Shadowing for Nonautonomous Dynamics

Lucas Backes; Davor Dragičević

doi:10.1515/ans-2018-2033

Artikel Open Access

Shadowing for Nonautonomous Dynamics

Lucas Backes und Davor Dragičević

Veröffentlicht/Copyright: 31. Oktober 2018

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Aus der Zeitschrift Advanced Nonlinear Studies Band 19 Heft 2

Abstract

We prove that whenever a sequence of bounded operators (Am)m∈ℤ acting on a Banach space X admits an exponential dichotomy and a sequence of differentiable maps fm:X→X, m∈ℤ, has bounded and Hölder derivatives, the nonautonomous dynamics given by xm+1=Am⁢xm+fm⁢(xm), m∈ℤ, has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman–Hartman theorem.

Keywords: Shadowing; Nonautonomous Systems; Exponential Dichotomies; Nonlinear Perturbations

MSC 2010: 37C50; 34D09; 34D10

1 Introduction

Given a dynamical system f:M→M acting on an arbitrary metric space (M,d), a δ-pseudotrajectory for f is any sequence of points (yn)n∈ℤ⊂M satisfying

d ⁢ ( y n + 1 , f ⁢ ( y n ) ) ≤ δ for every ⁢ n ∈ ℤ .

An important question that naturally arises is to understand whether a pseudotrajectory can be approximated by a real trajectory for f. More precisely, we ask if there exists a point x∈M such that d⁢(yn,fn⁢(x)) is small for every n∈ℤ. If this happens, we say that xshadows the pseudotrajectory (yn)n∈ℤ. Furthermore, if every pseudotrajectory has a shadowing point, we say that the system (f,M) has the shadowing property (see [21, 26] for more precise definitions).

We emphasize that there is a clear motivation for studying shadowing properties. For instance, a pseudotrajectory can be viewed as a result of measurements in a real system (that are usually subjected to round-off error or noise). Then to study the shadowing problem is to try to understand the relationship between the real trajectories and the approximate trajectories obtained by such measurements. If the system has the shadowing property, we have that the numerically obtained trajectories actually reflect the real behavior of trajectories of f. From a more theoretical perspective, the shadowing property has proved to be very fruitful when dealing with problems concerned with topological stability and construction of symbolic dynamics [8].

It is well known that many classes of dynamical systems exhibit the shadowing property. Indeed, Bowen [8] and Anosov [1] proved that uniformly hyperbolic dynamical systems with both discrete and continuous time have the shadowing property. Furthermore, Katok [16] proved that nonuniformly hyperbolic dynamical systems also possess certain types of the shadowing property. More recently, Pilyugin [25, 26] proved that structurally stable systems exhibit the shadowing property. Finally, we recall that it has been noticed that partially hyperbolic dynamical systems have no shadowing property [7].

The original proofs that uniformly hyperbolic dynamical systems have the shadowing property (due to Bowen and Anosov) rely on the existence of invariant stable and unstable manifolds. Later, Palmer [23] and independently Mayer and Sell [20] gave quite ingenious and very simple analytic proofs of the shadowing lemma that do not use the invariant manifold theory. Their approach also inspired versions of the shadowing lemma that deal with maps on Banach spaces (see [9, 15]). In addition, an analytic proof of the shadowing lemma for nonuniformly hyperbolic dynamics was given in [12]. Finally, in the recent paper [6], Bernardes Jr. et al. have developed a shadowing theory for linear operators on an arbitrary Banach space. We refer to [21, 26] for more discussion and further references related to shadowing theory. In particular, we recommend [18] for a nice survey devoted to the study of shadowing in the context of ordinary differential equations.

We emphasize that all the above mentioned results deal with autonomous dynamics and the main objective of the present paper is to develop a shadowing theory for nonautonomous systems acting on an arbitrary Banach space. More precisely, starting with a linear dynamics

(1.1) x m + 1 = A m ⁢ x m , m ∈ ℤ ,

where the sequence (Am)m∈ℤ admits an exponential dichotomy (which represents a version of the notion of hyperbolicity for time-varying dynamics), we prove that a small nonlinear perturbation of (1.1) has the shadowing property. In particular, the results from [6] correspond to a special case when (Am)m∈ℤ in (1.1) is a constant sequence of operators and when linear dynamics is not perturbed. Moreover, we propose a unified approach (inspired by [10]) that allows us to measure the error in the notion of a pseudotrajectory as well as its deviation from a trajectory in a variety of ways that include the concept of standard shadowing (as well as other concepts of shadowing studied in [6] and elsewhere) as a very particular case. We note that our methods are inspired by the above mentioned analytic proofs of the shadowing lemma.

Finally, as a nontrivial application of our results, we give a simple proof of the nonautonomous version of the Grobman–Hartman theorem in the discrete-time setting. We emphasize that the first version of this result goes back to Palmer [22] who considered finite-dimensional dynamics with continuous time. More recent versions are due to Barreira and Valls [5, 4] (see also [2]) who deal with discrete-time dynamics on an arbitrary Banach space that admits a nonuniform exponential dichotomy. Although our version of the nonautonomous Grobman–Hartman theorem works under more restrictive assumptions than those we just mentioned (see Remark 4.3 for a detailed discussion), our proof differs from theirs and is inspired by the classical construction of the conjugacy between Anosov diffeomorphism and its small perturbation (see [17]).

The paper is organized as follows. In Section 2 we introduce the class of sequence spaces that will be used to present a general framework that will unify various types of shadowing. We also recall the classical notion of an exponential dichotomy. Section 3 contains the main results of our paper, while in Section 4 we present some applications.

2 Preliminaries

2.1 Banach Sequence Spaces

In this subsection we present some basic definitions and properties from the theory of Banach sequence spaces. The material is taken from [10, 27], where the reader can also find more details.

Let 𝒮⁢(ℤ) be the set of all sequences 𝐬=(sn)n∈ℤ of real numbers. We say that a linear subspace B⊂𝒮⁢(ℤ) is a normed sequence space (over ℤ) if there exists a norm ∥⋅∥B:B→ℝ0+ such that if 𝐬′=(sn′)n∈ℤ∈B and |sn|≤|sn′| for n∈ℤ, then 𝐬=(sn)n∈ℤ∈B and ∥𝐬∥B≤∥𝐬′∥B. If in addition (B,∥⋅∥B) is complete, we say that B is a Banach sequence space.

Let B be a Banach sequence space over ℤ. We say that B is admissible if:

χ { n } ∈ B and ∥χ{n}∥B>0 for n∈ℤ, where χA denotes the characteristic function of the set A⊂ℤ,
for each 𝐬=(sn)n∈ℤ∈B and m∈ℤ, the sequence 𝐬m=(snm)n∈ℤ, defined by snm=sn+m, belongs to B and ∥𝐬m∥B=∥𝐬∥B.

Note that it follows from the definition that for each admissible Banach space B over ℤ, we have that ∥χ{n}∥B=∥χ{0}∥B for each n∈ℤ. Throughout this paper we will assume for the sake of simplicity that ∥χ{0}∥B=1.

We recall some explicit examples of admissible Banach sequence spaces over ℤ (see [10, 27]).

Example 2.1.

The set l∞={𝐬=(sn)n∈ℤ∈𝒮⁢(ℤ):supn∈ℤ⁡|sn|<∞} is a Banach sequence space when equipped with the norm ∥𝐬∥=supn∈ℤ⁡|sn|.

Example 2.2.

The set c0={𝐬=(sn)n∈ℤ∈𝒮⁢(ℤ):lim|n|→∞⁡|sn|=0} is a Banach sequence space when equipped with the norm ∥⋅∥ from the previous example.

Example 2.3.

For each p∈[1,∞), the set

l p = { 𝐬 = ( s n ) n ∈ ℤ ∈ 𝒮 ⁢ ( ℤ ) : ∑ n ∈ ℤ | s n | p < ∞ }

is a Banach sequence space when equipped with the norm

∥ 𝐬 ∥ = ( ∑ n ∈ ℤ | s n | p ) 1 / p .

Example 2.4 (Orlicz Sequence Spaces).

Let φ:(0,+∞)→(0,+∞] be a nondecreasing nonconstant left-continuous function. We set ψ⁢(t)=∫0tφ⁢(s)⁢𝑑s for t≥0. Moreover, for each 𝐬=(sn)n∈ℤ∈𝒮⁢(ℤ), let Mφ⁢(𝐬)=∑n∈ℤψ⁢(|sn|). Then

B = { 𝐬 ∈ 𝒮 ⁢ ( ℤ ) : M φ ⁢ ( c ⁢ 𝐬 ) ⁢ < + ∞ ⁢ for some ⁢ c > ⁢ 0 }

is a Banach sequence space when equipped with the norm

∥ 𝐬 ∥ = inf ⁡ { c > 0 : M φ ⁢ ( 𝐬 / c ) ≤ 1 } .

2.2 Important Construction

Let us now introduce sequence spaces that will play important role in our arguments. Let X be an arbitrary Banach space and B any Banach sequence space over ℤ with norm ∥⋅∥B. Set

X B := { 𝐱 = ( x n ) n ∈ ℤ ⊂ X : ( ∥ x n ∥ ) n ∈ ℤ ∈ B } .

Finally, for 𝐱=(xn)n∈ℤ∈XB, we define

(2.1) ∥ 𝐱 ∥ B := ∥ ( ∥ x n ∥ ) n ∈ ℤ ∥ B .

Remark 2.5.

We emphasize that in (2.1) we slightly abuse the notation, since the norms on B and XB are denoted in the same way. However, this will cause no confusion, since in the rest of the paper we will deal with spaces XB.

Example 2.6.

Let B=l∞ (see Example 2.1). Then

X B = { 𝐱 = ( x n ) n ∈ ℤ ⊂ X : sup n ∈ ℤ ⁡ ∥ x n ∥ < ∞ } .

The proof of the following result is straightforward (see [10, 27]).

Proposition 2.7.

( X B , ∥ ⋅ ∥ B ) is a Banach space.

2.3 Exponential Dichotomies

We now recall the notion of an exponential dichotomy which goes back to the landmark work of Perron [24]. Let X=(X,∥⋅∥) be an arbitrary Banach space and (Am)m∈ℤ a sequence of bounded linear operators on X such that

(2.2) sup m ∈ ℤ ⁡ ∥ A m ∥ < ∞ .

Set

𝒜 ⁢ ( m , n ) = { A m - 1 ⁢ ⋯ ⁢ A n if m > n , Id if m = n .

We say that the sequence (Am)m∈ℤ admits an exponential dichotomy if

there exists a sequence (Pm)m∈ℤ of projections on X satisfying

(2.3) P m + 1 ⁢ A m = A m ⁢ P m , m ∈ ℤ ,

such that each map Am|Im⁡Pm:ImPm→ImPm+1 is invertible;
there exist C,λ>0 such that

(2.4) ∥ 𝒜 ⁢ ( m , n ) ⁢ P n ∥ ≤ C ⁢ e - λ ⁢ ( m - n ) for m ≥ n

and

(2.5) ∥ 𝒜 ⁢ ( m , n ) ⁢ ( Id - P n ) ∥ ≤ C ⁢ e - λ ⁢ ( n - m ) for m ≤ n ,

where

𝒜 ( m , n ) := ( 𝒜 ( n , m ) | Im ⁡ P m ) - 1 : Im P n → Im P m for m < n .

Let B be an admissible Banach sequence space. We define a bounded linear operator 𝔸:XB→XB by

( 𝔸 ⁢ 𝐱 ) n = A n - 1 ⁢ x n - 1 for n ∈ ℤ and 𝐱 = ( x n ) n ∈ ℤ ∈ X B .

It follows easily from (2.2) that 𝔸 is a well-defined and bounded linear operator on XB.

The following result is only a particular case of the results established by Sasu [27].

Theorem 2.8.

For a sequence (Am)m∈Z of bounded linear operators on X satisfying (2.2) and an admissible Banach sequence space B, the following statements are equivalent:

( A m ) m ∈ ℤ admits an exponential dichotomy,
Id - 𝔸 is an invertible operator on X B .

One can use the previous result to establish the following one (see [14] for example) which tells us that the notion of an exponential dichotomy is robust under small linear perturbations.

Theorem 2.9.

Let B be an admissible Banach sequence space and assume that (Am)m∈Z is a sequence of bounded operators on X that admits an exponential dichotomy and satisfies (2.2). Then there exists c>0 such that for any sequence (Bm)m∈Z of bounded linear operators on X satisfying

sup m ∈ ℤ ⁡ ∥ A m - B m ∥ ≤ c ,

we have that (Bm)m∈Z also admits an exponential dichotomy. Furthermore, there exists K>0 such that ∥(Id-B)-1∥≤K, where the operator B:XB→XB is given by

( 𝔹 ⁢ 𝐱 ) n = B n - 1 ⁢ x n - 1 for n ∈ ℤ and 𝐱 = ( x n ) n ∈ ℤ ∈ X B .

Remark 2.10.

In fact, by carefully inspecting the proof of Theorem 2.8 one can conclude that we can choose constants C,λ>0 in the notion of an exponential dichotomy uniformly over all sequences (Bm)m∈ℤ satisfying the assumptions of Theorem 2.9.

2.4 A Fixed Point Theorem

We will also use the following classical consequence of the Banach fixed point theorem (see [19] for example).

Theorem 2.11.

Let Z be a Banach space and A a differentiable map defined on a neighborhood of 0∈Z. Furthermore, let Γ be a bounded linear operator on Z such that Id-Γ is invertible. Finally, suppose that there exist ρ>0 and κ∈(0,1) such that

for each z ∈ Z satisfying ∥ z ∥ ≤ ρ , ∥(Id-Γ)-1∥⋅∥dz⁢A-Γ∥≤κ,
∥ ( Id - Γ ) - 1 ∥ ⋅ ∥ A ⁢ ( 0 ) ∥ ≤ ( 1 - κ ) ⁢ ρ .

Then A has a unique fixed point in {z∈Z:∥z∥≤ρ}.

3 Main Result

3.1 Setup

Let B be an admissible Banach sequence space, X a Banach space and (Am)m∈ℤ a sequence of bounded linear operators on X that admits an exponential dichotomy and satisfies (2.2). Moreover, let c>0 be a constant given by Theorem 2.9. Finally, let fn:X→X, n∈ℤ be a sequence of differentiable maps such that

(3.1) ∥ d x ⁢ f n ∥ ≤ c for x ∈ X and n ∈ ℤ ,

sup m ∈ ℤ ⁡ ∥ f m ∥ sup < ∞ , where ∥ f m ∥ sup := sup x ∈ X ⁡ ∥ f m ⁢ ( x ) ∥ ,

and for each ε>0, there exists r=r⁢(ε)>0 such that for every x,y∈X satisfying ∥x-y∥≤r,

(3.2) ∥ d x ⁢ f n - d y ⁢ f n ∥ ≤ ε for every ⁢ n ∈ ℤ .

We consider a nonautonomous and nonlinear dynamics defined by the equation

(3.3) x n + 1 = F n ⁢ ( x n ) , n ∈ ℤ ,

where

F n := A n + f n .

We now introduce a notion of a pseudotrajectory associated with system (3.3). Given δ>0, the sequence (yn)n∈ℤ⊂X is said to be an (δ,B)-pseudotrajectory for (3.3) if (yn+1-Fn⁢(yn))n∈ℤ∈XB and

(3.4) ∥ ( y n + 1 - F n ⁢ ( y n ) ) n ∈ ℤ ∥ B ≤ δ .

Remark 3.1.

When B=l∞ (see Example 2.1), condition (3.4) reduces to

sup n ∈ ℤ ⁡ ∥ y n + 1 - F n ⁢ ( y n ) ∥ ≤ δ .

The above requirement represents a usual definition of a pseudotrajectory in the context of smooth dynamics (see [21, 26]).

We say that (3.3) has the B-shadowing property if for every ε>0, there exists δ>0 so that for every (δ,B)-pseudotrajectory (yn)n∈ℤ, there exists a sequence (xn)n∈ℤ satisfying (3.3) and such that (xn-yn)n∈ℤ∈XB, together with

(3.5) ∥ ( x n - y n ) n ∈ ℤ ∥ B ≤ ε .

Moreover, if there exists L>0 such that δ can be chosen as δ=L⁢ε, we say that (3.3) has the B-Lipschitz shadowing property.

Remark 3.2.

In the case when B=l∞, the above definition of shadowing is the classical one [21, 26]. On the other hand, if B=c0 (see Example 2.2), we speak about limit shadowing, while in the case when B=lp (see Example 2.3), we speak about lp-shadowing. Hence, our approach offers a unified treatment of various concepts of shadowing.

Our main objective is to show that under the above assumptions, (3.3) has the B- Lipschitz shadowing property. We begin with some simple auxiliary results.

3.2 Lemmata

Lemma 3.3.

Let δ>0 and assume that (yn)n∈Z is a (δ,B)-pseudotrajectory for (3.3). Furthermore, we define a map A:XB→XB by

( 𝐀 ⁢ ( 𝐱 ) ) n = F n - 1 ⁢ ( y n - 1 + x n - 1 ) - y n for n ∈ ℤ and 𝐱 = ( x n ) n ∈ ℤ ∈ X B .

Then A is a well-defined map.

Proof.

Observe that

( 𝐀 ⁢ ( 𝐱 ) ) n = F n - 1 ⁢ ( y n - 1 + x n - 1 ) - y n

= F n - 1 ⁢ ( y n - 1 + x n - 1 ) - F n - 1 ⁢ ( y n - 1 ) + F n - 1 ⁢ ( y n - 1 ) - y n

= A n - 1 ⁢ x n - 1 + f n - 1 ⁢ ( y n - 1 + x n - 1 ) - f n - 1 ⁢ ( y n - 1 ) + F n - 1 ⁢ ( y n - 1 ) - y n

for every n∈ℤ. Then it follows from (3.1) that

∥ ( 𝐀 ⁢ ( 𝐱 ) ) n ∥ ≤ ( sup m ∈ ℤ ⁡ ∥ A m ∥ ) ⋅ ∥ x n - 1 ∥ + c ⁢ ∥ x n - 1 ∥ + ∥ F n - 1 ⁢ ( y n - 1 ) - y n ∥

for each n∈ℤ and 𝐱=(xn)n∈ℤ∈XB. In a view of (2.2) and (3.4), we conclude that 𝐀⁢(𝐱)∈XB. ∎

Lemma 3.4.

The map A:XB→XB is differentiable and

( d 𝐱 ⁢ 𝐀 ⁢ 𝝃 ) n = A n - 1 ⁢ ξ n - 1 + d x n - 1 + y n - 1 ⁢ f n - 1 ⁢ ξ n - 1 for 𝐱 = ( x n ) n ∈ ℤ , 𝝃 = ( ξ n ) n ∈ ℤ ∈ X B .

Proof.

Let us fix 𝐱=(xn)n∈ℤ∈XB and define a linear operator L:XB→XB by

( L ⁢ 𝝃 ) n = A n - 1 ⁢ ξ n - 1 + d x n - 1 + y n - 1 ⁢ f n - 1 ⁢ ξ n - 1 for 𝝃 = ( ξ n ) n ∈ ℤ ∈ X B .

It follows readily from (2.2) and (3.1) that L is a well-defined and bounded operator. In addition, for 𝐡=(hn)n∈ℤ∈XB, we have that

( 𝐀 ⁢ ( 𝐱 + 𝐡 ) - 𝐀 ⁢ ( 𝐱 ) - L ⁢ 𝐡 ) n = f n - 1 ⁢ ( x n - 1 + y n - 1 + h n - 1 ) - f n - 1 ⁢ ( x n - 1 + y n - 1 ) - d x n - 1 + y n - 1 ⁢ f n - 1 ⁢ h n - 1

= ∫ 0 1 d x n - 1 + y n - 1 + t ⁢ h n - 1 ⁢ f n - 1 ⁢ h n - 1 ⁢ 𝑑 t - ∫ 0 1 d x n - 1 + y n - 1 ⁢ f n - 1 ⁢ h n - 1 ⁢ 𝑑 t .

Take an arbitrary ε>0 and the corresponding r=r⁢(ε)>0 such that (3.2) holds. Also, let 𝐡=(hn)n∈ℤ∈XB be such that ∥𝐡∥B≤r. Hence, ∥t⁢hn∥≤r for every t∈[0,1] and n∈ℤ. Therefore, it follows from (3.2) that

∥ ( 𝐀 ( 𝐱 + 𝐡 ) - 𝐀 ( 𝐱 ) - L 𝐡 ) n ∥ ≤ ∫ 0 1 ∥ d x n - 1 + y n - 1 + t ⁢ h n - 1 f n - 1 h n - 1 - d x n - 1 + y n - 1 f n - 1 h n - 1 ∥ d t ≤ ε ∥ h n - 1 ∥

for each n∈ℤ. Consequently, we have that

∥ 𝐀 ⁢ ( 𝐱 + 𝐡 ) - 𝐀 ⁢ ( 𝐱 ) - L ⁢ 𝐡 ∥ B ≤ ε ⁢ ∥ 𝐡 ∥ B .

Hence, we have proved that

lim 𝐡 → 0 ⁡ ∥ 𝐀 ⁢ ( 𝐱 + 𝐡 ) - 𝐀 ⁢ ( 𝐱 ) - L ⁢ 𝐡 ∥ B ∥ 𝐡 ∥ B = 0 ,

which implies the desired conclusion. ∎

Set Γ:=d𝟎⁢𝐀. It follows from Lemma 3.4 that

( Γ ⁢ 𝝃 ) n = A n - 1 ⁢ ξ n - 1 + d y n - 1 ⁢ f n - 1 ⁢ ξ n - 1 for 𝝃 = ( ξ n ) n ∈ ℤ ∈ X B and n ∈ ℤ .

Lemma 3.5.

We have that Id-Γ is an invertible operator on XB. Moreover, there exists a constant K>0, independent on the pseudotrajectory y, such that

∥ ( Id - Γ ) - 1 ∥ ≤ K .

Proof.

The statement follows directly from Theorems 2.8 and 2.9 together with (3.1) (and the choice of c). ∎

3.3 Main Results

The following is our main result.

Theorem 3.6.

System (3.3) has an B-Lipschitz shadowing property. Furthermore, if ε is sufficiently small, the sequence x=(xn)n∈Z satisfying (3.3) and (3.5) is unique.

Proof.

We wish to apply Theorem 2.11. Let K>0 be such that ∥(Id-Γ)-1∥≤K. As we noted in Lemma 3.5, we can choose K independently on 𝐲. Set r=r⁢(12⁢K) so that (3.2) holds with ε=12⁢K>0. It follows from (3.2) that

∥ ( d 𝐳 𝐀 - Γ ) 𝝃 ) n ∥ ≤ 1 2 ⁢ K ⋅ ∥ ξ n - 1 ∥ for any 𝝃 = ( ξ n ) n ∈ ℤ ∈ X B ,

and thus

∥ d 𝐳 ⁢ 𝐀 - Γ ∥ ≤ 1 2 ⁢ K for 𝐳 ∈ X B such that ∥ 𝐳 ∥ B ≤ r .

We conclude that

(3.6) ∥ ( Id - Γ ) - 1 ∥ ⋅ ∥ d z ⁢ A - Γ ∥ ≤ 1 2

for any 𝐳∈XB satisfying ∥𝐳∥B≤r.

Take now ε>0 such that ε≤r and set L=12⁢K. Thus, for δ=L⁢ε>0, we have that K⁢δ=ε/2 and

(3.7) ∥ ( Id - Γ ) - 1 ∥ ⋅ ∥ A ⁢ ( 0 ) ∥ ≤ K ⁢ δ = ε / 2 .

It follows from (3.6) and (3.7) that the assumptions of Theorem 2.11 are satisfied with κ=1/2 and ρ=ε, and consequently 𝐀 has a unique fixed point 𝐳∈XB such that ∥𝐳∥B≤ε. By setting 𝐱:=𝐲+𝐳, we obtain the desired conclusions. ∎

Remark 3.7.

We stress that Theorem 3.6 (in the particular case when B=l∞) is similar in nature to the main result in [9], with the important distinction that the maps Fn defined globally on X are unbounded.

We have the following simple consequence of Theorem 3.6.

Corollary 3.8 (Expansivity).

Let ε=r⁢(12⁢K)>0 and assume that x=(xn)n∈Z and z=(zn)n∈Z are sequences satisfying (3.3) and

∥ ( x n - z n ) n ∈ ℤ ∥ B ≤ ε .

Then x=z.

Proof.

Let ε>0 be as in the statement of the corollary and take the corresponding δ>0 as in the definition of the B-Lipschitz shadowing property. Obviously, 𝐳 is a (δ,B)-pseudotrajectory for (3.3), which is (ε,B)-shadowed by itself and 𝐱. Hence, the uniqueness part in Theorem 3.6 implies that 𝐱=𝐳, as claimed. ∎

Under certain periodicity assumption for system (3.3), we can also formulate a version of the Anosov closing lemma in our setting.

Corollary 3.9.

Assume that there exists N∈N such that Fn+N=Fn for each n∈N. Then there exists L>0 such that for any ε>0 sufficiently small and for every (L⁢ε,l∞)-pseudotrajectory y=(yn)n∈Z for (3.3) that satisfies yn=yn+N for n∈N, there exists a solution x=(xn)n∈Z of (3.3) such that supn∈Z⁡∥xn-yn∥≤ε and xn=xn+N for n∈N.

Proof.

Applying Theorem 3.6 (for B=l∞) we obtain the existence of a sequence 𝐱=(xn)n∈ℤ solving (3.3) such that supn∈ℤ⁡∥xn-yn∥≤ε. Let us define a new sequence 𝐱′=(xn′)n∈ℤ⊂X by

x n ′ = x n + N for n ∈ ℤ .

It is easy to verify that 𝐱′ solves (3.3) and it obviously satisfies supn∈ℤ⁡∥xn′-yn∥≤ε. Hence, the uniqueness in Theorem 3.6 implies that 𝐱=𝐱′, which immediately yields the desired conclusion. ∎

3.4 Shadowing of Linear Systems

In this subsection we deal with system (3.3) in the particular case when fn=0 for each n∈ℤ, i.e., when Fn=An for every n∈ℤ. Hence, we deal with linear dynamics

(3.8) x n + 1 = A n ⁢ x n , n ∈ ℤ .

Corollary 3.10.

System (3.8) has the B-Lipschitz shadowing property. Furthermore, for each ε>0, the sequence x=(xn)n∈Z satisfying (3.5) and (3.8) is unique.

Proof.

In view of Theorem 3.6, it only remains to establish the uniqueness of 𝐱. Assume that 𝐱=(xn)n∈ℤ and 𝐱′=(xn′)n∈ℤ satisfy (3.8), ∥𝐱-𝐲∥B≤ε and ∥𝐱′-𝐲∥B≤ε. Hence, ∥𝐱-𝐱′∥≤2⁢ε.

On the other hand, since 𝐱 and 𝐱′ satisfy (3.8), we have that

( Id - 𝔸 ) ⁢ ( 𝐱 - 𝐱 ′ ) = 0 ,

and thus Theorem 2.8 implies that 𝐱=𝐱′. ∎

Corollary 3.11.

For each sequence y=(yn)n∈Z⊂X such that

lim | n | → ∞ ⁡ ∥ y n + 1 - A n ⁢ y n ∥ = 0 ,

there exists a unique sequence x=(xn)n∈Z that solves (3.8) and satisfies

(3.9) lim | n | → ∞ ⁡ ∥ x n - y n ∥ = 0 .

Proof.

Take B=c0 (see Example 2.2). We note that 𝐲 is an (L⁢ε,B)-pseudotrajectory for (3.8) for some ε>0 (where L>0 comes from the definition of B-Lipschitz shadowing). By Corollary 3.10, we can find a sequence 𝐱=(xn)n∈ℤ that solves (3.8) and satisfies (xn-yn)n∈ℤ∈B (in fact, we also have that ∥(xn-yn)n∈ℤ∥B=supn∈ℤ⁡∥xn-yn∥≤ε), which immediately yields (3.9).

The uniqueness of 𝐱 can be established by using Theorem 2.8, as in the proof of Corollary 3.10. ∎

Corollary 3.12.

Take 1≤p<∞. For each sequence y=(yn)n∈Z⊂X such that

∑ n ∈ ℤ ∥ y n + 1 - A n ⁢ y n ∥ p < ∞ ,

there exists a unique sequence x=(xn)n∈Z that solves (3.8) and satisfies

∑ n ∈ ℤ ∥ x n - y n ∥ p < ∞ .

Proof.

The proof can be obtain by repeating the arguments in the proof of Corollary 3.11 and by using B=lp (see Example 2.3) instead of B=c0. ∎

Remark 3.13.

In a very particular case when (3.8) is an autonomous system, i.e., (An)n∈ℤ is a constant sequence of operators, Corollary 3.10 applied to B=l∞, together with Corollaries 3.11 and 3.12, gives us [6, Theorem A.].

4 Applications

In this section we present some applications of our main results.

4.1 A Nonautonomous Version of the Grobman–Hartman Theorem

Let (Am)m∈ℤ be a sequence of bounded and invertible linear operators on X as in Section 3.1. Furthermore, suppose that

a := sup m ∈ ℤ ⁡ ∥ A m - 1 ∥ < ∞ .

Let c>0 be given by Theorem 2.9 and fix D,r>0. Associated to these parameters by Theorem 3.6 (applied to B=l∞), consider ε>0 sufficiently small such that r⁢(3⁢ε)<c/2 (see (3.2)) and δ=L⁢ε>0. Moreover, suppose ε is so small that 3⁢ε still satisfies Theorem 3.6. Let (gn)n∈ℤ be a sequence of maps gn:X→X satisfying (3.1), with c/2 instead of c, and (3.2), and such that

∥ g n ∥ sup ≤ δ for each n ∈ ℤ .

We consider the difference equation

(4.1) y n + 1 = G n ⁢ ( y n ) , n ∈ ℤ ,

where Gn:=An+gn. By decreasing c (if necessary), we have that Gn is a homeomorphism for each n∈ℤ. Indeed, if c<a-1, then for each n∈ℤ and y∈X, the map Φy,n:X→X, defined by Φy,n⁢(x)=An-1⁢(y-gn⁢(x)), x∈X, is a contraction on X. This easily implies that Gn is a homeomorphism for each n∈ℤ. We define

𝒢 ⁢ ( m , n ) = { G m - 1 ∘ ⋯ ∘ G n if m > n , Id if m = n , G m - 1 ∘ ⋯ ∘ G n - 1 - 1 if m < n .

Theorem 4.1.

There exists a unique sequence hm:X→X, m∈Z, of homeomorphisms such that for each m∈Z,

(4.2) h m + 1 ∘ G m = A m ∘ h m

and

(4.3) ∥ h m - Id ∥ sup = sup x ∈ X ⁡ ∥ h m ⁢ ( x ) - x ∥ ≤ ε .

Proof.

Fix m∈ℤ and y∈X and define a sequence 𝐲=(yn)n∈ℤ by yn=𝒢⁢(n,m)⁢y for n∈ℤ. Note that 𝐲 is a solution of (4.1). Then

sup n ∈ ℤ ⁡ ∥ y n + 1 - A n ⁢ y n ∥ = sup n ∈ ℤ ⁡ ∥ g n ⁢ ( y n ) ∥ ≤ δ .

Hence, it follows from Corollary 3.10 (applied to the case when B=l∞) that there exists a unique sequence 𝐱=(xn)n∈ℤ such that xn+1=An⁢xn for n∈ℤ and supn∈ℤ⁡∥xn-yn∥≤ε. Set

h m ⁢ ( y ) = h m ⁢ ( y m ) =: x m .

It is easy to verify that (4.2) holds. Furthermore,

∥ h m ⁢ ( y ) - y ∥ = ∥ x m - y m ∥ ≤ ε ,

which yields (4.3). We will now prove that each hm is a homeomorphism. Let us start with the following simple auxiliary result.

Lemma 4.2.

Let x=(xn)n∈Z and x~=(x~n)n∈Z be two sequences such that

x n + 1 = A n ⁢ x n 𝑎𝑛𝑑 x ~ n + 1 = A n ⁢ x ~ n

for every n∈Z. Then, for every ρ>0, there exists N∈N such that if ∥xn-x~n∥≤3⁢ε for every |n|≤N, then we have that ∥x0-x~0∥≤ρ. In addition, an analogous property holds for solutions of (4.1).

Proof.

It follows from (2.3) and (2.4) that

∥ P 0 ⁢ ( x 0 - x ~ 0 ) ∥ = ∥ 𝒜 ⁢ ( 0 , - N ) ⁢ P - N ⁢ ( x - N - x ~ - N ) ∥ ≤ C ⁢ e - λ ⁢ N ⁢ ∥ x - N - x ~ - N ∥ .

Similarly, it follows from (2.3) and (2.5) that

∥ ( Id - P 0 ) ⁢ ( x 0 - x ~ 0 ) ∥ = ∥ 𝒜 ⁢ ( 0 , N ) ⁢ ( Id - P N ) ⁢ ( x N - x ~ N ) ∥ ≤ C ⁢ e - λ ⁢ N ⁢ ∥ x N - x ~ N ∥ .

Hence,

∥ x 0 - x ~ 0 ∥ ≤ C ⁢ e - λ ⁢ N ⁢ ( ∥ x - N - x ~ - N ∥ + ∥ x N - x ~ N ∥ ) ,

and therefore, by choosing N so that 6⁢C⁢e-λ⁢N⁢ε<ρ, we obtain the desired conclusion. As for the nonlinear case, let 𝐲=(yn)n∈ℤ and 𝐲~=(y~n)n∈ℤ be sequences satisfying (4.1) and suppose ∥yn-y~n∥≤3⁢ε for every |n|≤N. Setting zn=yn-y~n, we get that

z n + 1 = G n ⁢ ( y n ) - G n ⁢ ( y ~ n ) = d y ~ n ⁢ G n ⁢ z n + G n ⁢ ( y n ) - G n ⁢ ( y ~ n ) - d y ~ n ⁢ G n ⁢ z n .

Thus,

z n + 1 = ( L n + T n ) ⁢ z n ,

where Ln:=dy~n⁢Gn=An+dy~n⁢gn and

T n := ∫ 0 1 ( d t ⁢ y n + ( 1 - t ) ⁢ y ~ n ⁢ G n - d y ~ n ⁢ G n ) ⁢ 𝑑 t .

Now, using (3.2) we get that ∥Tn∥≤D⁢∥yn-y~n∥r≤D⁢(3⁢ε)r<c/2 for every |n|≤N. Therefore, it follows from Theorem 2.9 that the sequence (Bn)n∈ℤ, given by Bn=Ln+Tn for |n|≤N and Bn=Ln for |n|>N, admits an exponential dichotomy with constants C and λ as in Section 2.3 depending only on (An)n∈ℤ and c (see Remark 2.10). Thus, choosing N so that 6⁢C⁢e-λ⁢N⁢ε<ρ and proceeding as in the linear case, we get the desired result. ∎

Let us now establish the continuity of h0 (the same argument applies for every hm). For ρ>0, take N∈ℕ given by Lemma 4.2. By the continuity of the maps gn, there exists η>0 such that for every y,z∈X satisfying ∥y-z∥<η, we have that

∥ 𝒢 ⁢ ( n , 0 ) ⁢ y - 𝒢 ⁢ ( n , 0 ) ⁢ z ∥ < ε for every | n | ≤ N .

Take y,z∈X satisfying ∥y-z∥<η, and consider yn=𝒢⁢(n,0)⁢y, zn=𝒢⁢(n,0)⁢z, xn=𝒜⁢(n,0)⁢h0⁢(y) and x~n=𝒜⁢(n,0)⁢h0⁢(z) for n∈ℤ. For every n∈ℤ satisfying |n|≤N, we have that

∥ x n - x ~ n ∥ = ∥ 𝒜 ⁢ ( n , 0 ) ⁢ h 0 ⁢ ( y ) - 𝒜 ⁢ ( n , 0 ) ⁢ h 0 ⁢ ( z ) ∥

= ∥ h n ⁢ ( 𝒢 ⁢ ( n , 0 ) ⁢ y ) - h n ⁢ ( 𝒢 ⁢ ( n , 0 ) ⁢ z ) ∥

≤ ∥ h n ⁢ ( 𝒢 ⁢ ( n , 0 ) ⁢ y ) - 𝒢 ⁢ ( n , 0 ) ⁢ y ∥ + ∥ 𝒢 ⁢ ( n , 0 ) ⁢ y - 𝒢 ⁢ ( n , 0 ) ⁢ z ∥ + ∥ 𝒢 ⁢ ( n , 0 ) ⁢ z - h n ⁢ ( 𝒢 ⁢ ( n , 0 ) ⁢ z ) ∥

= ∥ h n ⁢ ( y n ) - y n ∥ + ∥ 𝒢 ⁢ ( n , 0 ) ⁢ y - 𝒢 ⁢ ( n , 0 ) ⁢ z ∥ + ∥ z n - h n ⁢ ( z n ) ∥

≤ ε + ε + ε = 3 ⁢ ε .

Therefore, since (xn)n∈ℤ and (x~n)n∈ℤ are solutions of xn+1=An⁢xn, n∈ℤ, it follows from Lemma 4.2 that ∥h0⁢(y)-h0⁢(z)∥≤ρ proving that h0 is continuous.

We now prove that hm is injective for each m∈ℤ. Suppose that there exist y,z∈X such that hm⁢(y)=hm⁢(z). We define sequences (yn)n∈ℤ, (zn)n∈ℤ and (xn)n∈ℤ, respectively, by yn=𝒢⁢(n,m)⁢y, zn=𝒢⁢(n,m)⁢y and xn=𝒜⁢(n,m)⁢hm⁢(y)=𝒜⁢(n,m)⁢hm⁢(z), n∈ℤ. Then, by the definition of hm, we have that

sup n ∈ ℤ ⁡ ∥ x n - y n ∥ ≤ ε and sup n ∈ ℤ ⁡ ∥ x n - z n ∥ ≤ ε .

In particular,

sup n ∈ ℤ ⁡ ∥ y n - z n ∥ ≤ 2 ⁢ ε .

Then Corollary 3.8 (applied for B=l∞) implies that yn=zn for every n∈ℤ. Consequently, y=ym=zm=z, and thus hm is injective.

Let us now establish the surjectivity of hm. Take x∈X and consider a sequence xn=𝒜⁢(n,m)⁢x. Then

sup n ∈ ℤ ∥ x n + 1 - G n x n ∥ = sup n ∈ ℤ ∥ A n x n - G n x n ∥ = sup n ∈ ℤ ∥ g n ( x n ) ∥ ≤ δ .

Hence, (xn)n∈ℤ is a (δ,l∞)-pseudotrajectory for (4.1). In particular, by Theorem 3.6 applied to (Gn)n∈ℤ, there exists a unique sequence (yn)n∈ℤ satisfying (4.1) and such that supn∈ℤ⁡∥xn-yn∥≤ε. Therefore, hm⁢(ym)=xm=x, proving that hm is surjective.

The proof of the continuity of hm-1 is completely analogous to the proof of the continuity of hm and therefore we omit it.

Finally, it remains to establish the uniqueness of hm. Indeed, suppose that (h~m)m∈ℤ is a sequence satisfying (4.2) and (4.3). Take y∈X, m∈ℤ and consider a sequence yn=𝒢⁢(n,m)⁢y, n∈ℤ. Then

h n + 1 ⁢ ( y n + 1 ) = A n ⁢ h n ⁢ ( y n ) and h ~ n + 1 ⁢ ( y n + 1 ) = A n ⁢ h ~ n ⁢ ( y n ) .

Moreover, ∥hn⁢(yn)-yn∥≤ε and ∥h~n⁢(yn)-yn∥≤ε for every n∈ℤ. Thus, by the uniqueness in Theorem 3.6, it follows that hn⁢(yn)=h~n⁢(yn) for every n∈ℤ. Consequently, hm⁢(y)=h~m⁢(y). Since y was arbitrary, we conclude that hm=h~m and the proof of the theorem is completed. ∎

Remark 4.3.

Theorem 4.1 can be described as a nonautonomous version of the classical Grobman–Hartman theorem [13]. The first result of this type has been established by Palmer [22] for finite-dimensional dynamics with continuous time. Subsequent generalizations for infinite-dimensional dynamics that admit a nonuniform exponential dichotomy are due to Barreira and Valls [4, 5] (see also [2] for a simple proof).

When compared with main results in [4, 5, 2], our Theorem 4.1 works under stronger assumptions that the maps gn are differentiable and that (3.2) holds.

However, our goal was not to refine results from those papers (which in fact seem to be rather optimal) but rather to offer a new approach to the problem of linearization of a nonautonomous dynamics based on the shadowing theory we developed in the previous section. As we mentioned in the introduction, this approach is well known in the context of autonomous dynamics but to the best of our knowledge it has not appeared earlier in the study of nonautonomous dynamics.

Finally, we note that the sufficient conditions under which the conjugacies hm are smooth were discussed for the first time in [11].

4.2 Preservation of Positive Lyapunov Exponents

Using our main results, we can also formulate conditions under which positive Lyapunov exponents associated with linear dynamics remain unchanged under small nonlinear perturbations. For more general results related to the preservation of Lyapunov exponents under perturbations, we refer to [3, Section 7] and the references therein. We continue to use the same notation as in the previous subsection.

Corollary 4.4.

Assume that (yn)n∈Z is a solution of (4.1) such that

λ := lim sup n → ∞ ⁡ 1 n ⁢ log ⁡ ∥ y n ∥ > 0 .

Then there exists a solution (xn)n∈Z of (3.8) such that

(4.4) λ = lim sup n → ∞ ⁡ 1 n ⁢ log ⁡ ∥ x n ∥ .

Proof.

Observe that

sup n ∈ ℤ ⁡ ∥ y n + 1 - A n ⁢ y n ∥ = sup n ∈ ℤ ⁡ ∥ g n ⁢ ( y n ) ∥ ≤ δ .

Hence, it follows from Corollary 3.10 (applied to the case when B=l∞) that there exists a unique sequence 𝐱=(xn)n∈ℤ such that xn+1=An⁢xn for n∈ℤ and supn∈ℤ⁡∥xn-yn∥≤ε. It readily follows that (4.4) holds. ∎

Communicated by Kenneth Palmer

Funding statement: L. Backes was partially supported by a CAPES-Brazil postdoctoral fellowship under Grant No. 88881.120218/2016-01 at the University of Chicago. D. Dragičević was supported in part by the Croatian Science Foundation under the project IP-2014-09-2285 and by the University of Rijeka under the project number 17.15.2.2.01.

Acknowledgements

We would like to express our gratitude to the anonymous referee for useful comments that helped us improve our paper.

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Received: 2018-07-28

Revised: 2018-08-22

Accepted: 2018-09-17

Published Online: 2018-10-31

Published in Print: 2019-05-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

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Shadowing; Nonautonomous Systems; Exponential Dichotomies; Nonlinear Perturbations

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