proposition 2.1, we have ImPt = Sc_k(t) for t ∈ R. Hence, each v∈E_k(s)belongs to ImP_sand so, by (2.9),

lim sup

t→+_∞

1

t logkT(t,s)vk ≤c_k−λ< c_k. Lettingc_k &b_k+1, we obtain

lim sup

t→+_∞

1

t logkT(t,s)vk ≤b_k+1.

Similarly, sincec_k+₁ ∈/Σ, there exist projections P_t⁰ fort ∈_R satisfying (2.1), a constantµ>0 and for eachε>0 a constantD= D(ε)>0 such that

kT(t,s)P_s⁰k ≤De⁽^c^k⁺¹⁻^µ⁾⁽^t⁻^s⁾⁺^ε^|^s^| fort≥s and

kT(t,s)Q⁰_sk ≤De⁻⁽^µ⁺^c^k⁺¹⁾⁽^s⁻^t⁾⁺^ε^|^s^| fort≤s, (2.10) where Q⁰_t = Id−P_t⁰. By Proposition2.1, we have ImQ⁰_t = Uc_k+1(t) for t ∈ R. Hence, each v∈E_k(s)belongs to ImQ⁰_sand so, by (2.10),

kvk ≤De⁻⁽^µ⁺^c^k⁺¹⁾⁽^t⁻^s⁾⁺^ε^|^t^|kT(t,s)vk fort ≥s.

Takingε sufficiently small, we obtain lim inf

t→+_∞

1

t logkT(t,s)vk ≥µ+c_k₊₁−ε>c_k₊₁

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and lettingc_k+₁ %a_k+₁ yield that lim inf

t→+_∞

1

t logkT(t,s)vk ≥a_k+1. This completes the proof of the theorem.

A similar argument yields a corresponding statement for negative time.

Theorem 2.8. Assume thatΣis neither∅norR. For each k=1, 2, . . ., s ∈_Rand v∈E_k(s)\ {0}, there exists a function x: (−_∞,s] → X such that x(s) = v, x(t₁) = T(t₁,t₂)x(t₂)for s ≥ t₁ ≥ t₂ and "

lim inf

t→−_∞

1

t logkx(t)k, lim sup

t→−_∞

1

t logkx(t)k

#

⊂ [a_k+1,b_k+1]. WhenΣ∩_R⁺is bounded, this statement also holds for k =0.

The following example illustrates Theorems2.7and2.8.

Example 2.9. Consider the evolution familyT(t,s)obtained from the nonautonomous linear equationx⁰ = A(t)xwith

A(t) =

1 0 0 3t²

.

For eacha> 1, the evolution familyT_a(t,s) =e⁻^a⁽^t⁻^s⁾T(t,s)admits a nonuniform dichotomy with projections Pt(x,y) = (x, 0)(see Example3.1 below for details). On the other hand, for a < 1 the evolution family T_a(t,s)admits a nonuniform dichotomy with projections P_t = 0.

Clearly,T₁(t,s)does not admit a nonuniform dichotomy and soΣ={1}. Now takec₁<1<c₀ (which corresponds to takea₁=b₁=1). Then

E₀(t) =S_c₀(t)∩U_c₁(t) = (_R× {0})∩_R²=_R× {0}

and by Theorems2.7and2.8, fors∈_Randv= (x, 0)∈_R× {0}with x6=0, we have

t→±lim_∞ 1

t logkT(t,s)vk=1.

2.4 Nonlinear perturbations

It turns out that the asymptotic behavior described in Theorem2.7persists under sufficiently small nonlinear perturbations. Given an evolution family T(t,s) on a Banach space X, we consider the nonlinear equation

x(t) =T(t,s)x(s) +

Z _t

s T(t,τ)f(τ,x(τ))dτ (2.11) for some continuous map f: R×X → X. Repeating arguments in the proof of Theorem 6 in [2] we obtain the following result.

Theorem 2.10. For an evolution family T(t,s)on a Banach space such thatΣis neither∅norR, let x(t)be a nonzero global solution of equation(2.11)such that

a_p≤lim inf

t→+_∞

1

t→+_∞

1

t logkx(t)k ≤supΣ

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for some integer p and

t→+lim_∞ Z _t₊₁

t e^δτkf(τ,x(τ))k

kx(τ)k ^dτ=0 for some δ>0. Then there exists i ∈ {1, . . . ,p}such that

a_i ≤lim inf

t→+_∞

1

t→+_∞

1

t logkx(t)k ≤b_i, with the convention that b₁= +_∞when I₁= [a₁,+_∞).

3 Examples

In this section we provide explicit examples of all possible forms of the nonuniform spectrum Σ given by Theorem 2.2. Let X = `²(_N) be a separable infinite-dimensional Hilbert space with the orthonormal basis{e₁,e₂, . . .}.

Example 3.1. Consider the evolution familyT(t,s)onX given by T(t,s)e_n=

(e^t³⁻^s³e₁, n=1, e^s³⁻^t³en, n≥2.

It is obtained from the linear equation x⁰ = A(t)x, where A(t)e₁ =3t²e₁ and A(t)e_n =−3t²e_n forn≥2.

We claim thatΣ= ∅. We first consider the evolution familyT¹(t,s) = e^t³⁻^s³ onR. Given a∈_Randλ>0, consider the functiong: R→_Rgiven by

g(t) =−at+t³−λt.

There existsC>0 such thatgis increasing on the intervals(−_∞,−C)and(C,+_∞). Hence, e⁻^a⁽^t⁻^s⁾⁺^t³⁻^s³⁺^λ⁽^s⁻^t⁾ =e^g⁽^t⁾⁻^g⁽^s⁾≤1

whenevert ≤s<−CorC< t≤s. This implies that there existsD>0 such that e⁻^a⁽^t⁻^s⁾⁺^t³⁻^s³⁺^λ⁽^s⁻^t⁾ ≤D

fort ≤sand so

(T¹)_a(t,s)≤De⁻^λ⁽^s⁻^t⁾

for t ≤ s. Hence, (T¹)_a(t,s) = e⁻^a⁽^t⁻^s⁾T¹(t,s)admits a nonuniform dichotomy with projections P_t =0. Now we consider the evolution familyT²(t,s) =e^s³⁻^t³. Proceeding as above, one can show that(T²)_a(t,s)admits a nonuniform dichotomy with projections P_t =Id. Therefore, Ta(t,s)admits a nonuniform dichotomy with projections Pt given by Pte₁ = 0 and Pten = en

forn≥2.

Example 3.2. Consider the evolution familyT(t,s)onX given by T(t,s)e_n =

(e^ct^cos^t⁻^cs^cos^s⁻^c^sint⁺^c^sinse₁, n=_1, e^s³⁻^t³e_n, n≥2,

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where c > 0. It is obtained from the linear equation x⁰ = A(t)x, where A(t)e₁ = −ctsinte₁ andA(t)en=−3t²en forn≥2.

We claim thatΣ=R. For this it is sufficient to prove that the nonuniform spectrum of the evolution family

T¹(t,s) =e^ct^cos^t⁻^cs^cos^s⁻^c^sin^t⁺^c^sins

is R. Take a ∈ _R and assume that the evolution family (T¹)_a(t,s) admits a nonuniform dichotomy with projections P_t. There are two possibilities: either P_t = _{Id for all} t ∈ _R _or Pt = 0 for all t ∈ R. In the first case, there exist λ > 0 and for each ε > 0 a constant D= D(ε)>0 such that

e⁻^a⁽^t⁻^s⁾T¹(t,s)≤De⁻^λ⁽^t⁻^s⁾⁺^ε^|^s^| fort ≥s.

In particular, fort=2lπ ands = (2l−1)πwithl∈ N, we obtain e⁽^λ⁻^a⁺^c⁾^π⁺^2cs≤ De^εs,

which is impossible for ε < 2c. In the second case, there exist λ > 0 and for each ε > _{0 a} constantD=D(ε)>0 such that

e⁻^a⁽^t⁻^s⁾T¹(t,s)≤De⁻^λ⁽^s⁻^t⁾⁺^ε^|^s^| fort ≤s.

Takingt=2lπands= (2l+1)πwithl∈_N, we obtain e⁽^a⁻^c⁺^λ⁾^π⁺^2cs≤ De^εs,

which again is impossible for ε < 2c. In other words, for each a ∈ _R the evolution family (T¹)_a(t,s)does not admit a nonuniform dichotomy. Thus,Σ=_R.

Example 3.3. Take numbers

b₁ ≥a₁>b₂≥ a₂ >b₃ ≥a₃>· · · >b_k ≥a_k

for some integerk ≥1. For each j∈ {1, . . . ,k}, letφ_j: R→_R be a smooth function such that φ_j(t) =a_j fort ≤ −1 andφ_j(t) =b_j fort ≥1. We consider a linear equationx⁰ = A(t)x onX, where A(t)e_j =a_j(t)e_j for each j, taking

a_j(t) =







φ_j(t) + ¹

2√

1+tsint+√

1+tcost, t ≥0, φ_j(t)− ¹

2√

1−tsint+√

1−tcost, t <₀

for 1≤ j≤ k anda_j(t) =−3t² for j> k. The corresponding evolution family T(t,s)satisfies T(t,s)e_j = T^j(t,s)e_j for each j, where

T^j(t,s) =











e^b^j⁽^t⁻^s⁾⁺

√1+tsint−√

1+ssins, t,s≥0, e^b^j^t⁻^a^j^s⁺

√1+tsint−√

1+|s|sins, t ≥0,s<0, e^a^j⁽^t⁻^s⁾⁺

√

1+|t|sint−√

1+|s|sins, t,s<0

(3.1)

for 1≤ j≤ kandT^j(t,s) =e^t³⁻^s³ for j> k.

We claim that for each j∈ {1, . . . ,k}anda∈/[a_j,b_j], the evolution family(T^j)_a(t,s)admits a nonuniform dichotomy. Takea >b_j. Sincea_j ≤ b_j, we have

e⁻^a⁽^t⁻^s⁾T^j(t,s)≤e⁻⁽^a⁻^b^j⁾⁽^t⁻^s⁾⁺

√

1+|t|+√

1+|s|

(3.2)

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fort ≥s. Moreover, since

p1+|t|

|t| →0 when|t| →+_∞, givenδ>0, there exists D=D(δ)>0 such that

e

√

1+|t| ≤ De^δ^|^t^| fort ∈_R.

Hence, it follows from (3.2) that

e⁻^a⁽^t⁻^s⁾T^j(t,s)≤D²e⁻⁽^a⁻^b^j⁾⁽^t⁻^s⁾⁺^δ^|^t^|+^δ^|^s^|

≤D²e⁻⁽^a⁻^b^j⁻^δ⁾⁽^t⁻^s⁾⁺^2δ^|^s^|

for t ≥ s. Sincea−b_j > 0 and δ is arbitrary, this shows that(T^j)_a(t,s)admits a nonuniform dichotomy with projectionsP_t =Id. Similarly, fora< a_j andt ≤s, we have

e⁻â⁽^t⁻^s⁾T^j(t,s)≤ D²e⁽â^j⁻â⁻^δ⁾⁽^t⁻^s⁾⁺^2δ^|^s^|.

Hence, (T^j)_a(t,s) admits a nonuniform dichotomy with projections P_t = 0. We also show that for each j ∈ {1, . . . ,k} and a ∈ [a_j,b_j], the evolution family (T^j)_a(t,s) does not admit a nonuniform dichotomy. Sinceb_j−a≥0, the first branch of

(T^j)_a(t,s) =











e⁽^b^j⁻^a⁾⁽^t⁻^s⁾⁺

√1+tsint−√

1+ssins, t,s≥0, e⁽^b^j⁻â⁾^t⁻⁽â^j⁻â⁾^s⁺

√1+tsint−√

1+|s|sins, t≥0,s<0, e⁽^a^j⁻^a⁾⁽^t⁻^s⁾⁺

√

1+|t|sint−√

1+|s|sins, t,s<0

precludes the existence of a nonuniform dichotomy with projections P_t =Id. Moreover, since a_j−a ≤ 0, the third branch precludes the existence of a nonuniform dichotomy with projections Pt =0. We conclude that for each j∈ {1, . . . ,k}, the evolution operator(T^j)_a(t,s)admits a nonuniform dichotomy if and only ifa∈/[a_j,b_j]. On the other hand, for eachj> kanda∈_R, the evolution family (T^j)_a(t,s)admits a nonuniform dichotomy with projections P_t =_Id.

Finally, we show that Σ = ^S^k_j₌₁[a_j,b_j]_{. Take} a ∈ _R\^S^k_j₌₁[a_j,b_j]. From what is proved, it follows that for a > b₁ the evolution family Ta(t,s)admits a nonuniform dichotomy with projectionsP_t = Id. Moreover, fora < a_k it admits a nonuniform dichotomy with projections P_tgiven by

Pte_j=0 for 1≤ j≤k and Pte_j =e_j forj>k.

Finally, take j ∈ {1, . . . ,k−1} such that a_j > a > b_j+1. Then T_a(t,s) admits a nonuniform dichotomy with projectionsPtgiven byPte_i =0 for 1≤i≤ jandPte_i =e_i fori> j. Therefore, Σ⊂ ^S^k_j₌₁[a_j,b_j]. Conversely, takea ∈[a_j,b_j]for some j=1, . . . ,kand assume thata∈/_{Σ. Since} Ta(t,s)admits a nonuniform dichotomy, the same happens to(T^j)_a(t,s), but this is impossible since a∈[a_j,b_j].

A similar construction can be effected for the case when the spectrum has unbounded connected components.

Example 3.4. Take numbersanandbnas in (2.6) with lim_j→+_∞a_j =−∞. We consider the evolution familyT(t,s)given byT(t,s)e_j = T^j(t,s)e_jforj∈_NwithT^j(t,s)as in (3.1). Proceeding as in Example 3.3, one can show that for each a > b₁ the evolution family T_a(t,s) _{admits a}

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nonuniform dichotomy with projections P_t = Id. Moreover, for a ∈ (b_j+1,a_j)with j ∈ _N it admits a nonuniform dichotomy with projectionsPt given by

P_te_i =0 for 1≤i≤ j and P_te_i =e_i fori≥ j+1.

Finally, in a similar manner to that in Example3.3, we have[a_j,b_j]⊂_Σfor each j∈ _Nand so Σ=^S^∞_n₌₁[_a_n_,_b_n]. A similar construction can be effected for the case when I1 = [_a₁_,+_∞)_. Example 3.5. Take numbersa_nandb_n as in (2.6) with lim_j→+_∞a_j = a_∞ ∈ R. For eachn∈ _N, letφ_n:R→_Rbe a smooth function such thatφ_n(t) =a_nfort≤ −1 andφ_n(t) =b_nfort ≥1.

We consider the linear equationx⁰ = A(t)x onX, where A(t)e_j =a_j(t)e_j and a_j(t) =







φ_j(t) + ¹

2√

1+tsint+√

1+tcost, t ≥0, φ_j(t)− ¹

2√

1−tsint+√

1−tcost, t <0

forj∈N. The corresponding evolution family T(t,s)satisfiesT(t,s)e_j = T^j(t,s)e_j, for j∈ _N, with T^j(t,s)as in (3.1). Proceeding as in Example3.3, one can show that for each a > b₁ the evolution familyT_a(t,s)admits a nonuniform dichotomy with projectionsP_t =Id. Moreover, fora∈(b_j+1,a_j)with j∈_Nit admits a nonuniform dichotomy with projections P_t given by

P_te_i =0 for 1≤i≤ j and P_te_i =e_i fori≥ j+1.

As in Example3.3, we have[a_j,b_j] ⊂_Σ for eachj∈ N. Finally, by Lemma2.5, (−_∞,a_∞]⊂ _Σ and soΣ= ^S^∞_n₌₁[a_n,b_n]∪(−_∞,a_∞]. Again, a similar construction can be effected for the case whenI₁ = [a₁,+_∞).

Acknowledgment

This research was supported by FCT/Portugal through UID/MAT/04459/2013.

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