Nonautonomous equations and almost reducibility sets

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Nonautonomous equations and almost reducibility sets

Luís Barreira

^B

and Claudia Valls

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

Received 12 May 2020, appeared 23 February 2021 Communicated by Mihály Pituk

Abstract. For a nonautonomous differential equation, we consider the almost reducibility property that corresponds to the reduction of the original equation to an autonomous equation via a coordinate change preserving the Lyapunov exponents. In particular, we characterize the class of equations to which a given equation is almost reducible. The proof is based on a characterization of the almost reducibility to an autonomous equation with a diagonal coefficient matrix. We also characterize the notion of almost reducibility for an equation x⁰ = A(t,θ)x depending continuously on a real parameterθ. In particular, we show that the almost reducibility set is always anF_σδ-set and for anyF_σδ-set containing zero we construct a differential equation with that set as its almost reducibility set.

Keywords: almost reducibility, nonautonomous equations.

2020 Mathematics Subject Classification: 37D99.

1 Introduction

We first describe the reducibility property and the type of problems considered in the paper.

Let A(t)and B(t)be q×qmatrices varying continuously with t ≥ 0 and consider the linear equations

x⁰ = _A(_t)_x _and _y⁰ = _B(_t)_y. _(1.1) Let T(t,s)andS(t,s)be the corresponding evolution families such that

T(t,s)x(s) =x(t) and S(t,s)y(s) =y(t)

for any solutions x and y of the equations in (1.1) and for any t,s ≥ 0. We say that the equations areequivalent via a coordinate change U(t)given by invertibleq×qmatrices if

U(t)⁻¹T(t,s)U(s) =S(t,s) _{for all}t,s≥_0. _(1.2)

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

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More generally, one can also consider piecewise continuous functions A(t) and B(t) (see Section2), in which case the evolution familiesT(t,s)andS(t,s)are still continuous in(t,s).

In this paper we consider the class of equations that are equivalent to an autonomous equation. Namely, we say that the equationx⁰ = A(t)x isreducible via a coordinate change U(t) if it is equivalent to some autonomous equationy⁰ = By. Moreover, we say that the equation x⁰ = A(t)xisalmost reducibleif it is equivalent to some autonomous equation via a Lyapunov coordinate changeU(t), that is, a coordinate change satisfying

tlim→_∞

1

t logkU(t)k= lim

t→_∞

1

t logkU(t)⁻¹k=0. (1.3) The Lyapunov coordinate changes are the only coordinate changes that preserve simulta- neously the Lyapunov exponents of all sequences of invertible matrices with a finite Lyapunov exponent. More precisely, for eachv ∈_R^qlet

λ_A(v) = lim

t→_∞

1

t logkT(t, 0)vk

be the Lyapunov exponent associated with the equationx⁰ = A(t)x, with the convention that log 0= −∞. The Lyapunov exponent λ_B(v)for the equation y⁰ = B(t)y is defined similarly.

The former statement on the preservation of the Lyapunov exponents means that a coordinate changeU(t)is a Lyapunov coordinate change if and only if the evolution families of any two equivalent equations as in (1.1) that satisfy (1.2) also satisfy

λ_A(U(0)v) =λ_B(v) for allv∈_R^q.

This causes that the almost reducibility property occurs naturally whenever we want to reduce the original dynamics to a simpler one without changing the asymptotic behavior given by the Lyapunov exponents.

A first notion of reducibility is due to Lyapunov [5] (see [7] for an English translation).

He considered instead bounded coordinate changes with bounded inverses, that is, transfor- mations satisfying

sup

t≥0

kU(t)k<+_∞ and sup

t≥0

kU(t)⁻¹k<+_∞. (1.4) We refer the reader to [4,6,8,9] and the references therein for some early results as well as to the book [3] for a global panorama of the area in 1980. While the coordinate changes satisfying (1.4) are appropriate to study uniform Lyapunov stability (because bounded coordinate changes preserve this type of stability), in order to study nonuniform Lyapunov stability it is crucial to consider Lyapunov coordinate changes as in (1.3).

We first give a characterization of the almost reducibility of an equation to an autonomous equation with a diagonal coefficient matrix (see Theorem2.1).

Theorem 1.1. For an equation x⁰ = A(t)x on R^q such that the Lyapunov exponent λ_A is finite on R^q\ {0}, we have

tlim→_∞

1 t

Z _t

0 trA(s)ds=_inf

∑

q j=1

λ_A(v_j)

with the infimum taken over all bases v₁, . . . ,v_q forR^q if and only if the equation is almost reducible to an equation y⁰ =By with B diagonal.

We shall use this result to characterize the autonomous equations to which a given equation is almost reducible (see Theorem2.2).

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Theorem 1.2. x⁰ = A(t)x is almost reducible to y⁰ = By and y⁰ = Cy if and only if the eigenvalues of B and C, counted with multiplicities and eventually up to a permutation, have the same real parts.

We also characterize completely the notion of almost reducibility for continuous 1-parameter families of linear differential equations. Namely, we consider equations x⁰ = A(t,θ)x depending continuously on a real parameter θ. The almost reducibility set of this equation is the set of all θ ∈ _R for which the equation is almost reducible. We have the following result (see Theorem3.1).

Theorem 1.3. The almost reducibility set of x⁰ = A(t,θ)x is an F_σδ-set.

Finally, we establish a partial converse of Theorem1.3. Namely, we construct a differential equation with given Fσδ-set containing zero as its almost reducibility set (see Theorem4.1).

Theorem 1.4. Given an integer q≥2and an F_σδ-set M containing zero, there exists an equation x⁰ = A(t,θ)x whose almost reducibility set is equal to M. Moreover, given an unbounded nondecreasing functionρ(t)≥0, we may require that

kA(t,θ)k ≤ρ(t)(₁+|θ|) for all t ≥0andθ∈_R.

The proof of Theorem1.4is partly inspired by arguments in [1].

2 The notion of almost reducibility

We introduce the notion of almost reducibility for the class of nonautonomous linear equations and we establish some of its basic properties. In particular, we characterize completely the class of autonomous equations to which a given nonautonomous equation is almost reducible.

Let Mq be the set of allq×qmatrices with real entries and let GLq ⊂ Mq be the subset of all invertible matrices. Consider a piecewise continuous function A: R⁺₀ → M_q. We say that the equation

x⁰ = A(t)x (2.1)

is almost reducible to an equation x⁰ = Bx for some matrix B ∈ M_q if there exist matrices U(t)∈ GLqfort ≥0 satisfying

tlim→_∞

1

t logkU(_t)k= _lim

t→_∞

1

t logkU(_t)⁻¹k=₀ _(2.2) such that

U(t)⁻¹T(t,s)U(s) =e^B⁽^t⁻^s⁾ fort,s ≥_0, _(2.3) where T(t,s)is the evolution family associated with equation (2.1). This means that we have T(t,s)x(s) =x(t)for any solution x = x(t)of the equation x⁰ = A(t)xand all t,s ≥ 0. Then we also say that equation (2.1) is almost reducible. The family (U(t))_t_≥₀ is called a Lyapunov coordinate change.

We start by describing when a nonautonomous equation is almost reducible to an autonomous equation with a diagonal coefficient matrix. The Lyapunov exponent λ: R^q → [−_∞,+_∞]associated with equation (2.1) is defined by

λ(v) = lim

t→_∞

1

t logkT(t, 0)vk,

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with the convention that log 0=−∞. We shall always assume thatλtakes only finite values onR^q\ {0}. It follows from the theory of Lyapunov exponents that these finite values are say λ₁< · · ·<λ_p for some positive integer p≤qand that the sets

E_i =v∈_R^q:λ(v)≤ λ_i

are linear subspaces fori=1, . . . ,p. A basisv₁, . . . ,v_qforR^qis said to benormal(with respect to equation (2.1)) if for eachi=1, . . . ,psome elements of{v₁, . . . ,vq}form a basis forE_i. Theorem 2.1. Let x⁰ = A(t)x be an equation on R^q whose Lyapunov λtakes only finite values on R^q\ {0}. Then

tlim→_∞

1 t

Z _t

0 trA(s)ds=

∑

q j=1

λ(v_j) (2.4)

for some normal basis v₁, . . . ,v_q for R^q if and only if the equation x⁰ = A(t)x is almost reducible to an autonomous equation with a diagonal coefficient matrix, whose entries on the diagonal are then necessarilyλ(v₁), . . . ,λ(v_q), up to a permutation.

Proof. Assume first that (2.4) holds for some normal basis v₁, . . . ,v_q for R^q. Let U(0) be the matrix with columnsv₁, . . . ,vqand for eacht>0, let

U(t) =T(t, 0)U(0)diag e⁻^λ⁽^v¹⁾^t, . . . ,e⁻^λ⁽^v^q⁾^t . Then

U(t)⁻¹T(t,s)U(s) =diag e^λ⁽^v¹⁾⁽^t⁻^s⁾, . . . ,e^λ⁽^v^q⁾⁽^t⁻^s⁾ , that is, property (2.3) holds taking

B=diag λ(v₁), . . . ,λ(v_q).

In order to show that (U(t))_t_≥₀ is a Lyapunov coordinate change, notice that the columns ofU(t)are the vectors

T(t, 0)v₁e⁻^λ⁽^v¹⁾^t, . . . ,T(t, 0)vqe⁻^λ⁽^v^q⁾^t. Since

tlim→_∞

1

t log kT(t, 0)v_ike⁻^λ⁽^vⁱ⁾^t

=0, (2.5)

we obtain

tlim→_∞

1

t logkU(_t)k ≤0.

Now we consider the matrices

U(t)⁻¹=diag e^λ⁽^v¹⁾^t, . . . ,e^λ⁽^v^q⁾^t

(T(t, 0)U(0))⁻¹. We have

(T(t, 0)U(₀))⁻¹ =C(t)_{/ det}(T(t, 0)U(₀))

for some matricesC(t)with(i,j)entry given by(−1)ⁱ⁺^j_∆^ji(t), where∆^ji(t)is the determinant of the matrix obtained fromT(t, 0)U(0)erasing itsjth line andith column. Then

U(t)⁻¹= D(t) ^exp^∑

q

j=1λ(v_j)t

det(T(t, 0)U(₀))^, ^(2.6)

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where

D(t) =diag e⁻^∑^j⁶⁼¹^λ⁽^v^j⁾^t, . . . ,e⁻^∑^j⁶⁼^q^λ⁽^v^j⁾⁽^m⁻¹⁾ C(t). By Liouville’s theorem we have

detT(t, 0) =exp Z _t

0 trA(s)ds (2.7)

and so it follows from (2.4) that

tlim→_∞

1

t log detT(t, 0) =

∑

q j=₁

λ(v_j). Therefore,

tlim→_∞

1

t log exp∑^q_j=1λ(v_j)t

|det(T(t, 0)U(0))| =0. (2.8) The (i,j)entry ofD(t)is given by(−1)ⁱ⁺^j_∆^¯^ji(t), where ¯∆^ji(t)is the determinant of the matrix obtained from T(t, 0)U(0) dividing each kth column by e^λ⁽^v^k⁾^t and then erasing the jth line and theith column. It follows from (2.5) that

tlim→_∞

1

t log|_∆^¯^ji(t)| ≤0 and so lim

t→_∞

1

t logkD(t)k ≤0.

Therefore, by (2.6) and (2.8), we obtain lim

t→_∞

1

t log kU(t)⁻¹k⁻¹=−lim

t→_∞

1

t logkU(t)⁻¹k ≥0, which shows that(U(t))_t_≥₀is a Lyapunov coordinate change.

Now assume that the equationx⁰ = A(t)xis almost reducible to an autonomous equation with a diagonal coefficient matrix, that is,

U(t)⁻¹T(t,s)U(s) =diag e^a¹⁽^t⁻^s⁾, . . . ,e^a^q⁽^t⁻^s⁾

(2.9) for some matricesU(t)∈GL_q, fort≥0, satisfying (2.2) and some numbersa₁, . . . ,a_q∈_R. Let v₁, . . . ,vqbe the columns ofU(0). Then

kU(t)⁻¹T(t, 0)v_ik=e^aⁱ^t.

By (2.2), this implies that the basis v₁, . . . ,vqis normal withλ(v_i) = a_i fori =1, . . . ,q. More- over, again by (2.9), we have

det(U(t)⁻¹)_detT(t, 0)_detU(₀) =e^∑^q^j⁼¹^λ⁽^vⁱ⁾^t. (2.10) Since detU(t)is a sum of products of the entries ofU(t), by (2.2) we have

tlim→_∞

1

t log|detU(t)|=0 and so it follows from (2.7) and (2.10), that identity (2.4) holds.

We use Theorem2.1 to characterize the class of autonomous equations to which an equationx⁰ = A(t)x is almost reducible.

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Theorem 2.2. Let x⁰ = A(t)x be an equation onR^qthat is almost reducible to an equation x⁰ = Bx.

Then the equation x⁰ = A(t)x is almost reducible to an equation x⁰ =Cx if and only if the eigenvalues λ_i(B)andλ_i(C), respectively, of B and C counted with multiplicities, satisfy

Reλ_i(B) =Reλ_i(C) for i=1, . . . ,q, eventually up to a permutation.

Proof. First assume that the equation x⁰ = A(t)x is almost reducible to both x⁰ = Bx and x⁰ =Cx. Consider Lyapunov coordinate changes(U(t))_t_≥₀ _and(V(t))_t_≥₀ _{such that}

U(t)⁻¹T(t,s)U(s) =e^B⁽^t⁻^s⁾ and V(t)⁻¹T(t,s)V(s) =e^C⁽^t⁻^s⁾ fort,s≥0. Then

W(t)⁻¹e^B⁽^t⁻^s⁾W(s) =e^C⁽^t⁻^s⁾

fort,s ≥0, where the matricesW(t) =U(t)⁻¹V(t)form again a Lyapunov coordinate change.

It follows readily from the identity

W(t)⁻¹e^BtW(0) =e^Ct

that the Lyapunov exponentsλ^B and λ^C associated, respectively, with the equations x⁰ = Bx andx⁰ =Cx satisfy

λ^B(W(0)v) =λ^C(v) for allv∈_R^q. (2.11) The values ofλ^B andλ^Care, respectively, Reλ_i(B)and Reλ_i(C)fori=1, . . . ,q, counted with their multiplicities and so it follows readily from (2.11) that

Reλ_i(B) =Reλ_i(C) fori=1, . . . ,q, (2.12) eventually up to a permutation.

Now assume that property (2.12) holds, eventually up to a permutation. Again, the values of the Lyapunov exponentsλ^B andλ^Care, respectively, Reλ_i(B)and Reλ_i(C)fori=1, . . . ,q, counted with their multiplicities. Therefore, condition (2.4) holds for the differential equations x⁰ =Bxandx⁰ =Cx. By Theorem2.1, there exist Lyapunov coordinate changes(U^¯(t))_t_≥₀and (V^¯(t))_t_≥₀such that

U¯(t)⁻¹e^B⁽^t⁻^s⁾U¯(s) =diag Reλ₁(B), . . . , Reλ_q(B)^t⁻^s and

V¯(t)⁻¹e^C⁽^t⁻^s⁾V¯(s) =diag Reλ₁(C), . . . , Reλ_q(C)^t⁻^s fort ≥0. By (2.12), we obtain

U¯(t)⁻¹e^B⁽^t⁻^s⁾U¯(s) =V^¯(t)⁻¹e^C⁽^t⁻^s⁾V¯(s) fort ≥_{0 and so}

W(t)⁻¹T(t,s)W(s) =e^C⁽^t⁻^s⁾ fort,s≥0, where

W(t) =U(t)U^¯(t)V^¯(t)⁻¹

for eacht ≥0. Since(W(t))_t_≥₀is a Lyapunov coordinate change, we conclude thatx⁰ = A(t)x is almost reducible to the equationx⁰ =Cx.

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3 Characterization of almost reducibility sets

In this section we give a characterization of the almost reducibility sets of a differential equationx⁰ = A(t,θ)x depending on a real parameterθ. Namely, we show that any such set is an F_σδ-set. More precisely, letMbe the set of all equations x⁰ = A(t,θ)xsuch that the map

R⁺₀ ×_R3(t,θ)7→ A(t,θ)∈ M_q

is piecewise continuous in t and continuous in θ. We denote by T_θ(t,s) the corresponding evolution family. Thealmost reducibility set of an equationx⁰ = A(t,θ)x is the set of allθ ∈_R for which the equation is almost reducible.

Theorem 3.1. The almost reducibility set of any equation x⁰ = A(t,θ)x inMis an F_σδ-set.

Proof. Let M be the almost reducibility set of the equation. For each n ∈ _N and ε > 0, we define a function g_n,ε: M_q×GL_q×_R→[0,n]by

g_n,ε(B,C,θ) =sup

t≥0

min{n,h_t(B,C,θ)}, where

f_t(B,C,θ) =_maxke^BtCT_θ(_0,t)ke⁻^εt,kT_θ(t, 0)C⁻¹e⁻^Btke⁻^εt . The functiong_n,ε is lower semicontinuous in(B,C,θ)since the functions

ke^BtCT_θ(0,t)ke⁻^εt and kT_θ(t, 0)C⁻¹e⁻^Btke⁻^εt

are continuous (in view of the continuous dependence of a solution on a parameter) and the supremum of any number of continuous functions is lower semicontinuous. Therefore, the set

Dn,ε =g⁻_n,ε¹(−_∞,n/2] is closed for each n∈_Nandε>0.

Lemma 3.2. The equation x⁰ = A(t,θ)x is almost reducible to the equation x⁰ = Bx if and only if there exists C∈ GLqsuch that for eachε>0we have

g_n,ε(B,C,θ)≤n/2 for some n∈_N. (3.1) Proof of the lemma. First assume that the equation x⁰ = A(t)x is almost reducible to the equation x⁰ = Bx. Then there exists a Lyapunov coordinate change (U(t))_t_≥₀ satisfying (2.3).

By property (2.2), for eachε>0 we have

−ε< −¹

t logkU(t)⁻¹k ≤ ¹

t logkU(t)k<ε for any sufficiently largetand so there existsc=c(ε)>0 such that

c⁻¹e⁻^εt <kU(t)⁻¹k⁻¹ ≤ kU(t)k<ce^εt (3.2) for all t≥0. Now takeC=U(0)⁻¹. By (2.3) with s=0 we have

U(t) =T_θ(t, 0)C⁻¹e⁻^Bt and U(t)⁻¹= e^BtCT_θ(_0,t)_.

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Hence, it follows readily from (3.2) that sup

t≥0

ke^BtCT_θ(0,t)ke⁻^εt+kT_θ(t, 0)C⁻¹e⁻^Btke⁻^εt

<_∞ and so property (3.1) holds.

Now assume that there exists C ∈ GL_q satisfying (3.1) for each ε > 0. Then there exists n∈_Nsuch that

max

ke^BtCT_θ(0,t)ke⁻^εt,kT_θ(t, 0)C⁻¹e⁻^Btke⁻^εt ≤n/2 for allt≥0 and so

tlim→_∞

1

t logke^BtCT_θ(0,t)k ≤0 (3.3) and

tlim→_∞

1

t logkT_θ(t, 0)C⁻¹e⁻^Btk ≤_0. _(3.4) Finally, let

U(t) =T_θ(t, 0)C⁻¹e⁻^Bt fort≥0.

Note thatU(0) =C⁻¹. Therefore, e^B⁽^t⁻^s⁾=e^Bte⁻^Bs

=U(t)⁻¹T_θ(t, 0)U(0) U(0)⁻¹T_θ(0,s)U(s)

=U(t)⁻¹T_θ(t,s)U(s)_. Moreover, since

U(t)⁻¹=e^BtCT_θ(_0,t)_, it follows readily from (3.3) and (3.4) that

0≤ lim

t→_∞

1

t log(kU(t)⁻¹k⁻¹)≤ lim

t→_∞

1

t logkU(t)k ≤0 and so condition (2.2) also holds.

By Lemma 3.2, the equation x⁰ = A(t,θ)x is almost reducible if and only if there exist B∈ M_qandC ∈GL_q such that

(B,C,θ)∈D_ε := ^[

n∈_N

D_n,ε for eachε>0. Therefore, the almost reducibility set is

M= ^\

ε>0

π(Dε),

whereπ: M_q×GL_q×_R→_Ris the projection onto the third component. Fork∈ _N_let E_k =(B,C,θ)∈ M_q×GL_q×_R:kBk ≤k, k⁻¹ ≤ |detC| ≤k, |θ| ≤k . Then each setD_n,ε∩E_k is compact and

Dε = ^[

n∈_N

Dn,ε = ^[

n,k∈_N

(Dn,ε∩E_k). Therefore,

M= ^\

ε>0

π(D_ε) = ^\

p∈_N [

n,k∈_N

π(D_n,1/p∩E_k)

and since the map π is continuous, each set π(D_n,1/p∩E_k)is compact. This shows that the almost reducibility setM is anF_σδ-set.

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4 Construction of families of equations

We also construct (as explicitly as possible) a differential equation inM with a given F_σδ-set containing zero as its almost reducibility set.

Theorem 4.1. Given an integer q ≥ 2 and an F_σδ-set M containing zero, there exists an equation x⁰ = A(t,θ)x in M whose almost reducibility set is equal to M. Moreover, given an unbounded nondecreasing functionρ(t)≥0, we may require that

kA(t,θ)k ≤ρ(t)(₁+|θ|) for all t ≥₀andθ∈_R.

Proof. We start by describing some auxiliary notions that will be used in the proof. Given a,b,c,θ ∈R, we consider the 2×2 matrices

B(u,θ) =

aθ c(1−θ) +bθ

−c(1−θ)−bθ −aθ

, (4.1)

whereu = (a,b,c)_and

ν=ν(u,θ) = q

(a²−(b−c)²)θ²−2c(b−c)θ−c². (4.2) Then B(u,θ)has eigenvalues±ν. Givenr,s∈_Rwithrs>0 andd∈_R⁺, we define

a=d(s−r), b=d(2rs−r−s), c=2drs. (4.3) Then

a²−(b−c)²=−4d²rs <0 and one can show thatθ ∈[r,s]if and only if

P(u,θ):= (a²−(b−c)²)θ²−2c(b−c)θ−c² ≥0. (4.4) SinceM is anF_σδ-set containing zero, one can write

R\M = ^[

w∈_N

H^w, where H^w= ^\

i∈_N

U_i^w

for some nonempty open setsU_i^w⊂_R\ {0}satisfyingU_i^w₊₁ ⊂U_i^wfor eachw,i∈_N. Moreover, U_i^w=^S_m_∈_NI_im^w for some nonempty open finite intervals I_im^w ⊂_R\ {0}with the property that eachθ ∈U_i^w belongs to at most two intervals I_im^w (for eachw,i∈_N).

We still need an additional decomposition. For each intervalI_im^w = (_α,β), we consider the sequence (c_l)_l_∈_Zdefined recursively as follows. Takec₀= (α+β)/2. For each l∈_{N, let}

c_2l = ^c^2l⁻²+β

2 , c₋_2l = ^c⁻^2l⁺²+α 2 and

c_2l₋₁ = ^c^2l⁻²+c_2l

2 , c₋_2l+₁= ^c⁻^2l⁺²+c₋_2l

2 .

We define J_iml^w = [c_l,c_l+2]forl∈_Zand so I_im^w = ^[

l∈_Z

J_iml^w .

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Note that each point θ ∈ U_i^w belongs to at most three intervals J_iml^w (for each w,i,m ∈ _N).

Moreover, given θ ∈ I_im^w, there exists l = l(θ) ∈ _Z with θ ∈ J_iml^w such that θ is at least at a distance|J_iml^w |/6 from each endpoint of J_iml^w (where|I|denotes the length of the interval I).

Now let ι: N → _N³×_Z be a bijection. Writing J_iml^w = [r,s] and η = ι⁻¹(w,i,m,l), we consider the uniqued=d(η)∈ _R⁺such that

max

θ∈_R P(u(η),θ) =d²rs(r−s)² = ¹

w, (4.5)

withu(η) = (a,b,c)given by (4.3). Then P(u(η),θ)≥ ⁵

9d²rs(r−s)²= ⁵

9w forθ ∈

r+ ^s−r

6 ,s− ^s−r 6

. (4.6)

Consider the functionσ(t) = min{ρ(t),t}fort ≥0. Moreover, consider a strictly increas- ing sequence of positive integers(`_j)_j_∈_Nsuch that`₁ =1,

`_3j₋₂

`_3j₋₁

j−1 i

∑

=1

σ(`_3i₋₁)< ¹

j, `_3j₋₁

`_3j < ¹

j, `_3j₊₁ =₂`_3j−`_3j₋₁ _(4.7) and

σ(`_3j₋₁)≥2κku(j)k (4.8)

for allj∈_{N, where}κ>0 is fixed constant such that

^{a b}

c d

≤κk(a,b,c,d)k for any a,b,c,d∈_R.

Finally, let∆j = [`_j,`_j₊₁)for each j∈_Nand define

A(t,θ) =











B(u(j),θ) ift ∈_∆_3j for somej∈_N,

−B(u(j),θ) ift ∈_∆_3j₋₁ for somej∈_N, Id ift ∈_∆_3j₋₂ for somej∈_N.

By (4.1) together with (4.8), we obtain

kA(t,θ)k ≤ kB(u(j),θ)k ≤κku(j)k

≤σ(`_3j₋₁)(1+|θ|)

≤σ(t)(1+|θ|)≤ρ(t)(1+|θ|). Lemma 4.2. x⁰ = A(t,θ)x is not almost irreducible forθ ∈_R\M.

Proof of the lemma. Take w ∈ _N such that θ ∈ U_i^w for all i ∈ _N. For each i ∈ _N there exists m ∈ _N such that θ ∈ I_im^w. Moreover, let l = l(θ) ∈ _Z be the integer introduced before (4.5) and write ι⁻¹(w,i,m,l) = r_i. For each j ∈ _∆_3r_i₋₁∪_∆_3r_i the matrices ±B(u(3r_i),θ) have real eigenvalues. Denoting their (common) top eigenvalue by ν_i, it follows readily from (4.2) and (4.4) together with (4.5) and (4.6) that

1 2√

w ≤ν_i ≤ √¹ w.

Denoting by T_θ(t,s)the evolution family associated with the equation x⁰ = A(t,θ)x, we have T_θ(`_3r_i₋₁, 0) =Id and so

T_θ(`_3r_i_{, 0}) =T_θ(`_3r_i_,`_3r_i₋₁)_.

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Therefore,

kTθ(`_3r_i, 0)k=kTθ(`_3r_i,`_3r_i₋₁)k

=e^(`³^ri^−`³^ri⁻¹⁾^B⁽^u⁽^3rⁱ⁾^,θ⁾

≥e^νⁱ^(`³^ri^−`³^ri⁻¹⁾

≥exp

`_3r_i−`_3r_i₋₁ 2√

w

which in view of (4.7) gives

ilim→_∞

1

`_3r_i ^logkT_θ(`_3r_i, 0)k ≥ lim

i→_∞

1 2√

w

1− `_3r_i₋₁

`_3r_i

= ¹

2√

w >0. (4.9) Now we assume that the equation x⁰ = A(t,θ)x is almost reducible to an equationx⁰ = Bx.

Then there exist matricesU(t)satisfying (2.2) and (2.3). Since T_θ(`_3r_i₋₁, 0) =Id, we have e^B^`³^ri⁻¹ =U(`_3r_i₋₁)⁻¹U(0)

and

e⁻^B^`³^ri⁻¹ =U(0)⁻¹U(`_3r_i₋₁)

for alli∈N. Therefore, for each ε>0 there existsc₀ =c₀(ε)>0 such that max

ke^B^`³^ri⁻¹k,ke⁻^B^`³^ri⁻¹k ≤c0e^ε^`³^ri⁻¹

for alli ∈ _{N. Since} `_3r_i₋₁ → _∞ wheni → _∞andε is arbitrary, all eigenvalues ofB have real part equal to 0 and so

ke^Btk ≤c₁(1+|t|) for somec₁>0 and any t≥0.

On the other hand, by (2.3) we have

T_θ(t, 0) =U(t)e^BtU(0)⁻¹ and so

kT_θ(`_3r_i, 0)k ≤ kU(0)⁻¹k · kU(`_3r_i)k · ke^B^`³^rik

≤ c₁(1+|`_3r_i|)kU(0)⁻¹k · kU(`_3r_i)k. Finally, taking into account thatU(t)satisfies (2.2) we obtain

ilim→_∞

1

`_3r_i ^logkT_θ(`_3r_i, 0)k ≤0,

which contradicts (4.9). This shows that the equationx⁰ = A(t,θ)xis not almost reducible.

Lemma 4.3. x⁰ = _A(_t,θ)x is almost reducible forθ ∈ M.

Proof of the lemma. Sinceθ 6∈H^wfor everyw∈_N,θ belongs to at most finitely many setsU_i^w, i ∈ _N _(because U_i^w₊₁ ⊂ U_i^w for each w,i ∈ N) and since each element of U_i^w belongs to at most two intervals I_im^w with m ∈ _N and to at most three closed intervals J_iml^w with l ∈ _{Z, we} conclude thatθ belongs to finitely many closed intervalsJ_iml^w withi,m∈_Nandl∈ _Zfor each w ∈ N. This implies that for each w ∈ _N there exists N = N_w ∈ _N such that for η ≥ N

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with ι(η) = (w_η,i_η,m_η,l_η)we have θ 6∈ J_i^w^η

ηmηlη and so also P(u(η),θ) < 0 whenever w_η ≤ w.

In particular, forη≥ Nwithw_η ≤ wwe have ν=iν¯ with ¯ν∈_Rand so ke^B⁽^u⁽^η⁾^,θ⁾^tk ≤1+2kB(u(η),θ)k · |t|

(see for example [2, p. 65]). For the values ofη≥N withw_η >w, in view of (4.5) we have P(u(η),θ)≤ ¹

w_η ≤ ¹ w+1. Takew∈_{N. If}η≥ N, then

kT_θ(t,`_3η₋₁)k ≤e^B⁽^u⁽^η⁾^,θ⁾⁽^t^−`^3η⁻¹⁾

≤ ₁+_2σ(`_3η₋₁)(₁+|θ|)(t−`_3η₋₁)_exp

t−`_3η₋₁

√w+1

≤ 1+2σ(`_3η₋₁)(1+|θ|)t exp

t−`_3η₋₁

√w+1

(4.10)

fort ∈_∆_3η₋₁∪_∆_3η. Now take

t∈_∆_3η₋₁∪_∆_3η∪_∆_3η₊₁ withη∈_N. SinceA(t,θ) =Id fort ∈_∆_3η₋₂and

T_θ(t,`_3N₋₁) =T_θ(t,`_3η₋₁)

η−1 i

∏

=N

T_θ(`_3i₊₁,`_3i₋₁), using (4.10) we obtain

kT_θ(t,`_3N₋₁)k ≤ kT_θ(t,`_3η₋₁)k

η−1 i

∏

=N

kT_θ(`_3i₊₁,`_3i₋₁)k

≤ 1+2σ(`_3η₋₁)(1+|θ|)t

η−1 i

∏

=N

1+2σ(`_3i₋₁)(1+|θ|)`_3i₊₁

×exp 1

√w+₁ (t−`_3η₋₁) +

η−1 i

∑

=N

(`_3i₊₁−`_3i₋₁)

!!

≤ 1+2σ(`_3η₋₁)(1+|θ|)t

η−1

∏

i=1

1+2σ(`_3i₋₁)(1+|θ|)`_3i₊₁

×exp

t−`_3η₋₁+`_3η₋₂−`_3N₋₁

√w+1

≤ 1+2σ(`_3η₋₁)(1+|θ|)t

η−1

∏

i=1

1+2σ(`_3i₋₁)(1+|θ|)`_3i₊₁exp t

√w+1. Then

tlim→_∞

1

t logkT_θ(t,`_3N₋₁)k ≤ √ ¹

w+1 +lim

t→_∞

1

t log 1+2σ(`_3η₋₁)(1+|θ|)t +_lim

t→_∞

1 t

η−1

∑

i=1

log 1+_2σ(`_3i₋₁)(₁+|θ|)`_3i₊₁_.

(4.11)

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Since

σ(`_3η₋₁) =min{ρ(`_3η₋₁),`_3η₋₁} ≤`_3η₋₁ ≤t, we obtain

tlim→_∞

1

t log 1+2σ(`_3η₋₁)(1+|θ|)t

=0. (4.12)

Moreover, since log(1+x)≤x for allx≥ 0, it follows from (4.7) that 1

t

η−1 i

∑

=1

log 1+2σ(`_3i₋₁)(1+|θ|)`_3i₊₁≤ ²

`_3η₋₁

η−1 i

∑

=1

σ(`_3i₋₁)(1+|θ|)`_3i₊₁

≤ ²(1+|θ|)`_3η₋₂

`_3η₋₁

η−1 i

∑

=1

σ(`_3i₋₁)

≤ ²(1+|θ|)

η .

Therefore,

tlim→_∞

1 t

η−1 i

∑

=1

log 1+2σ(`_3i₋₁)(1+|θ|)`_3i₊₁ =0 (4.13) since η→_∞whent→∞. By (4.12) and (4.13), it follows from (4.11) that

tlim→_∞

1

t logkT_θ(t,`_3N₋₁)k ≤ √ ¹ w+1 for any w∈ _Nand so

tlim→_∞

1

nlogkT_θ(t, 0)k ≤0.

One can also show that

tlim→_∞

1

nlogkT_θ(0,t)k ≤0

interchangingB(u,θ)with−B(u,θ)in the definition of A(t,θ). This implies that lim

t→_∞

1

t logkT_θ(t, 0)k ≥ lim

t→_∞

1

t log kT_θ(0,t)k⁻¹

=−lim

t→_∞

1

t logkT_θ(0,t)k ≥0 and so

lim

t→_∞

1

t logkT(t, 0)^±¹k=_0.

ForUθ(t) =Tθ(t, 0)we have

U_θ(t)⁻¹T_θ(t, 0)U_θ(t) =T_θ(0,t)T_θ(t, 0) =Id.

So, identity (2.3) holds with B = 0. This shows that the differential equationx⁰ = A(t,θ)x is almost reducible.

In order to construct an equationx⁰ = A^˜(t,θ)x on R^q with almost reducibility set M for q>2, it suffices to take

A˜(t,θ) =diag(A(t,θ), 0). This concludes the proof of the theorem.

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Acknowledgment

This research was partially supported by FCT/Portugal through UID/MAT/04459/2019.

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