Spectra and Normal Forms (SpringerBriefs in Mathematics) 3031518969, 9783031518966

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SpringerBriefs in Mathematics Luís Barreira · Claudia Valls

Spectra and Normal Forms

SpringerBriefs in Mathematics Series Editors Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Palle Jorgensen, Iowa, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, Canada Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York, USA George Yin, Detroit, USA Ping Zhang, Kalamazoo, USA

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Luís Barreira • Claudia Valls

Spectra and Normal Forms

Luís Barreira Departamento de Matemática, Instituto Superior Técnico Universidade de Lisboa Lisbon, Portugal

Claudia Valls Departamento de Matemática, Instituto Superior Técnico Universidade de Lisboa Lisbon, Portugal

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-031-51896-6 ISBN 978-3-031-51897-3 (eBook) https://doi.org/10.1007/978-3-031-51897-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

This small book is a streamlined exposition of the notions and results leading to the construction of normal forms and, ultimately, to the construction of smooth conjugacies for the perturbations of tempered exponential dichotomies. These are exponential dichotomies for which the exponential growth rates of the underlying linear dynamics never vanish. In other words, its Lyapunov exponents are all nonzero. We consider mostly difference equations, although we also briefly consider the case of differential equations. The main components of the exposition are tempered spectra, normal forms, and smooth conjugacies. The first two lie at the core of the theory and have an importance that undoubtedly surpasses the construction of conjugacies. Indeed, the theory is very rich and developed in various directions that are also of interest by themselves. This includes the study of dynamics with discrete and continuous time, of dynamics in finite and infinite-dimensional spaces, as well as of dynamics depending on a parameter. This led us to make an exposition not only of tempered spectra and subsequently of normal forms, but also briefly of some important developments in those other directions. Afterward, we continue the presentation with the construction of stable and unstable invariant manifolds and, consequently, of smooth conjugacies, while using most of the former material. The text can be naturally divided into three parts. The first part (Chapters 1, 2, and 3, with emphasis on the basic theory) is dedicated to the tempered spectrum and the construction of normal forms. In Chapter 1, we introduce the notion of (tempered) spectrum in terms of the notion of tempered exponential dichotomy. The chapter also includes a description of all possible forms of the spectrum and detailed examples of all of them. We continue in Chapter 2 with the description of the Lyapunov exponents, which always belong to some connected component of the spectrum. We also consider exponentially decaying perturbations and show that again the Lyapunov exponents of the nonlinear dynamics belong to some connected component. Finally, in Chapter 3, starting with a block-diagonal preparation of the linear part, we construct normal forms for the tempered perturbations of a linear dynamics using an appropriate nonautonomous notion of resonance.

v

vi

Preface

The second part (Chapters 4, 5, and 6, with emphasis on further developments) is dedicated to the discussion of some additional topics related to tempered spectra and normal forms. Although strictly speaking the material is not necessary for the third part, these developments are important by themselves and the presentation would be quite poorer without them. In Chapter 4, we consider dynamics depending on a parameter. In particular, we describe how the tempered spectrum may vary with a parameter-dependent linear perturbation, and we establish the regularity of the normal forms when the perturbation depends smoothly on the parameter. Chapter 5 is a brief presentation of the notions and results in the former chapters for differential equations. In Chapter 6, we consider the infinite-dimensional setting with the study of linear and nonlinear dynamics defined by sequences of compact linear operators and their perturbations. The study of perturbations depending on a parameter is of utmost importance and is the main theme for example of bifurcation theory. Normal forms play a crucial role in the study of bifurcations since they reduce the nonlinear part of the dynamics to the simplest possible form. Besides difference and differential equations, it is also important to consider infinite-dimensional systems both for discrete and continuous time. This includes partial differential equations and functional differential equations, although these topics clearly fall out of the scope of our book. For details we refer instead to the notes at the end of each chapter. Finally, the third part (Chapters 7 and 8, with emphasis on smooth linearization) is dedicated to the construction of smooth conjugacies between a tempered exponential dichotomy and its tempered perturbations in the absence of resonances. This requires a detailed preparation in Chapter 7 with the construction of stable and unstable invariant manifolds together with crucial bounds. These are used in Chapter 8 to make a preparation of the dynamics so that the manifolds become the stable and unstable spaces. Finally, also in Chapter 8, we use the material in the former chapters on tempered spectra, formal forms, and invariant manifolds to construct smooth conjugacies when there are no resonances, or even when there are no resonances up to a given order. We note that the notion of tempered spectrum is naturally adapted to the study of nonautonomous dynamics. The reason for this is that any autonomous linear dynamics with a tempered exponential dichotomy has automatically a uniform exponential dichotomy. We emphasize that in strong contrast to what happens with a uniform exponential dichotomy, for a tempered exponential dichotomy the stability along the stable direction when time goes forward and along the unstable direction when time goes backward need not be uniform. In other words, it may depend on the initial time. This causes important changes and the need for adaptations of the classical theory as well as for new ideas. Most notably, the spectra defined in terms of tempered exponential dichotomies and uniform exponential dichotomies are distinct in general. More precisely, the tempered spectrum may be smaller, which causes that it may lead to less resonances and thus to simpler normal forms (an explicit example is given in Chapter 3). Another important aspect is the need for Lyapunov norms in the study of exponentially decaying perturbations (see Chapter 2) and in the study of parameter-dependent dynamics (see Chapter 4). Other characteristics are the need for a spectral gap to obtain the regularity of the normal

Preface

vii

forms on a parameter in Chapter 4 and the need for a careful control of the small exponential terms in the construction of invariant manifolds in Chapter 7 and of smooth conjugacies in Chapter 8. The following diagram is a summary of the relation between the chapters. A solid arrow means that there is a strong dependence between the two chapters, while a dotted arrow means that there is a dependence but small. The central line of the exposition with the discussion of tempered spectra, normal forms, and smooth conjugacies is marked in gray. Chapter 2

Chapter 1 Chapter 5

Chapter 4

Chapter 3

Chapter 6

Chapter 7

Chapter 8

The text is self-contained, and all proofs have been simplified or even rewritten on purpose for the book so that all is as streamlined as possible. Moreover, all chapters are supplemented by detailed notes discussing the origins of the notions and results as well as their proofs, together with the discussion of the proper context, also with references to precursor results and further developments. The book is aimed at researchers and graduate students who wish to have a sufficiently broad view of the area, without the discussion of accessory material. It can also be used as a basis for graduate courses on spectra, normal forms, and smooth conjugacies. Lisbon, Portugal November 2023

Luís Barreira Claudia Valls

Contents

1

Spectra and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Tempered Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples of Tempered Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 7

2

Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Linear Dynamics and Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Exponentially Decaying Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 23

3

Resonances and Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Block-Diagonalization of a Linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Construction of Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 36

4

Parameter-Dependent Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Linear Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nonlinear Perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 General Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 50 51

5

The Case of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Tempered Spectrum and Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Examples of Tempered Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Block-Diagonalization of a Linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Construction of Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 67 71 74

6

Infinite-Dimensional Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Structure of the Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Examples of Tempered Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 81 85

7

Stable and Unstable Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2 Invariant Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.3 Invariant Foliations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

ix

x

8

Contents

Construction of Smooth Conjugacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Bounds for the Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Smooth Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 117 119

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Chapter 1

Spectra and Examples

In this chapter we introduce the notions of tempered exponential dichotomy and of tempered spectrum for a sequence of .d × d matrices that need not be invertible. The tempered spectrum can be thought of as a nonautonomous version of the usual notion of spectrum for a single matrix. We also describe all possible forms of the tempered spectrum and we give explicit examples of all of them. More precisely, for each possible form we describe explicitly a sequence of invertible matrices with that tempered spectrum.

1.1 Tempered Spectrum We first introduce the notion of tempered exponential dichotomy. Let .(An )n∈Z be a sequence of .d × d matrices (not necessarily invertible). For each .m, n ∈ Z with .m ≥ n, we define ⎧ Am,n =

.

Am−1 · · · An

if m > n,

Id

if m = n.

Definition 1.1 A sequence of .d × d matrices .(An )n∈Z is said to have a tempered exponential dichotomy if: 1. There are projections .Pn : Rd → Rd for .n ∈ Z satisfying An Pn = Pn+1 An

.

for each n ∈ Z

(1.1)

such that the map An |ker Pn : ker Pn → ker Pn+1

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 L. Barreira, C. Valls, Spectra and Normal Forms, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-51897-3_1

(1.2)

1

2

1 Spectra and Examples

is onto and invertible. 2. There are a constant .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that for each .m, n ∈ Z we have ||Am,n Pn || ≤ De−λ(m−n)+ε|n|

for m ≥ n

(1.3)

||A¯ m,n Qn || ≤ De−λ(n−m)+ε|n|

for m ≤ n,

(1.4)

.

and .

where .Qn = Id − Pn and )−1 ( A¯ m,n = An,m |ker Pm : ker Pn → ker Pm

.

for m ≤ n.

(1.5)

Then we shall also say that .(An )n∈Z has a tempered exponential dichotomy with constants .λ and D. A sequence of positive numbers .(Dn )n∈Z is said to be upper tempered if .

lim sup n→±∞

1 log Dn ≤ 0. |n|

Note that this happens if and only if given .ε > 0, there is .D = D(ε) > 0 such that Dn ≤ Deε|n|

.

for all n ∈ Z.

(1.6)

Thus, a sequence .(An )n∈Z has a tempered exponential dichotomy if and only if there are projections .Pn for .n ∈ Z satisfying (1.1) such that each map in (1.2) is onto and invertible, and there are .λ > 0 and an upper tempered sequence .(Dn )n∈Z such that .

||Am,n Pn || ≤ Dn e−λ(m−n)

for m ≥ n

||A¯ m,n Qn || ≤ Dn e−λ(n−m)

for m ≤ n.

and .

We shall also say that .(An )n∈Z has a tempered exponential contraction if it has a tempered exponential dichotomy with .Pn = Id for all .n ∈ Z and that .(An )n∈Z has a tempered exponential expansion if it has a tempered exponential dichotomy with .Pn = 0 for all .n ∈ Z. For any tempered exponential dichotomy, the sets En = Pn (Rd )

.

and

Fn = Qn (Rd )

are called, respectively, the stable and unstable spaces at time n. They satisfy

1.1 Tempered Spectrum

3

Rd = En ⊕ Fn

.

for n ∈ Z

and can be univocally characterized as follows. Proposition 1.1 Assume that the sequence of .d × d matrices .(An )n∈Z has a tempered exponential dichotomy. For each .n ∈ Z, we have ⎧ ⎫ En = v ∈ Rd : sup ||Am,n v|| < +∞

.

m≥n

and .Fn is the set of all .v ∈ Rd for which there is a bounded sequence .(xm )m≤n in d .R such that xn = v

.

and

xm = Am−1 xm−1 for m ≤ n.

(1.7)

Proof Take .v ∈ En . By (1.3) we have .

sup ||Am,n v|| < +∞.

(1.8)

m≥n

Now assume that .v ∈ Rd satisfies (1.8). It follows from (1.3) that .

sup ||Am,n Qn v|| = sup ||Am,n (v − Pn v)|| < +∞.

m≥n

(1.9)

m≥n

On the other hand, by (1.4), for .m ≥ n we have ||Qn v|| ≤ De−λ(m−n)+ε|m| ||Am,n Qn v||,

.

which is equivalent to ||Am,n Qn v|| ≥

.

1 λ(m−n)−ε|m| e ||Qn v||. D

If .Qn v /= 0, then taking .ε < λ we obtain .

sup ||Am,n Qn v|| = +∞,

m≥n

which contradicts (1.9). Hence, .Qn v = 0 and so .v ∈ En . Now we consider a vector .v ∈ Fn and the sequence .xm = A¯ m,n v for .m ≤ n. Then property (1.7) holds and by (1.4) we have .supm≤n ||xm || < +∞. Finally, assume that .(xm )m≤n is a sequence with the properties in the proposition. It follows from (1.1) and (1.3) that ||Pn v|| = ||An,m Pm xm || ≤ De−λ(n−m)+ε|m| ||xm ||

.

4

1 Spectra and Examples

for .m ≤ n. Taking .ε < λ and letting .α = supm≤n ||xm ||, we obtain ||Pn v|| ≤ De−λ(n−m)+ε|m| α → 0

.

when .m → −∞. Hence, .Pn v = 0 and so .v ∈ Fn .

⨆ ⨅

The notion of tempered spectrum is defined in terms of the notion of tempered exponential dichotomy. Definition 1.2 The tempered spectrum (or, simply, the spectrum) of a sequence of d × d matrices .A = (An )n∈Z is the set .Σ = Σ (A) of all numbers .a ∈ R such that the sequence .(e−a An )n∈Z does not have a tempered exponential dichotomy.

.

We note that the tempered spectrum of a constant sequence of matrices .An = B for .n ∈ Z is the set of absolute values of the eigenvalues of B. Given .a ∈ R and .n ∈ Z, let ⎧ ⎫ ( −a(m−n) ) a d .En = v ∈ R : sup e ||Am,n v|| < +∞ m≥n

and let .Fna be the set of all .v ∈ Rd for which there is a sequence .(xm )m≤n in .Rd satisfying (1.7) such that .

sup (e−a(m−n) ||xm ||) < +∞.

m≤n

Clearly, if .a < b, then Ena ⊂ Enb

.

and

Fnb ⊂ Fna

(1.10)

for .n ∈ Z. Now take .a ∈ R \ Σ . By Proposition 1.1, Rd = Ena ⊕ Fna

.

for n ∈ Z

(1.11)

is the splitting into stable and unstable spaces of the tempered exponential dichotomy of the sequence .(e−a An )n∈Z . Because of the invertibility of the maps in (1.2), the dimensions .dim Fna (and so, by (1.11), also the dimensions .dim Ena ) are independent of n. We shall denote their common value by .dim F a . Finally, the following result describes all possible forms of the tempered spectrum for an arbitrary sequence of matrices. For .−∞ ≤ a ≤ b ≤ +∞, let |a, b| = R ∩ [a, b].

.

More precisely,

1.1 Tempered Spectrum

|a, b| =

.

5

⎧ ⎪ [a, b] ⎪ ⎪ ⎪ ⎨(−∞, b] ⎪ [a, +∞) ⎪ ⎪ ⎪ ⎩ R

if a, b ∈ R, if a = −∞, b ∈ R, if a ∈ R, b = +∞, if a = −∞, b = +∞.

Theorem 1.1 For a sequence of .d × d matrices .(An )n∈Z , either .Σ = ∅ or .Σ = Uk i=1 |ai , bi | for some numbers .

− ∞ ≤ a1 ≤ b1 < a2 ≤ b2 < · · · < ak ≤ bk ≤ +∞

(1.12)

and some positive integer .k ≤ d. Proof The statement will follow from some auxiliary results. Lemma 1.1 For each .a ∈ R \ Σ and all b in some neighborhood of a, we have b ∈ R \ Σ with

.

Ena = Enb

.

and

Fna = Fnb

for n ∈ Z.

(1.13)

Proof of the lemma Given .a ∈ R \ Σ , the sequence .(e−a An )n∈Z has a tempered exponential dichotomy. Hence, there are projections .Pn for .n ∈ Z satisfying (1.1), a constant .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that ||e−a(m−n) Am,n Pn || ≤ De−λ(m−n)+ε|n|

for m ≥ n

||e−a(m−n) A¯ m,n Qn || ≤ De−λ(n−m)+ε|n|

for m ≤ n,

.

and .

with .A¯ m,n as in (1.5). Therefore, for each .b ∈ R we have .

||e−b(m−n) Am,n Pn || ≤ De−(λ−a+b)(m−n)+ε|n|

for m ≥ n

||e−b(m−n) A¯ m,n Qn || ≤ De−(λ+a−b)(n−m)+ε|n|

for m ≤ n.

and .

This implies that .b ∈ / Σ whenever .|a − b| < λ. More precisely, the sequence (e−b An )n∈Z has a tempered exponential dichotomy, with the same projections .Pn . This shows that (1.13) holds for all .n ∈ Z, whenever .|a − b| < λ. ⨆ ⨅

.

Lemma 1.1 shows that .Σ is closed. Now take .c1 , c2 ∈ R \ Σ with .c1 < c2 . It follows from (1.10) that .dim F c1 ≥ dim F c2 .

6

1 Spectra and Examples

Lemma 1.2 We have .[c1 , c2 ] ∩ Σ /= ∅ if and only if .dim F c1 > dim F c2 . Proof of the lemma Assume first that .[c1 , c2 ] ∩ Σ /= ∅. If .dim F c1 = dim F c2 , then it follows from the inclusions in (1.10) that .Enc1 = Enc2 and .Fnc1 = Fnc2 for .n ∈ Z. Hence, there are projections .Pn for .n ∈ Z satisfying (1.1), a constant .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that for .i = 1, 2 we have ||e−ci (m−n) Am,n Pn || ≤ De−λ(m−n)+ε|n|

for m ≥ n

(1.14)

||e−ci (m−n) A¯ m,n Qn || ≤ De−λ(n−m)+ε|n|

for m ≤ n.

(1.15)

.

and .

For each .a ∈ [c1 , c2 ], it follows from (1.14) that ||e−a(m−n) Am,n Pn || ≤ De−λ(m−n)+ε|n|

.

for m ≥ n,

and it follows from (1.15) that ||e−a(m−n) A¯ m,n Qn || ≤ De−λ(n−m)+ε|n|

.

for m ≤ n.

Hence, .[c1 , c2 ] ⊂ R \ Σ , which contradicts the initial assumption. Now assume that .dim F c1 > dim F c2 and let } { b = inf a ∈ R \ Σ : dim F a = dim F c2 .

.

(1.16)

Since .dim F c1 > dim F c2 , by Lemma 1.1 we have .c1 < b < c2 . Assume that .b /∈ Σ . • If .dim F b = dim F c2 , then it follows from Lemma 1.1 that there is .ε > 0 such that .dim F c = dim F c2 and .c ∈ R\Σ for .c ∈ (b−ε, b], which contradicts (1.16). • If .dim F b /= dim F c2 , then it follows from Lemma 1.1 that there is .ε > 0 such that .dim F c /= dim F c2 and .c ∈ R \ Σ for .c ∈ [b, b + ε), which also contradicts (1.16). Therefore, .b ∈ Σ and so .[c1 , c2 ] ∩ Σ /= ∅.

⨆ ⨅

Recall that the set .R \ Σ is open. We claim that it has at most .d + 1 components. Indeed, if it had at least .d + 2 connected components, then there would be points .c1 , . . . , cd+2 ∈ R \ Σ , say with c1 < c2 < · · · < cd+2 ,

.

such that [ci , ci+1 ] ∩ Σ /= ∅ for i = 1, . . . , d + 1.

.

1.2 Examples of Tempered Spectra

7

It follows from Lemma 1.2 that dim F c1 > dim F c2 > · · · > dim F cd+2 ,

.

which implies that .dim F c1 ≥ d + 1. But this is impossible since .Fnc1 ⊂ Rd . Hence, .R \ Σ is an open set with at most .d + 1 connected components. Therefore, either .Σ = ∅ or .Σ is the union of at most d closed intervals, either bounded or unbounded (which gives the second alternative in the theorem). In addition, it follows readily from Lemma 1.2 that for any .a, b ∈ R in the same connected component of .R \ Σ we have Ena = Enb

and

.

Fna = Fnb

for n ∈ Z.

(1.17)

1.2 Examples of Tempered Spectra In this section we give explicit examples of sequences of .d × d matrices for all possible forms of the tempered spectrum .Σ described in Theorem 1.1. It turns out that it suffices to consider sequences of invertible matrices to obtain all spectra. When all matrices .An are invertible, we also define .Am,n = A−1 n,m for .m ≤ n, and so ⎧ ⎪ ⎪ ⎨Am−1 · · · An

Am,n =

Id ⎪ ⎪ ⎩A−1 · · · A−1 m n−1

.

if m > n, if m = n,

(1.18)

if m < n.

Example 1.1 (.Σ = ∅) For each .n ∈ Z, let .An be the .d × d matrix defined by An = e(n+1)

.

3 −n3

Id.

We claim that for each .a ∈ R, the sequence .(e−a An )n∈Z has a tempered exponential expansion. Indeed, for any .m, n ∈ Z we have 3 −n3

Am,n = em

.

Id = e(m−n)(m

2 +mn+n2 )

Id

and since m2 + mn + n2 ≥ (m2 + n2 )/2,

.

one can show that for each .a ∈ R there are .D, λ > 0 such that e−a(m−n) em

.

3 −n3

≤ De−λ(n−m)

for m ≤ n.

(1.19)

8

1 Spectra and Examples

In fact a stronger statement holds. Note that taking .D = 1 the inequality in (1.19) is equivalent to m2 + mn + n2 ≥ a + λ.

.

Given a and .λ, this holds for all but finitely many pairs .(m, n) such that .||(m, n)|| < c for some .c = c(a, λ) ∈ R. Therefore, for each .a ∈ R and .λ > 0 there is .D > 0 satisfying (1.19). Hence, .Σ = ∅. Example 1.2 (.Σ = R) For each .n ∈ Z, let .An be the .d × d matrix defined by An = e(n+1) cos(n+1)−n cos n Id.

.

For any .m, n ∈ Z we have Am,n = em cos m−n cos n Id.

(1.20)

.

Now take .a ∈ R and assume that .(e−a An )n∈Z has a tempered exponential contraction. Then there are .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that e−a(m−n) ||Am,n || ≤ De−λ(m−n)+ε|n|

.

for m ≥ n,

which is equivalent to em cos m−n cos n ≤ De(a−λ)(m−n)+ε|n|

.

for m ≥ n.

Take .m = 2π l and .n = 2π l − π with .l ∈ N. Then e4π l−π ≤ De(a−λ)π +ε(2π l−π ) ,

.

which fails for .ε < 2 and l sufficiently large. Therefore, .(e−a An )n∈Z does not have a tempered exponential contraction. Proceeding analogously, one can show that the sequence also does not have a tempered exponential expansion and so .Σ = R. Example 1.3 (.Σ nonempty and bounded) Take real numbers .ai and .bi as in (1.12) for some positive integer .k ≤ d. For each .n ∈ Z, let .An be the .d × d diagonal matrix defined by An = diag(cn1 , . . . , cnd ),

.

with

(1.21)

1.2 Examples of Tempered Spectra

⎧ i .cn

=

9

√

√ n+1 cos(n+1)− n cos n √ √ eai + |n+1| cos(n+1)− |n| cos n

e bi +

if n ≥ 0, if n < 0

(1.22)

for .i = 1, . . . , k, and .cni = cnk for .i > k. For each .i = 1, . . . , k and .m ≥ n, let

i .cm,n

=

⎧ i i ⎪ ⎪ ⎨cm−1 · · · cn

if m > n, if m = n,

1 ⎪ ⎪ ⎩(ci

i −1 n−1 · · · cm )

if m < n.

It is easy to verify that

i cm,n =

.

⎧ √ √ bi (m−n)+ m cos m− n cos n ⎪ ⎪ ⎨e

√ √ ebi m−ai n+ m cos m− |n| cos n ⎪ √ √ ⎪ ⎩eai (m−n)+ |m| cos m− |n| cos n

if m, n ≥ 0, if m ≥ 0, n < 0,

(1.23)

if m, n < 0.

Since .ai ≤ bi , we have √ √ |m| cos m− |n| cos n

i cm,n ≤ ebi (m−n)+

.

for m ≥ n.

Take .a > bi . Then √ √ |m|+ |n|

i e−a(m−n) cm,n ≤ e−(a−bi )(m−n)+

.

for m ≥ n.

(1.24)

√ Since . |n|/|n| → 0 when .|n| → +∞, given .ε > 0, there is .D = D(ε) > 0 such that √

e

.

|n|

≤ Deε|n|

for n ∈ Z.

(1.25)

Thus, it follows from (1.24) that i e−a(m−n) cm,n ≤ D 2 e−(a−bi )(m−n)+ε|m|+ε|n| .

≤ D 2 e−(a−bi −ε)(m−n)+2ε|n|

(1.26)

for .m ≥ n. Since .a − bi > 0 and .ε is arbitrary, the sequence .(e−a cni )n∈Z has a tempered exponential contraction. Now take .a < ai . By (1.25), for .m ≤ n we have √ √ |m|− |n|

i e−a(n−m) cn,m ≥ e−(a−ai )(n−m)− .

and so,

≥ D −2 e−(a−ai +ε)(n−m)−2ε|n|

(1.27)

10

1 Spectra and Examples i e−a(m−n) cm,n ≤ D 2 e(a−ai +ε)(n−m)+2ε|n| .

.

Since .a − ai < 0 and .ε is arbitrary, the sequence .(e−a cni )n∈Z has a tempered exponential expansion. We also consider the case when .a ∈ [ai , bi ]. Note that ⎧ √ √ (bi −a)(m−n)+ m cos m− n cos n ⎪ if m, n ≥ 0, ⎪ ⎨e √ √ −a(m−n) i m cos m− (b −a)m−(a −a)n+ |n| cos n i .e cm,n = e i if m ≥ 0, n < 0, ⎪ √ √ ⎪ ⎩e(ai −a)(m−n)+ |m| cos m− |n| cos n if m, n < 0. Since .bi − a ≥ 0, the formula for .m, n ≥ 0 shows that .(e−a cni )n∈Z does not have a tempered exponential contraction, and since .ai − a ≤ 0, the formula for .m, n < 0 shows that .(e−a cni )n∈Z does not have a tempered exponential expansion. Hence, this sequence has a tempered exponential dichotomy if and only if .a ∈ R \U [ai , bi ]. Finally, we find the tempered spectrum of .(An )n∈Z . Take .a ∈ R \ ki=1 [ai , bi ] and write .x = (x1 , . . . , xd ). When .bi < a < ai+1 for some .1 ≤ i < k let Pn x = (x1 , . . . , xi , 0, . . . , 0),

.

when .a > bk let .Pn = Id, and when .a < a1 let .Pn = 0. Then .(e−a An )n∈Z has a tempered exponential dichotomy with these projections. Indeed, if .bi < a < ai+1 , then for .j ≤ i and .m ≥ n proceeding as in (1.26) we obtain e−a(m−n) cm,n ≤ D 2 e−(a−bj −ε)(m−n)+2ε|n| j

.

≤ D 2 e−(a−bi −ε)(m−n)+2ε|n| ,

while for .j > i and .m ≤ n proceeding as in (1.27) we obtain e−a(m−n) cm,n ≤ D 2 e(a−aj +ε)(n−m)+2ε|n| j

.

≤ D 2 e(a−ai +ε)(n−m)+2ε|n| .

When .a > bk or .a < a1 one can proceed analogously, which gives .Σ ⊂ Uk [a i=1 i , bi ]. Now assume that for some i there is .a ∈ [ai , bi ] \ Σ . For .j /= i the sequence −a cj ) .(e n n∈Z has a tempered exponential contraction or a tempered exponential expansion. On the other hand, as shown above, .(e−a cni )n∈Z has neither a tempered exponential contraction nor a tempered exponential expansion. This contradiction shows that Σ =

k | |

.

i=1

[ai , bi ].

1.2 Examples of Tempered Spectra

11

We proceed with the construction of unbounded tempered spectra. Example 1.4 (.Σ unbounded from above) Take .2 ≤ k ≤ d. For each .n ∈ Z, let An be the .d × d matrix defined by (1.21), with .cni as in (1.22) for .i = 1, . . . , k − 1, and ⎧ √ √ e2n+1+ n+1 cos(n+1)− n cos n if n ≥ 0, i √ √ (1.28) .cn = eak + |n+1| cos(n+1)− |n| cos n if n < 0

.

for .i ≥ k. Using Example 1.3, it suffices to study the sequence .cnk . Clearly,

k cm,n =

.

⎧ √ √ (m+n)(m−n)+ m cos m− n cos n ⎪ ⎪ ⎨e √ √ 2 em −ak n+ m cos m− |n| cos n ⎪ ⎪ ak (m−n)+√|m| cos m−√|n| cos n ⎩ e

if m, n ≥ 0, if m ≥ 0, n < 0,

(1.29)

if m, n < 0.

Now take .a < ak . By (1.25), for .m ≤ n we have √ √ |m|− |n|

k ¯ −(a−ak )(n−m)− e−a(n−m) cn,m ≥ De .

¯ −2 e−(a−ak +ε)(n−m)−2ε|n| ≥ DD

for some constant .D¯ > 0. Since .a − ak < 0 and .ε is arbitrary, the sequence (e−a cnk )n∈Z has a tempered exponential expansion. Proceeding as in Example 1.3, one can show that

.

Σ =

k−1 | |

.

[ai , bi ] ∪ [ak , +∞).

i=1

Similarly, one can consider the .d × d matrices .An = cnk Id, with .cnk as in (1.28) taking .i = k. Proceeding as before we find that .Σ = [ak , +∞). Example 1.5 (.Σ unbounded from below) One can also obtain examples with an unbounded interval .(−∞, b1 ]. Take .2 ≤ k ≤ d. For each .n ∈ Z, let .An be the .d × d matrix defined by (1.21), with ⎧ 1 .cn

=

√

√ n+1 cos(n+1)− n cos n √ √ e2n+1+ |n+1| cos(n+1)− |n| cos n

e b1 +

if n ≥ 0, if n < 0,

cni as in (1.22) for .i = 2, . . . , k, and .cni = cnk for .i > k. Clearly,

.

(1.30)

12

1 Spectra and Examples

1 .cm,n

⎧ √ √ b1 (m−n)+ m cos m− n cos n ⎪ ⎪ ⎨e √ 2 √ = eb1 m−n + m cos m− |n| cos n ⎪ √ √ ⎪ ⎩e(m+n)(m−n)+ |m| cos m− |n| cos n

if m, n ≥ 0, if m ≥ 0, n < 0,

(1.31)

if m, n < 0.

Now take .a > b1 . By (1.25), for .m ≥ n we have √ √ |m|+ |n|

1 ¯ −(a−b1 )(m−n)+ e−a(m−n) cm,n ≤ De .

¯ 2 e−(a−b1 −ε)(m−n)+2ε|n| ≤ DD

for some constant .D¯ > 0. Since .a − b1 > 0 and .ε is arbitrary, the sequence −a c1 ) .(e n n∈Z has a tempered exponential contraction. Again proceeding as in Example 1.3, one can show that Σ = (−∞, b1 ] ∪

k | |

.

[ai , bi ].

i=2

Similarly, one can consider the .d × d matrices .An = cn1 Id, with .cn1 as in (1.30). Proceeding as before we find that .Σ = (−∞, b1 ]. Example 1.6 (.Σ /= R unbounded from above and below) Take .2 ≤ k ≤ d. For each .n ∈ Z, let .An be the .d × d matrix defined by (1.21), with .cn1 as in (1.30), .cni as in (1.22) for .i = 2, . . . , k − 1, and .cni as in (1.28) for .i ≥ k. Proceeding as in Example 1.3, one can show that Σ = (−∞, b1 ] ∪

k−1 | |

.

[ai , bi ] ∪ [ak , +∞).

i=2

Finally, for each .n ∈ Z, let .An be the .d × d matrix defined by (1.21), with .cn1 as in (1.30), and .cni as in (1.28) for .i ≥ 2. Proceeding as in Example 1.3, we find that Σ = (−∞, b1 ] ∪ [ak , +∞).

.

Notes The notion of tempered exponential dichotomy in Definition 1.1 (also called a nonuniform exponential dichotomy with an arbitrarily small nonuniform part) describes the behavior of a linear dynamics .xn+1 = An xn for .n ∈ Z when all Lyapunov exponents (that is, the exponential growth rates of the dynamics) are nonzero. It can also be seen as a simple version of the notion of nonuniform hyperbolicity for cocycles over an invertible map. A principal example of the latter is any cocycle over an invertible measure-preserving map with nonzero Lyapunov exponents almost everywhere (see for example Theorem 3.3.4 in [10], although the details are out of the scope of this book). The theory goes back to important works

1.2 Examples of Tempered Spectra

13

of Oseledets [70] and Pesin [77–79] in the context of smooth ergodic theory. See also the important developments of Katok in [51]. In strong contrast, the notion of tempered exponential dichotomy considers a single trajectory and thus has no measure-theoretic nature. The notion of tempered spectrum in Definition 1.2 goes back to work of Sacker and Sell in [92] with the development of a nonautonomous version of the notion of spectrum of a linear map. More precisely, they introduced a spectrum for linear cocycles over a flow using uniform exponential dichotomies. Similar ideas were then used by Siegmund [97] and Aulbach and Siegmund [6] to introduce spectra, respectively, for nonautonomous linear differential equations and linear difference equations. We mimic their definitions using now tempered exponential dichotomies (a similar version for one-sided sequences of invertible matrices first appeared in [9]). For further developments and additional references, see [20, 72], as well as [49] for the relation to ergodic theory, [2, 57, 60] for the study of random dynamical systems, and [21, 61, 93] for the infinite-dimensional case. The notions of tempered exponential dichotomy and tempered spectrum are considered in Chapter 6 for sequences of bounded linear operators on a Banach space. The proof of Theorem 1.1 giving all possible forms of the tempered spectrum is based on the proof of a corresponding statement in [9] for one-sided sequences of invertible matrices, although the necessary changes are simple (see also [6] as well as [5] for the noninvertible case). The main element of the argument consists of showing that given two numbers .c1 and .c2 outside the tempered spectrum .Σ , the interval between them contains a point in .Σ if and only if the stable and unstable spaces of the sequences .(e−c1 An )n∈Z and .(e−c2 An )n∈Z have different dimensions. Since the ambient space is finite-dimensional, this implies that the set .R\Σ has only finitely many connected components. Theorem 1.1 is analogous to a result obtained by Sacker and Sell in [92] for a linear cocycle over a flow with compact base (that cannot be written as a disjoint union of nonempty compact invariant sets). The examples in Section 1.2 of sequences of matrices for all possible forms of the tempered spectrum are adapted from examples in [8] in the infinite-dimensional setting (see also Chapter 6). We emphasize that all forms of the tempered spectrum described in Theorem 1.1 are realized by sequences of invertible matrices. There are various variants of the notion of exponential dichotomy (essentially going back to Perron in [76]) that can also be used to define corresponding spectra. For example, one can consider nonuniform exponential dichotomies with a small nonuniform part (but not necessarily arbitrarily small) as in [23], or strong exponential dichotomies as in [7]. The first notion occurs naturally in the context of the nonuniform hyperbolicity theory (see the monographs [10, 18] for details). Roughly speaking, even though in the context of ergodic theory the nonuniform part of an exponential dichotomy can be made arbitrarily small for almost all trajectories, for some important results, such as the existence of stable and unstable invariant manifolds, it is sufficient to assume a sufficiently small nonuniform part (but not necessarily arbitrarily small). On the other hand, the notion of strong exponential dichotomy (which can also be uniform, nonuniform with a small nonuniform part, or tempered) requires not only upper bounds along the stable direction when the time

14

1 Spectra and Examples

goes forward and along the unstable direction when the time goes backward (as in Definition 1.1) but also lower bounds in both situations. Again it is ubiquitous in the context of ergodic theory (see [10, 18]). For details and references on exponential dichotomies we refer the reader to the books [20, 31, 42, 46, 64, 94]. In particular, in a series of papers Sacker and Sell [89–91, 93] (see also [88]) gave sufficient conditions for the existence of exponential dichotomies. These notions of exponential dichotomy lead to corresponding spectra that are distinct in general. For example, any uniform exponential dichotomy is also a tempered exponential dichotomy, and so the spectrum defined in terms of uniform exponential dichotomies may in general be larger than the tempered spectrum (see [8] for explicit examples). Similarly, one can also ask for descriptions of all possible forms of these spectra. For example, [23] characterizes a spectrum defined in terms of exponential dichotomies with a small nonuniform part, while [7] describes a spectrum defined in terms of strong exponential dichotomies. See also [26, 105] for the study of stochastic differential equations and random dynamical systems. The infinite-dimensional setting of linear operators acting on a Banach space is considered in Chapter 6. In that setting the spectra can take additional forms to those described in Theorem 1.1. They can have for example infinitely many connected components.

Chapter 2

Asymptotic Behavior

In this chapter we show that all lower and upper Lyapunov exponents, both to the future and to the past, belong to some connected component of the tempered spectrum. We also consider the asymptotic behavior of a linear dynamics under exponentially decaying perturbations that can either be linear or nonlinear. It turns out that all Lyapunov exponents of the perturbed dynamics still belong to some connected component of the tempered spectrum of the linear dynamics. This result depends strongly on the use of Lyapunov norms, which are also used in other parts of the book.

2.1 Linear Dynamics and Lyapunov Exponents In this section we show that to each connected component I of the tempered spectrum of an invertible sequence of matrices one can associate a subspace of .Rd such that the Lyapunov exponents of all nonzero vectors in this space belong to I . Let .(An )n∈Z be a sequence of invertible .d × d matrices with nonempty tempered spectrum. By Theorem 1.1 we have Σ=

k | |

.

|ai , bi |

i=1

for some numbers .ai and .bi as in (1.12) and some positive integer .k ≤ d. Note that a1 can be .−∞ and that .bk can be .+∞. Take .ci ∈ (bi , ai+1 ) for .i = 1, . . . , k − 1. Moreover, take .c0 < a1 if .a1 /= −∞ and .c0 = −∞ otherwise, and take .ck > bk if .bk /= +∞ and .ck = +∞ otherwise. We define .

c

Hni = Enci ∩ Fn i−1

.

for i = 1, . . . , k,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 L. Barreira, C. Valls, Spectra and Normal Forms, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-51897-3_2

(2.1)

15

16

2 Asymptotic Behavior

with the convention that En+∞ = Fn−∞ = Rd .

(2.2)

.

In addition, we define Hn0 = Enc0

.

and

Hnk+1 = Fnck ,

(2.3)

with the convention that En−∞ = Fn+∞ = {0}.

(2.4)

.

By property (1.17), the subspaces .Hni are independent of the numbers .ci . Moreover, Am,n Hni = Hmi

.

for m, n ∈ Z.

Now we consider the lower and upper Lyapunov exponents, both to the future and to the past. These are the numbers .

lim inf

m→±∞

1 log||Am,n v|| and m

lim sup m→±∞

1 log||Am,n v|| m

for each .v ∈ Rd \ {0}. Theorem 2.1 Let .(An )n∈Z be a sequence of invertible .d×d matrices with nonempty tempered spectrum. Then for each .n ∈ Z the following properties hold: ⊕ i 1. .Rd = k+1 i=0 Hn . i 2. Given .v ∈ Hn \ {0} with .i ∈ {1, . . . , k}, we have .

1 1 log||Am,n v||, lim sup log||Am,n v|| ∈ |ai , bi |. m→±∞ m m→±∞ m lim inf

3. Given .v ∈ Hn0 \ {0}, we have .

lim inf

m→±∞

1 1 log||Am,n v|| = lim sup log||Am,n v|| = −∞, m m m→±∞

and given .v ∈ Hnk+1 \ {0}, we have .

lim inf

m→±∞

1 1 log||Am,n v|| = lim sup log||Am,n v|| = +∞. m m→±∞ m

2.1 Linear Dynamics and Lyapunov Exponents

17

Proof Note that c

c

Enci = (Eni−1 ⊕ Fn i−1 ) ∩ Enci .

c

c

c

= Eni−1 ⊕ (Fn i−1 ∩ Enci ) = Eni−1 ⊕ Hni

for .i = 1, . . . , k + 1, since (E + F ) ∩ G = E + (F ∩ G)

.

whenever E, F , and G are subspaces of .Rd with .E ⊂ G. Hence, one can use (finite) induction to obtain the first statement in the theorem. Now we prove the other statements. Assume that .ci ∈ R \ Σ. Then the sequence −ci A ) .(e n n∈Z has a tempered exponential dichotomy and so there are projections .Pn for .n ∈ Z satisfying (1.1), a constant .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that ||Am,n Pn || ≤ De(ci −λ)(m−n)+ε|n|

.

for m ≥ n

(2.5)

and ||Am,n Qn || ≤ De−(λ+ci )(n−m)+ε|n|

.

for m ≤ n.

By Proposition 1.1, we have .Pn (Rd ) = Enci for .n ∈ Z. Hence, each vector .v ∈ Hni \ {0} belongs to .Pn (Rd ) and so it follows from (2.5) that .

lim sup m→+∞

1 log||Am,n v|| ≤ ci − λ < ci . m

When .i /= 0, letting .ci \ bi gives .

lim sup m→+∞

1 log||Am,n v|| ≤ bi . m

(2.6)

On the other hand, when .i = 0, letting .ci \ −∞ gives .

lim sup m→+∞

1 log||Am,n v|| = −∞. m

(2.7)

Now assume that .ci−1 ∈ R \ Σ. Then the sequence .(e−ci−1 An )n∈Z has a tempered exponential dichotomy and so there are projections .Pn for .n ∈ Z satisfying (1.1), a constant .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that ||Am,n Pn || ≤ De(ci−1 −λ)(m−n)+ε|n|

.

for m ≥ n

18

2 Asymptotic Behavior

and ||Am,n Qn || ≤ De−(λ+ci−1 )(n−m)+ε|n|

.

for m ≤ n.

(2.8)

c

By Proposition 1.1, we have .Qn (Rd ) = Fn i−1 for .n ∈ Z. Hence, each vector .v ∈ Hni \ {0} belongs to .Qn (Rd ) and so it follows from (2.8) that ||v|| ≤ De−(λ+ci−1 )(m−n)+ε|m| ||Am,n v||

.

for m ≥ n.

Taking .ε < λ gives .

1 log||Am,n v|| ≥ λ + ci−1 − ε > ci−1 . m→+∞ m lim inf

When .i /= k + 1, letting .ci−1 / ai gives .

lim inf

m→+∞

1 log||Am,n v|| ≥ ai . m

(2.9)

On the other hand, when .i = k + 1, letting .ci−1 / +∞ gives .

lim inf

m→+∞

1 log||Am,n v|| = +∞. m

(2.10)

By the definition of the spaces .Hni , the second statement in the theorem follows readily from (2.6) and (2.9), while the third statement is simply a restatement of (2.7) and (2.10). Proceeding similarly, we can also obtain the corresponding statements when .m → −∞. This completes the proof of the theorem. ⨆ ⨅ Before proceeding we go back to the examples of nonempty tempered spectra in Chapter 1 and we describe the spaces .Hni for each of them. Example 2.1 (.Σ = R; see Example 1.2) We have .k = 1, .c0 = −∞ and .c1 = +∞. Hence, it follows from (2.1) and (2.2) that Hn1 = En+∞ ∩ Fn−∞ = Rd .

.

By (1.20), for .v ∈ Hn1 \ {0} we obtain .

lim sup m→±∞

1 log||Am,n v|| = 1 and m

1 log||Am,n v|| = −1. m→±∞ m lim inf

Moreover, by (2.3) and (2.4) we have Hn0 = En−∞ = {0}

.

and

Hn2 = Fn+∞ = {0}.

2.1 Linear Dynamics and Lyapunov Exponents

19

Example 2.2 (.Σ nonempty and bounded; see Example 1.3) Since the spectrum is bounded, the numbers c0 < c1 < · · · < ck

.

(2.11)

are all finite. Moreover, it is shown in Example 1.3 that .(e−c0 An )n∈Z has a tempered exponential expansion, and that .(e−ck An )n∈Z has a tempered exponential contraction. Therefore, .Enc0 = Fnck = {0} and so by (2.3) we have Hn0 = Hnk+1 = {0}.

.

On the other hand, it follows readily from the construction that .

dim Hni = 1

for i = 1, . . . , k − 1

and .dim Hnk = d − k − 1. By (1.23), for .v ∈ Hni \ {0} we obtain .

lim inf

1 1 log||Am,n v|| = lim sup log||Am,n v|| = bi m→+∞ m m→+∞ m

(2.12)

1 1 log||Am,n v|| = lim sup log||Am,n v|| = ai . m→−∞ m m→−∞ m

(2.13)

and .

lim inf

Example 2.3 (.Σ unbounded from above; see Example 1.4) Take .2 ≤ k ≤ d. Now we have .ck = +∞ in (2.11). Hence, it follows from (2.4) that .Hnk+1 = {0}. Moreover, by Example 1.3 the sequence .(e−c0 An )n∈Z has a tempered exponential expansion, and so by (2.3) we have Hn0 = Enc0 = {0}.

.

The remaining spaces .Hni for .i = 1, . . . , k − 1 have the dimensions in Example 2.2. Again by (1.23), for .v ∈ Hni \ {0} with .1 ≤ i < k we have (2.12) and (2.13). Finally, by (1.29) for .v ∈ Hnk \ {0} we obtain .

lim inf

1 1 log||Am,n v|| = lim sup log||Am,n v|| = +∞ m→+∞ m m→+∞ m

(2.14)

1 1 log||Am,n v|| = lim sup log||Am,n v|| = ak . m→−∞ m m→−∞ m

(2.15)

and .

lim inf

The case of the spectrum .Σ = [ak , +∞) is analogous.

20

2 Asymptotic Behavior

Example 2.4 (.Σ unbounded from below; see Example 1.5) Take .2 ≤ k ≤ d. Now we have .c0 = −∞ in (2.11). Hence, it follows from (2.4) that .Hn0 = {0}. Moreover, by Example 1.3 the sequence .(e−ck An )n∈Z has a tempered exponential contraction, and so by (2.3) we have Hnk+1 = Fnck = {0}.

.

The remaining spaces .Hni for .i = 1, . . . , k − 1 have the dimensions in Example 2.2. Again by (1.23), for .v ∈ Hni \ {0} with .1 < i ≤ k we have (2.12) and (2.13). Finally, by (1.31) for .v ∈ Hn1 \ {0} we obtain .

1 1 log||Am,n v|| = lim sup log||Am,n v|| = b1 m→+∞ m m→+∞ m lim inf

(2.16)

and .

lim inf

m→−∞

1 1 log||Am,n v|| = lim sup log||Am,n v|| = −∞. m m→−∞ m

(2.17)

The case of the spectrum .Σ = (−∞, b1 ] is analogous. Example 2.5 (.Σ /= R unbounded from above and below; see Example 1.6) Take .2 ≤ k ≤ d. Now we have .c0 = −∞ and .ck = +∞ in (2.11). Hence, it follows from (2.4) that Hn0 = Hnk+1 = {0}.

.

The remaining spaces .Hni for .i = 1, . . . , k − 1 have the dimensions in Example 2.2. For .v ∈ Hni \ {0} with .1 < i < k we have (2.12) and (2.13). Finally, for .v ∈ Hn1 \ {0} we have (2.16) and (2.17), while for .v ∈ Hnk \ {0} we have (2.14) and (2.15). The case of the spectrum .Σ = (−∞, b1 ] ∪ [ak , +∞) is analogous. We also give an example for which the spaces .Hn0 and .Hnk+1 in (2.3) have positive dimension. Example 2.6 For each .n ∈ Z, let .An be the .3 × 3 matrix ( 3 3 3 3) An = diag en −(n+1) , 1, e(n+1) −n .

.

For any .m, n ∈ Z we have ( 3 3 3 3) Am,n = diag e−m +n , 1, em −n

.

and one can proceed in a similar manner to that in Example 1.1 to show that .Σ = {0}. We have .k = 1 and .c0 < 0 < c1 . Moreover, it follows from (2.1) and (2.3) that

2.1 Linear Dynamics and Lyapunov Exponents

.

dim Hni = 1

21

for i = 0, 1, 2.

We also obtain .

1 1 log||Am,n v|| = lim sup log||Am,n v|| = −∞ m→±∞ m m→±∞ m lim inf

for .v ∈ Hn0 \ {0}, .

lim inf

m→±∞

1 1 log||Am,n v|| = lim sup log||Am,n v|| = 0 m m→±∞ m

for .v ∈ Hn1 \ {0}, and .

1 1 log||Am,n v|| = lim sup log||Am,n v|| = +∞ m m m→±∞

lim inf

m→±∞

for .v ∈ Hn2 \ {0}. Now we consider a particular class of matrices. Definition 2.1 We say that a sequence of invertible .d × d matrices .(An )n∈Z has tempered growth if there are a constant .γ ≥ 0 and for each .ε > 0 a constant .K = K(ε) > 0 such that ||Am,n || ≤ Keγ |m−n|+ε|n|

.

for m, n ∈ Z.

For example, any sequence of invertible .d × d matrices .An for .n ∈ Z such that both .(An )n∈Z and .(A−1 n )n∈Z are bounded has tempered growth. This follows readily from the definition of .Am,n in (1.18). The tempered spectrum of any sequence of matrices with tempered growth is automatically bounded and nonempty. Proposition 2.1 Let .(An )n∈Z be a sequence of invertible .d × d matrices with tempered growth. Then .Σ is bounded and nonempty. Proof For .m ≥ n we have e−a(m−n) ||Am,n || ≤ Ke(−a+γ )(m−n)+ε|n| ,

.

which shows that .(e−a An )n∈Z has a tempered exponential contraction for .a > γ . Similarly, for .m ≤ n we have e−a(m−n) ||Am,n || = Ke(a+γ )(n−m)+ε|n| ,

.

22

2 Asymptotic Behavior

which shows that .(e−a An )n∈Z has a tempered exponential expansion for .a < −γ . Therefore, .Σ ⊂ [−γ , γ ] and the spectrum is bounded. It follows from Lemma 1.2 that .Σ is nonempty (since .dim F a = 0 for .a > γ , and .dim F a = d for .a< − γ ). ⨅ ⨆ For any sequence of invertible matrices with tempered growth, by Proposition 2.1 the spectrum is bounded and nonempty, and so it follows from Theorem 1.1 that Σ=

k | |

.

[ai , bi ]

(2.18)

i=1

for some real numbers .ai and .bi as in (1.12) and some positive integer .k ≤ d. Take c0 < a1 , .ci ∈ (bi , ai+1 ) for .i = 1, . . . , k − 1, and .ck > bk . We continue to consider the spaces .Hni introduced in (2.1).

.

Theorem 2.2 Let .(An )n∈Z be a sequenceUof invertible .d ×d matrices with tempered k growth and tempered spectrum .Σ = i=1 [ai , bi ]. Then for each .n ∈ Z the following properties hold: ⊕ 1. .Rd = ki=1 Hni . 2. Given .v ∈ Hni \ {0}, we have .

lim inf

m→±∞

1 1 log||Am,n v||, lim sup log||Am,n v|| ∈ [ai , bi ]. m m m→±∞

Proof We show in the proof of Proposition 2.1 that .(e−a An )n∈Z has a tempered exponential contraction (respectively, a tempered exponential expansion) for any sufficiently large a (respectively, any sufficiently small a). Therefore, .Rd = Enck = Fnc0 and so it follows from (2.3) that now .Hn0 = Hnk+1 = {0}. The desired statement ⨆ ⨅ is thus a simple consequence of Theorem 2.1. In fact we can strengthen property 2 in Theorem 2.2 by showing that the lower and upper Lyapunov exponents of any nonzero vector belong to some interval .[ai , bi ] of the tempered spectrum. Theorem 2.2 says that this happens for any nonzero vector inside each space .Hni . Theorem 2.3 Let .(An )n∈Z be a sequence U of invertible .d ×d matrices with tempered growth and tempered spectrum .Σ = ki=1 [ai , bi ]. Then for each .n ∈ Z and .v ∈ Rd \ {0} there is .i ∈ {1, . . . , k} such that .

1 1 log||Am,n v||, lim sup log||Am,n v|| ∈ [ai , bi ]. m→±∞ m m→±∞ m lim inf

Σk i Proof Take .v ∈ Rd \ {0} and write .v = i=1 vi with .vi ∈ Hn for each i. By property 2 in Theorem 2.2, given .ε > 0, if .vi /= 0, then

2.2 Exponentially Decaying Perturbations

23

e(ai −ε)m ||vi || ≤ ||Am,n vi || ≤ e(bi +ε)m ||vi ||

.

for any sufficiently large m. Now let .I = {i : vi /= 0} and .j = max I . Then ||Am,n v|| ≥ ||Am,n vj || −

Σ

||Am,n vi ||

i∈I \{j } .

≥ e(aj −ε)m ||vj || −

Σ

e(bi +ε)m ||vi ||

i∈I \{j }

for any sufficiently large m. Taking .ε such that .aj − ε > bi + ε for all .i ∈ I \ {j } we obtain 1 log||Am,n v|| ≥ aj − ε. m→+∞ m lim inf

.

Finally, since .ε is arbitrary, we conclude that .

lim inf

m→+∞

1 log||Am,n v|| ≥ aj . m

We also have ||Am,n v|| ≤

.

Σ Σ Σ ||Am,n vi || ≤ e(bi +ε)m ||vi || ≤ e(bj +ε)m ||vi || i∈I

i∈I

i∈I

for any sufficiently large m. Hence, .

lim sup m→+∞

1 log||Am,n v|| ≤ bj + ε m

and since .ε is arbitrary, we conclude that .

lim sup m→+∞

1 log||Am,n v|| ≤ bj . m

Proceeding analogously, we also obtain the statement when .m → −∞.

⨆ ⨅

2.2 Exponentially Decaying Perturbations In this section we consider the lower and upper Lyapunov exponents of the (linear or nonlinear) perturbations of a linear dynamics. We show that exponentially decaying perturbations give rise to Lyapunov exponents that belong necessarily to some interval of the tempered spectrum of the original linear dynamics.

24

2 Asymptotic Behavior

More precisely, given a sequence of invertible .d × d matrices .(An )n∈Z and continuous maps .fn : Rd → Rd for .n ∈ Z, we consider the dynamics xn+1 = An xn + fn (xn )

.

for n ∈ Z.

(2.19)

For each never vanishing sequence .(xn )n∈Z satisfying (2.19), we consider the lower and upper Lyapunov exponents, both to the future and to the past, given by 1 log||xn || and n→±∞ n lim inf

.

lim sup n→±∞

1 log||xn ||. n

We assume that ||fn (x)|| ≤ kn ||x||

.

for x ∈ Rd

(2.20)

for some sequence .(kn )n∈Z of positive numbers such that .

lim sup n→±∞

1 log kn < 0. |n|

(2.21)

This means that the perturbations .fn decay exponentially with .|n|. Theorem 2.4 Let .(An )n∈Z be a sequence U of invertible .d ×d matrices with tempered growth and tempered spectrum .Σ = ki=1 [ai , bi ]. If the maps .fn : Rd → Rd for .n ∈ Z satisfy (2.20) and (2.21), then for each never vanishing sequence .(xn )n∈Z satisfying (2.19) there is .i ∈ {1, . . . , k} such that .

lim inf

n→±∞

1 1 log||xn ||, lim sup log||xn || ∈ [ai , bi ]. n n n→±∞

Proof For simplicity we consider only the limits when .n → +∞ (the limits when n → −∞ can be considered analogously). Take .i ∈ {1, . . . , k − 1} and .ci ∈ (bi , ai+1 ). Then there are projections .Pn for .n ∈ Z satisfying (1.1), a constant .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that .

||Am,n Pn || ≤ De(ci −λ)(m−n)+ε|n|

for m ≥ n

(2.22)

||Am,n Qn || ≤ De(ci +λ)(m−n)+ε|n|

for m ≤ n.

(2.23)

.

and .

2.2 Exponentially Decaying Perturbations

25

We consider the Lyapunov norms ( ( ) ) ||x||n = sup e−(ci −λ)(m−n) ||Am,n Pn x|| + sup e−(ci +λ)(m−n) ||Am,n Qn x||

.

m≥n

m≤n

(2.24) for each .n ∈ Z and .x ∈ Rd . Then ||x|| ≤ ||x||n ≤ 2Deε|n| ||x||,

.

(2.25)

and for each .m ≥ n we have e−(ci −λ)(m−n) ||Am,n Pn x||m ≤ ||Pn x||n ≤ ||x||n

.

(2.26)

and e−(ci +λ)(n−m) ||An,m Qm x||n ≤ ||Qm x||m ≤ ||x||m .

.

(2.27)

We emphasize that the exponential terms .eε|n| in (2.22) and (2.23) no longer appear in the inequalities (2.26) and (2.27). We start with an auxiliary statement. Lemma 2.1 Either .

lim sup n→+∞

1 log ||xn || < ci n

(2.28)

1 log ||xn || > ci . n

(2.29)

or .

lim inf

n→+∞

Proof of the lemma Let yn = Pn xn

zn = Qn xn .

(2.30)

yn+1 = An Pn xn + Pn+1 fn (xn )

(2.31)

.

and

Then .

and zn+1 = An Qn xn + Qn+1 fn (xn ).

.

26

2 Asymptotic Behavior

By (2.25) and (2.27), we obtain ||zn+1 ||n+1 ≥ ||An Qn xn ||n+1 − ||Qn+1 fn (xn )||n+1 ≥ eci +λ ||zn ||n − 2Deε|n+1| ||Qn+1 fn (xn )||

.

¯ 2εn kn ||xn || ≥ eci +λ ||zn ||n − De for some constant .D¯ > 0 and all sufficiently large n. By (2.21), provided that .ε is sufficiently small we have e2εn kn → 0

.

(2.32)

when .n → +∞. Analogously, by (2.26) and (2.31) we have ¯ 2εn kn ||xn || ||yn+1 ||n+1 ≤ eci −λ ||yn ||n + De .

¯ 2εn kn ||xn ||n , ≤ eci −λ ||yn ||n + De

(2.33)

without loss of generality with the same constant .D¯ > 0. Take .δ > 0 such that δ
m. Now let By (2.32) there is .m ∈ N such that .De a = eci +λ

.

and

a = eci −λ .

For .n > m, it follows from (2.32) and (2.33) that .

||zn+1 ||n+1 ≥ (a − δ)||zn ||n − δ||yn ||n

(2.34)

||yn+1 ||n+1 ≤ (a + δ)||yn ||n + δ||zn ||n .

(2.35)

and .

We show that one of the following properties holds: 1. ||zn ||n ≤ ||yn ||n

for n ≥ m.

(2.36)

||yn ||n < ||zn ||n

for n ≥ l.

(2.37)

.

2. There is .l > m such that .

2.2 Exponentially Decaying Perturbations

27

Assume that (2.36) does not hold. Then .||yl ||l < ||zl ||l for some .l ≥ m. By (2.34) and (2.35) we have ||zl+1 ||l+1 ≥ (a − 2δ)||zl ||l

.

and ||yl+1 ||l+1 ≤ (a + 2δ)||zl ||l .

.

Therefore, .

||yl+1 ||l+1 ≤

a + 2δ ||zl+1 ||l+1 < ||zl+1 ||l+1 . a − 2δ

It follows by induction that .||yn ||n < ||zn ||n for all .n ≥ l, which establishes (2.37). In the remainder of the proof we consider the two conditions (2.36) and (2.37). Assume first that (2.36) holds. By (2.35), for .n ≥ m we have ||yn+1 ||n+1 ≤ (a + 2δ)||yn ||n ,

.

which gives ||yn ||n ≤ (a + 2δ)n−m ||ym ||m .

.

Hence, .

||xn || ≤ ||xn ||n = ||yn ||n + ||zn ||n ≤ 2||yn ||n ≤ 2(a + 2δ)n−m ||ym ||m

for .n ≥ m and so .

lim sup n→+∞

1 log ||xn || ≤ log(a + 2δ) < ci , n

since a + 2δ
ci − ε, n→+∞ n lim inf

(2.38)

since a − 2δ >

.

1 ci +λ 1 ci 1 ci +λ 1 1 e − e + (e + eci −λ ) > eci +λ + eci > eci . 2 2 2 2 2

The arbitrariness of .ε in (2.38) yields inequality (2.29).

⨆ ⨅

One can also take .c0 < a1 and .ck > bk . In the first case .Pn = 0 for .n ∈ Z and hence, using the notation in (2.30) we have .yn = 0 for all .n ∈ Z. Therefore, proceeding as in the proof of Lemma 2.1 we find that property (2.37) holds (recall that .xn never vanishes) and so also that lim inf

.

n→+∞

1 log||xn || > c0 . n

In the second case .Pn = Id for .n ∈ Z and hence, using the notation in (2.30) we have .zn = 0 for all .n ∈ Z. Therefore, property (2.36) holds and so also .

lim sup n→+∞

1 log||xn || < ck . n

Letting .c0 → a1 and .ck → bk yields that lim inf

1 log ||xn || ≥ a1 n

(2.39)

lim sup

1 log ||xn || ≤ bk . n

(2.40)

.

n→+∞

and .

n→+∞

Finally, we use Lemma 2.1 together with (2.39) and (2.40) to prove the theorem. Let a = lim sup

.

n→+∞

1 log||xn || n

and assume that .a ∈ (bi , ai+1 ) for some .i ∈ {1, . . . , k − 1}. For .ci = a it follows from Lemma 2.1 that

2.2 Exponentially Decaying Perturbations

.

lim inf

n→+∞

29

1 1 log||xn || > ci = lim sup log||xn ||. n n n→+∞

This contradiction implies that .a ∈ [ai , bi ] for some i. Similarly, let 1 log||xn || n→+∞ n

b = lim inf

.

and assume that .b ∈ (bi , ai+1 ) for some .i ∈ {1, . . . , k − 1}. For .ci = b it follows from Lemma 2.1 that .

lim sup n→+∞

1 1 log||xn || < ci = lim inf log||xn ||, n→+∞ n n

which implies that .b ∈ [ai , bi ] for some i. When the maps .fn are linear one can write .fn (x) = Bn x for some .d × d matrices .Bn and all .n ∈ Z. Then the dynamics in (2.19) becomes xn+1 = (An + Bn )xn

.

for n ∈ Z

and conditions (2.20) and (2.21) can simply be replaced by .

lim sup n→±∞

1 log ||Bn || < 0. |n|

Notes The proof of Theorem 2.1 on the possible values of the lower and upper Lyapunov exponents is based on the proof of a corresponding statement in [9] for one-sided sequences of invertible matrices. The result is analogous to one obtained by Sacker and Sell in [92] for a linear cocycle over a flow. The stronger version in Theorem 2.3 (of property 2 in Theorem 2.2) is taken from [17]. The splitting of the ambient space into the spaces .Hni in (2.1) was also obtained in [92], again for a linear cocycle over a flow. This property is reminiscent of an earlier result of Oseledets in [70] for a cocycle over an invertible measure-preserving map. The latter gives, for almost every point, a splitting into spaces obtained in a similar manner to that in (2.1) using the filtrations associated with the forward and backward Lyapunov exponents (instead of the stable and unstable spaces). The theory is beyond the scope of this book and instead we refer the reader to [10]. Theorem 2.4 on the Lyapunov exponents of the perturbations of a linear dynamics is taken from [17] and its proof is a simplification of the argument in that paper. To the possible extent, this follows a related approach in [12] (see also [81]). However, here the Lyapunov exponents need not be limits, which thus causes some modifications. Another important aspect is the use of Lyapunov norms, with respect to which the contraction and expansion of a tempered exponential dichotomy are transformed, respectively, in contraction and expansion in a single iteration. More precisely, while in (1.3) and (1.4), for a given .n ∈ N, the contraction occurs for .|m|

30

2 Asymptotic Behavior

sufficiently large, the contraction in (2.26) and (2.27) occurs after a single iteration (that is, for .m = n+1). We use these norms in the proof of Theorem 2.4 to show that for any exponentially decaying perturbation there is still contraction and expansion after a single step. We note that Lyapunov norms, and in fact also Lyapunov inner products, play an important role in the nonuniform hyperbolicity theory, particularly in the context of smooth ergodic theory (see [10] for details). For perturbations of autonomous difference equations, a version of Theorem 2.4 was established earlier by Coffman in [24] (see [28] for the case of continuous time). Some former related results are due Perron [75], Lettenmeyer [55] and Hartman and Wintner [45]. Corresponding work for autonomous delay equations is due to Pituk [80, 81] (for solutions with values in a finite-dimensional space and finite delay) and Matsui, Matsunaga and Murakami [65] (for solutions with values in a Banach space and infinite delay). See [12] for the study of a nonautonomous dynamics with discrete time when the Lyapunov exponents are assumed to be limits (such as for a Lyapunov regular system). We emphasize that the theory of Lyapunov exponents has many further developments, which are of interest by themselves, but these are out of the scope of this book. This includes the regularity theory of Lyapunov exponents (which goes back to Lyapunov himself) and the relation to ergodic theory. For details and references we refer the reader to [7, 10].

Chapter 3

Resonances and Normal Forms

In this chapter we construct normal forms for the perturbations of a nonautonomous linear dynamics. After introducing a notion of resonance in the general nonautonomous setting, the construction proceeds by removing all nonresonant terms up to a certain order. We start by making a preparation of the linear part of the dynamics by applying a coordinate change that brings it to a block-diagonal form.

3.1 Block-Diagonalization of a Linear Dynamics Let .(An )n∈Z be a sequence of invertible .d × d matrices with tempered growth (see Definition 2.1). By Proposition 2.1 the tempered spectrum is then nonempty and bounded, and so of the form (2.18) for some real numbers .ai and .bi as in (1.12) and some positive integer .k ≤ d. Mainly as a preparation for the construction of normal forms in Section 3.2, we show that any such sequence of matrices can be reduced to a sequence in block-diagonal form. The matrices .An induce a dynamics .xn+1 = An xn for .n ∈ Z. Now we consider a (nonautonomous) coordinate change .xn = Un yn for some invertible .d × d matrices .Un and all .n ∈ Z. Then −1 −1 yn+1 = Un+1 xn+1 = Un+1 An xn = Bn yn

.

for .n ∈ Z, where −1 Bn = Un+1 An Un

.

for n ∈ Z.

(3.1)

We shall consider coordinate changes of a certain type.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 L. Barreira, C. Valls, Spectra and Normal Forms, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-51897-3_3

31

32

3 Resonances and Normal Forms

Definition 3.1 A sequence of invertible .d × d matrices .(Un )n∈Z is said to be tempered if .

1 log||Un || = 0 and n→±∞ n lim

1 log||Un−1 || = 0. n→±∞ n lim

(3.2)

Two sequences of .d × d matrices .(An )n∈Z and .(Bn )n∈Z are said to be cohomologous if there is a tempered sequence .(Un )n∈Z satisfying (3.1). Proposition 3.1 Two cohomologous sequences of invertible .d × d matrices have the same tempered spectrum. Proof Let .Am,n and .Bm,n be defined as in (1.18) using, respectively, the matrices An and .Bn . By (3.1) we have

.

Bm,n = Um−1 Am,n Un .

.

(3.3)

Therefore, by (3.2) we have that .(e−a An )n∈Z has a tempered exponential dichotomy if and only if .(e−a Bn )n∈Z has a tempered exponential dichotomy. This yields the desired statement. ⨆ ⨅ Moreover, it follows readily from (3.2) and (3.3) that if one sequence has tempered growth, then the same happens to the other. The following theorem reduces any sequence of invertible matrices with tempered growth to a cohomologous sequence of matrices in block-diagonal form. Let ni = dim Hni

.

for i = 1, . . . , k

with the spaces .Hni as in (2.1). Theorem 3.1 Any sequence of invertible U .d × d matrices .(An )n∈Z with tempered growth and tempered spectrum .Σ = ki=1 [ai , bi ] is cohomologous to a sequence .(Bn )n∈Z with Bn = diag(Bn1 , . . . , Bnk ),

.

where each sequence .B i = (Bni )n∈Z is composed of .ni × ni matrices such that Σ(B i ) = [ai , bi ] for i = 1, . . . , k.

.

(3.4)

Proof We start with an auxiliary result. Lemma 3.1 If the sequence .(An )n∈Z has a tempered exponential dichotomy with projections .Pn , then it is cohomologous to a sequence Bn = diag(Bn1 , Bn2 )

.

for n ∈ Z,

(3.5)

3.1 Block-Diagonalization of a Linear Dynamics

33

where each .Bni is an .mi × mi matrix with .m1 = dim Pn (Rd ) and .m2 = d − m1 . Moreover, the sequence .(Bn )n∈Z has a tempered exponential dichotomy with projections .P = diag(IdRm1 , 0) that are independent of n. Proof of the lemma Let .Nn = An,0 S −1 , where S is an invertible .d × d matrix such that .SP0 S −1 = P . Now define In = P Nn∗ Nn P + (Id − P )Nn∗ Nn (Id − P ),

.

(3.6)

where .Nn∗ denotes the transpose of .Nn . One can easily verify that .In is symmetric and positive-definite and so it has a unique positive-definite square root .Jn . It follows from (3.6) that .In is a block matrix (with blocks of dimensions .m1 and .m2 ) and thus the same happens to .Jn . Finally, let Un = Nn Jn−1 .

.

(3.7)

Note that .Jn is a block matrix, and so P Jn = Jn P .

.

Moreover, by (3.6) we have P Nn∗ Nn P = P In = P Jn2 ,

.

which implies that P Un∗ Un P = P (Jn−1 )∗ Nn∗ Nn Jn−1 P = (Jn−1 )∗ P Nn∗ Nn P Jn−1 .

= (Jn−1 )∗ P Jn2 Jn−1 = (Jn−1 )∗ P Jn = P (Jn−1 )∗ Jn = P .

Analogously, (Id − P )Un∗ Un (Id − P ) = Id − P ,

.

which finally yields the identity Id = P Un∗ Un P + (Id − P )Un∗ Un (Id − P ).

.

For any vector .v ∈ Rd we thus have

(3.8)

34

3 Resonances and Normal Forms

( )2 ||Un v||2 ≤ ||Un P v|| + ||Un (Id − P )v|| .

≤ 2||Un P v||2 + 2||Un (Id − P )v||2 = 2||v||2 and so .||Un || ≤

√

2. On the other hand, by (3.7) we have

(Un−1 )∗ Un−1 = (Nn∗ )−1 Jn2 Nn−1 = (Nn∗ )−1 P Nn∗ Nn P Nn−1 + (Nn∗ )−1 (Id − P )Nn∗ Nn (Id − P )Nn−1

.

= Pn∗ Pn + (Id − Pn )∗ (Id − Pn ) and for any vector .v ∈ Rd we obtain ||Un−1 v||2 ≤ ||Pn v||2 + ||(Id − Pn )v||2 ≤ (||Pn ||2 + ||Id − Pn ||2 )||v||2 .

.

Hence, for each .ε > 0 there is .D = D(ε) > 0 such that ||Un−1 || ≤ (||Pn ||2 + ||Id − Pn ||2 )1/2 ≤ D(ε)eε|n|

.

for all .n ∈ Z. This shows that .

√ 1 −ε|n| e ≤ ||Un−1 ||−1 ≤ ||Un || ≤ 2 D(ε)

and so the sequence .(Un )n∈Z is tempered. Now observe that by (3.8) we also have Un−1 Pn Un = Jn Nn−1 Pn Nn Jn−1 = Jn P Jn−1 = P .

.

(3.9)

For .Bn as in (3.1) it follows from (3.9) that −1 P Bn = Un+1 Pn+1 Un+1 Bn −1 = Un+1 Pn+1 An Un .

−1 = Un+1 An Pn Un −1 = Un+1 An Un Un−1 Pn Un = Bn P .

This shows that .Bn has the block-diagonal form in (3.5). Moreover, since .Jn−1 has a block-diagonal form, we have

3.1 Block-Diagonalization of a Linear Dynamics

35

−1 Bn P = Un+1 An Un P −1 = Un+1 An An,0 S −1 Jn−1 P −1 = Un+1 An An,0 S −1 P Jn−1 .

−1 = Un+1 An An,0 P0 S −1 Jn−1

(3.10)

−1 = Un+1 An An,0 P0 A0,n Un −1 = Un+1 An Pn Un

and so also −1 Bn (Id − P ) = Un+1 An (Id − Pn )Un .

.

(3.11)

Since .(Un )n∈Z is tempered, this implies that the sequence .(Bn )n∈Z has a tempered exponential dichotomy with projections P independent of n (with the blocks .Bn1 and .Bn2 in (3.5) corresponding, respectively, to contraction and expansion). ⨆ ⨅ Take .c ∈ (b1 , a2 ). The sequence .(e−c An )n∈Z has a tempered exponential dichotomy. By Lemma 3.1 there is a tempered sequence .(Un )n∈Z such that −1 e−c Un+1 An Un = diag(Cn , Dn ) =: Xn ,

.

where .Cn and .Dn are blocks of dimensions, respectively, .n1 = dimHn1 and .d − n1 . Moreover, .(Xn )n∈Z has a tempered exponential dichotomy with projections .diag(IdRn1 , 0). On the other hand, it follows from Lemmas 1.1 and 1.2 (see (1.17)) that .Ena = Enb and .Fna = Fnb for a and b in the same connected component of .R \ Σ. Therefore, the matrix S and so also the matrices .Un in the proof of Lemma 3.1 can be chosen to be independent of c. This allows us to define Bn1 = ec Cn

.

for n ∈ Z

(since these matrices are thus independent of c). Now take .a ≥ c. Then (e−a Bn1 )n∈Z = (ec−a Cn )n∈Z has a tempered exponential contraction. Therefore, 1 1 .Σ(B ) ⊂ (−∞, c) and letting .c \ b1 we obtain .Σ(B ) ⊂ (−∞, b1 ]. Since .

Σ(B 1 ) ⊂ Σ(ec X) = Σ(A),

.

we conclude that .Σ(B 1 ) ⊂ [a1 , b1 ]. We continue with the construction of the matrices .Bni . Take .c¯ ∈ (b2 , a3 ). The sequence .(e−c¯ ec Dn )n∈Z has a tempered exponential dichotomy. By Lemma 3.1 there is a tempered sequence of .(d − n1 ) × (d − n1 ) matrices .(U¯ n )n∈Z such that −1 c e−c¯ U¯ n+1 e Dn U¯ n = diag(C¯ n , D¯ n ) =: X¯ n ,

.

36

3 Resonances and Normal Forms

where .C¯ n and .D¯ n are blocks of dimensions, respectively, .n2 = dimHn2 and .d − n1 −n2 . Moreover, .(X¯ n )n∈Z has a tempered exponential dichotomy with projections .diag(IdRn2 , 0). Analogously, one can show that the matrices Bn2 = ec¯ C¯ n

.

for n ∈ Z

are independent of .c¯ and .Σ(B 2 ) ⊂ [a2 , b2 ]. In addition, the sequence of matrices ¯ n ) for .n ∈ Z is tempered and .Vn = Un diag( IdRn1 , U −1 Vn+1 An Vn = diag(Bn1 , Bn2 , ec¯ D¯ n ).

.

One can continue this procedure to find in a finitely many steps a tempered sequence (V¯n )n∈Z such that

.

−1 An V¯n = diag(Bn1 , . . . , Bnk ) =: Bn V¯n+1

.

for some .ni × ni matrices .Bni with Σ(B i ) ⊂ [ai , bi ]

.

for i = 1, . . . , k.

Finally, by Proposition 3.1 we have .Σ(B) = Σ(A). Since .Σ(B) = follows from (3.12) that property (3.4) holds.

(3.12) Uk

i=1 Σ(B

i ),

it

3.2 Construction of Normal Forms In this section we construct normal forms, using the tempered spectrum and the block-diagonalization of a linear dynamics in Section 3.1. We continue to assume that .(An )n∈Z is a sequence of invertible .d × d matrices with tempered growth. Let .(Bn )n∈Z be the cohomologous sequence constructed in Theorem 3.1 and consider the nonlinear dynamics xn+1 = Bn xn + fn (xn )

.

for n ∈ Z,

(3.13)

where each map .fn : Rd → Rd is of class .C p , for some .p ≥ 1, with .fn (0) = 0 and d0 fn = 0 for .n ∈ Z. Given .x ∈ Rd , we write .x = (x 1 , . . . , x k ), where .x i ∈ Rni for i .i = 1, . . . , k with .ni = dim Hn (see (2.1)). Moreover, for each .r = (r1 , . . . , rk ) ∈ Nk0 we define .

|r| = r1 + · · · + rk

.

and

One can write each map .fn as a Taylor series

r! = r1 ! · · · rk !.

3.2 Construction of Normal Forms

37

Σ

fn (x) =

.

r∈Nk0 ,2≤|r|≤p

1 r ∂ fn x r + o(||x||p ), r! 0

(3.14)

where ∂ r fn = ∂xr11 · · · ∂xrkk fn

.

is a partial derivative and x r = (x 1 )r1 · · · (x k )rk

.

is a multivector in .Rr1 n1 +···+rk nk . More precisely, .(x i )ri is a notation for i i rn .(x , . . . , x ) ∈ R i i and we have | ∂0r fn x r = ∂xr11 · · · ∂xrkk fn |x=0 (x 1 )r1 · · · (x k )rk ,

.

where the .|r|-linear map .∂0r fn is applied to the multivector .x r . Now we introduce a notion of resonance in the present nonautonomous setting. Definition 3.2 Given .i ∈ {1, . . . , k} and .r ∈ Nk0 with .|r| ≥ 2, we say that the pair .(i, r) is a resonance of order .|r| if ai ≤

.

and

bi ≥ ,

(3.15)

where . is the standard inner product on .Rk . When .aj = bj for all .j = 1, . . . , k, condition (3.15) becomes .ai = , which recovers the classical (autonomous) notion of resonance. Writing fn = (fn1 , . . . , fnk ),

.

with .fni : Rd → Rni for .i = 1, . . . , k, the term .(1/r!)∂0r fni x r in (3.14) is said to be resonant if the pair .(i, r) is a resonance. Now we construct normal forms for the perturbations of a sequence of matrices in block-diagonal form. This corresponds to eliminate all nonresonant terms in (3.14) up to order p by making an appropriate coordinate change. Theorem 3.2 Let .(An )n∈Z be a sequence of invertible .d ×d matrices with tempered growth and let .(Bn )n∈Z be the cohomologous sequence constructed in Theorem 3.1. Moreover, let .fn : Rd → Rd be maps of class .C p , for some .p ≥ 1, with .fn (0) = 0 and .d0 fn = 0 for .n ∈ Z such that .

lim sup n→±∞

1 log||∂0r fn || ≤ 0 for 2 ≤ |r| ≤ p. |n|

(3.16)

38

3 Resonances and Normal Forms

Then: 1. There are polynomials .hn : Rd → Rd with .hn (0) = 0 and .d0 hn = 0 for .n ∈ Z satisfying .

lim sup n→±∞

1 log||hn (x)|| ≤ 0. |n|

(3.17)

2. Letting .xn = yn + hn (yn ) for .n ∈ Z in (3.13) we obtain yn+1 = Bn yn + gn (yn )

.

for n ∈ Z,

(3.18)

where .gn = (gn1 , . . . , gnk ) : Rd → Rd are maps of class .C p with .gn (0) = 0 and .d0 gn = 0. 3. .∂0r gni = 0 for all .n ∈ Z, .i = 1, . . . , k and .r ∈ Nk0 with .2 ≤ |r| ≤ p such that .(i, r) is not a resonance. Proof Assume that .(i, r) is not a resonance. It follows from (3.15) that either .ai > l . or .bi < . Let .Blm,n be as in (1.18) with each matrix .Am replaced by .Bm d d We define maps .hn : R → R by hn = (h1n , . . . , hkn ),

.

(3.19)

j

where .hn = 0 for .j /= i, i .hn (x)

=−

+∞ Σ

Bin,l+1

1 r i 1 1 r1 ∂ f (B x ) · · · (Bkl,n x k )rk r! 0 l l,n

(3.20)

Bin,l+1

1 r i 1 1 r1 ∂ f (B x ) · · · (Bkl,n x k )rk r! 0 l l,n

(3.21)

l=n

whenever .ai > and hin (x) =

n−1 Σ

.

l=−∞

whenever .bi < . To verify that the two series converge, take .ci < ai and .ci > bi for .i = 1, . . . , k such that ci > whenever ai >

.

and ci < whenever bi < ,

.

where

3.2 Construction of Normal Forms

c = (c1 , . . . , ck )

.

39

and

c = (c1 , . . . , ck ).

(3.22)

We observe that given .ε > 0, there is .D = D(ε) > 0 such that .

||Bim,n || ≤ Deci (m−n)+ε|n|

for m ≥ n

(3.23)

||Bim,n || ≤ Deci (m−n)+ε|n|

for m ≤ n.

(3.24)

and .

Indeed, by construction the sequences .(e−ci Bni )n∈Z and .(e−ci Bni )n∈Z have, respectively, a tempered exponential contraction and a tempered exponential expansion. In particular, since .(e−ci Bni )n∈Z has a tempered exponential contraction, there are a constant .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that ||e−ci (m−n) Bim,n || ≤ De−λ(m−n)+ε|n| ≤ Deε|n|

.

for .m ≥ n, which yields (3.23). Property (3.24) can be obtained similarly. On the other hand, by (3.16), for each .ε > 0 there is .J = J (ε) > 0 such that ||∂0r fn || ≤ J eε|n|

.

for 2 ≤ |r| ≤ p, n ∈ Z.

Now let || || 1 || || cnl : = ||Bin,l+1 ∂0r fli (B1l,n x 1 )r1 · · · (Bkl,n x k )rk || r! . 1 ≤ ||Bin,l+1 || J eε|l| ||B1l,n ||r1 ||x 1 ||r1 · · · ||Bkl,n ||rk ||x k ||rk r! and assume that .ai > . For .l ≥ n we have cnl ≤ .

1 J D |r|+1 eε|l|+ε|l+1|+|r|ε|n| e−ci +(ci −)(n−l) ||x|||r| r!

(3.25)

≤ dnr e−ci +(ci −−2ε)(n−l) ||x|||r| with 1 J D |r|+1 eε+ε(2+|r|)|n| . (3.26) r! Σ Since .ci − > 0, the series . l≥n cnl converges for any sufficiently small .ε. Now assume that .bi < . For .l ≤ n we have dnr =

.

40

3 Resonances and Normal Forms

cnl ≤ .

1 J D |r|+1 eε|l|+ε|l+1|+|r|ε|n| e−ci +(ci −)(n−l) ||x|||r| r!

(3.27)

≤ dnr e−ci +(ci −+2ε)(n−l) ||x|||r| , Σ again with .dnr as in (3.26). Since .ci − < 0, the series . l≤n cnl converges for any sufficiently small .ε. By construction each component .hin in (3.19) is a polynomial of degree 0 or .|r| in the variables .x 1 , . . . , x k and so the same happens to .hn . It follows from (3.25) and (3.27) that ||hn (x)|| ≤ max{e−ci , e−ci }dnr ||x|||r|

.

and hence, .

1 log||hn (x)|| ≤ ε(2 + |r|). |n|

lim sup n→±∞

Finally, the arbitrariness of .ε yields property (3.17). Taking .xn = yn + hn (yn ) we obtain (3.18), where gn (yn ) = Bn hn (yn ) + fn (yn + hn (yn )) − hn+1 (yn+1 ) .

= Bn hn (yn ) + fn (yn ) − hn+1 (yn+1 ) + o(||yn |||r| ).

When .ai > we have hin+1 (Bn y) = −

+∞ Σ

.

Bin+1,l+1

1 r i 1 B 1 y 1 )r1 · · · (Bkl,n+1 Bnk y k )rk ∂ f (B r! 0 l l,n+1 n

Bin+1,l+1

1 r i 1 1 r1 ∂ f (B y ) · · · (Bkl,n y k )rk r! 0 l l,n

l=n+1

=−

+∞ Σ l=n+1

= Bni hin (y) +

1 r i r ∂ f y , r! 0 n

and when .bi < we have hin+1 (Bn y) =

n Σ

Bin+1,l+1

1 r i 1 ∂ f (B B 1 y 1 )r1 · · · (Bkl,n+1 Bnk y k )rk r! 0 l l,n+1 n

Bin+1,l+1

1 r i 1 1 r1 ∂ f (B y ) · · · (Bkl,n y k )rk r! 0 l l,n

l=−∞ .

=

n Σ l=−∞

= Bni hin (y) +

1 r i r ∂ f y . r! 0 n

3.2 Construction of Normal Forms

41

Therefore, ⎞ ⎛ 1 hn+1 (Bn y) = Bn hn (y) + 0, . . . , 0, ∂0r fni y r , 0, . . . , 0 , r!

.

which gives ⎛

⎞ 1 r i r .gn (y) = fn (y) − 0, . . . , 0, ∂ f y , 0, . . . , 0 + o(||y|||r| ). r! 0 n This corresponds to eliminate the term .(1/r!)∂0r fni y r in .fn . To eliminate all nonresonant terms up to order p from each map .fn we apply successively coordinate changes as above, starting with those of lowest order (that is, from .|r| = 2 to .|r| = p). We note that each coordinate change only affects the corresponding resonant term and eventually terms of higher order. Therefore, by starting with the terms of the lowest order ensures that after eliminating a resonant term it cannot reappear afterwards. This leads, in finitely many steps, to a coordinate change xn = yn + h¯ n (yn )

.

for n ∈ Z

and some polynomials .h¯ n that transforms the dynamics in (3.13) to that in (3.18), ⨆ ⨅ now without nonresonant terms up to order .|r|. We also consider briefly the case when there are no resonances. The following result is a simple consequence of Theorem 3.2. Theorem 3.3 Let .(An )n∈Z be a sequence of invertible .d ×d matrices with tempered growth and let .(Bn )n∈Z be the cohomologous sequence constructed in Theorem 3.1. Moreover, let .fn : Rd → Rd be maps of class .C p , for some .p ≥ 1, with .fn (0) = 0 and .d0 fn = 0 for .n ∈ Z satisfying (3.16). If there are no resonances, then for the maps .gn in (3.18) we have .d0i gn = 0 for all .n ∈ Z and .0 ≤ i ≤ p. Proof By Theorem 3.2, using appropriate coordinate changes one can eliminate all terms in (3.14) up to order p (because there are no resonances). This yields the ⨆ ⨅ desired result. As already noted in the preface, in general the tempered spectrum may be smaller than the corresponding spectrum defined in terms of exponential dichotomies. The following example shows that indeed this may happen and lead to much simpler normal forms. Example 3.1 For each .n ∈ Z, let ⎞ ⎛ cn 0 , .An = 0 ecn

(3.28)

42

3 Resonances and Normal Forms

where cn = e

.

√

√ |n+1| cos(n+1)− |n| cos n

.

By Example 1.3 the tempered spectrum is .Σ = {0, 1}. In particular, the resonances of order 2 are .(1, r) with .r = (2, 0), and .(2, r) with .r = (1, 1). This corresponds, respectively, to the terms .(x 2 , 0) and .(0, xy) in the Taylor series of each perturbation .fn . Therefore, by Theorem 3.2 there is a near-identity coordinate change xn = yn + hn (yn )

.

for n ∈ Z

that eliminates all terms of order 2 in the dynamics (xn+1 , yn+1 ) = (cn xn , ecn yn ) + fn (xn , yn )

.

for n ∈ Z,

(3.29)

where ) ( fn (x, y) = k1n xy + k2n y 2 , k3n x 2 + k4n y 2

.

for some constants .k1n , k2n , k3n , k4n ∈ R such that .

1 log |kin | ≤ 0 for i = 1, 2, 3, 4. |n|

lim sup n→±∞

Now let .Σ ' be the set of all numbers .a ∈ R such that the sequence .(e−a An )n∈Z does not have a uniform exponential dichotomy. We recall that a sequence of matrices .(An )n∈Z (here for simplicity assumed to be invertible) is said to have a uniform exponential dichotomy if there are projections .Pn for .n ∈ Z satisfying (1.1) and constants .λ, D > 0 such that ||Am,n Pn || ≤ De−λ(m−n)

.

for m ≥ n

and ||Am,n (Id − Pn )|| ≤ De−λ(n−m)

.

for m ≤ n.

Clearly, any uniform exponential dichotomy is also a tempered exponential dichotomy, which implies that .Σ ⊂ Σ ' . In fact we show that .Σ ' = R for the sequence of matrices in (3.28). Note that Am,n =

.

where

⎞ ⎛ 0 cm,n , 0 em−n cm,n

3.2 Construction of Normal Forms

43

cm,n = e

.

√

√ |m| cos m− |n| cos n

.

Now take .a ∈ R. We show that .(e−a cn )n∈Z does not have a uniform exponential dichotomy with .Pn = Id for .n ∈ Z. Otherwise, we would have √

e

.

√ m cos m− n cos n

≤ De(a−λ)(m−n)

for m ≥ n ≥ 0.

Letting .m = 2π l and .n = 2π l − π with .l ∈ N, we obtain √

e

.

√ 2π l+ 2π l−π

≤ De(a−λ)π ,

(3.30)

which fails for l sufficiently large. Now we show that .(e−a cn )n∈Z does not have a uniform exponential dichotomy with .Pn = 0 for .n ∈ Z. Otherwise, √

e

.

√ |m| cos m− |n| cos n

≤ De(a−λ)(n−m)

for m ≤ n < 0.

Letting .m = −2π l and .n = −2π l + π with .l ∈ N, again we obtain (3.30), which fails for l sufficiently large. Similarly, one can show that for each .a ∈ R the sequence −a ec ) .(e n n∈Z does not have a uniform exponential dichotomy with .Pn = Id for .n ∈ Z, neither a uniform exponential dichotomy with .Pn = 0 for .n ∈ Z. This readily yields the desired property. Since .Σ ' = R, one cannot even start discussing a notion of resonance. In other words, the classical theory based on uniform exponential dichotomies cannot be applied to simplify the dynamics in (3.29). Notes The notions of tempered sequence and of cohomologous sequences (see Definition 3.1 and Proposition 3.1) are taken from [10]. Theorem 3.1 on the block-diagonalization of a linear dynamics is taken from [14]. The construction in the proof of Lemma 3.1, which is the center of the argument, is a modification of a construction of Coppel in [31] for ordinary differential equations (see also his former works [29, 30]). An analogous result to Theorem 3.1 was first obtained by Siegmund in [99], using the spectrum introduced in [97]. In the case of discrete time, Siegmund [96] (see also [74]) made an analogous construction using the spectrum introduced in [6]. We note that the spectra considered in all these works are defined in terms of uniform exponential dichotomies. Theorem 3.1 is analogous to an earlier result in ergodic theory known as the Oseledets–Pesin reduction theorem. The latter transforms a cocycle in another one in block-form via a tempered coordinate change (see [10] for details). The notion of resonance in Definition 3.2 is taken from Siegmund [100] (see also [39]), although we use the tempered spectrum, which is defined in terms of tempered exponential dichotomies. As we already noted in Chapter 1, in general the tempered spectrum and the spectrum defined in terms of uniform exponential dichotomies may be distinct and so they may lead to different resonances.

44

3 Resonances and Normal Forms

The construction of normal forms in Theorem 3.2 is taken from [14]. The first construction of normal forms for a nonautonomous dynamics with discrete time is due to Siegmund [100], again using a spectrum defined in terms of uniform exponential dichotomies, and to the possible extent our proof follows his approach. The estimates contain extra terms because we consider tempered exponential dichotomies and tempered perturbations (see (3.16)). For a different and earlier approach we refer the reader to work of Guysinsky and Katok in [39]. They construct normal forms for extensions of homeomorphisms of compact metric spaces by smooth contractions on .Rd , using an analogous notion of subresonance and a narrow band condition. We recall that when the intervals of the tempered spectrum reduce to points, that is, when .ai = bi for .i = 1, . . . , k, the notion of resonance in Definition 3.2 coincides with the classical notion of resonance. On the other hand, for an arbitrary nonautonomous linear dynamics, a priori the lower and upper Lyapunov exponents may take any value in the intervals .[ai , bi ] (see Theorems 2.1 and 2.3) and so in general the new notion is distinct. When those intervals are not single points, the series in (3.20) and (3.21) may not converge unless use the new notion of resonance. Of course, some specific examples may require less restrictive conditions. The notion of normal form goes back to Poincaré in [83] in connection with his study of the integrability of nonlinear systems and nowadays has many applications. For details and references see for example the books [4, 38, 48, 104]. The idea is always to eliminate successively terms of progressively higher order using nearidentity coordinate changes. The infinite-dimensional setting of linear operators acting on a Banach space is considered in Chapter 6. In particular, we construct normal forms for the tempered perturbations of a sequence of compact linear operators.

Chapter 4

Parameter-Dependent Dynamics

This chapter is dedicated to the study of parameter-dependent perturbations of a linear dynamics. We first consider a parameter-dependent linear perturbation of a linear dynamics and we describe how the tempered spectrum may vary. Then we study the smooth dependence of a normal form on a parameter when the nonlinear perturbation depends smoothly on the parameter.

4.1 Linear Perturbations In this section we describe how the tempered spectrum may vary under sufficiently small parameter-dependent linear perturbations. We start with an auxiliary result concerning the constants in the notion of tempered exponential dichotomy. Given .δ > 0, let .Σδ be the .δ-neighborhood of .Σ. Proposition 4.1 Let .(An )n∈Z be a sequence ofUinvertible .d × d matrices with tempered growth and tempered spectrum .Σ = ki=1 [ai , bi ]. Given .δ > 0 such that .Σδ has k connected components, there are a constant .κ(δ) > 0 and for each .ε > 0 a constant .C(δ) = C(δ, ε) > 0 such that for each .a ∈ R \ Σδ the sequence −a A ) .(e n n∈Z has a tempered exponential dichotomy with constants .κ(δ) and .C(δ). Proof Take .i = 1, . . . , k − 1 and .a ∈ [bi + δ, ai+1 − δ]. Note that indeed bi + δ < ai+1 − δ since .Σδ has k connected components. Then the sequence −a A ) .(e n n∈Z has a tempered exponential dichotomy with projections .Pn = Pn (a) for .n ∈ Z satisfying (1.1), a constant .λ(a) > 0 and for each .ε > 0 a constant .D(a) = D(a, ε) > 0 satisfying the usual exponential bounds. It follows readily from the proof of Lemma 1.1 that for each .

( ) b ∈ a − λ(a), a + λ(a) ∩ (R \ Σ)

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 L. Barreira, C. Valls, Spectra and Normal Forms, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-51897-3_4

45

46

4 Parameter-Dependent Dynamics

the sequence .(e−b An )n∈Z also has a tempered exponential dichotomy, with the same projections .Pn and constant .D(b) = D(a) for each .ε > 0. Covering the compact set .[bi + δ, ai+1 − δ] by intervals .(a − λ(a), a + λ(a)) and extracting a finite subcover, we obtain projections .Pn = Pn,i for .n ∈ Z satisfying (1.1), a constant .λi > 0 and for each .ε > 0 a constant .Di = Di (ε) > 0 such that for each .a ∈ [bi + δ, ai+1 − δ] we have ||e−a(m−n) Am,n Pn,i || ≤ Di e−λi (m−n)+ε|n|

.

for m ≥ n

and ||e−a(m−n) Am,n (Id − Pn,i )|| ≤ Di e−λi (n−m)+ε|n|

.

for m ≤ n.

On the other hand, there are constants .λ0 , D0 > 0 such that ||e−(a1 −δ)(m−n) Am,n || ≤ D0 e−λ0 (n−m)+ε|n|

.

for .m ≤ n. Given .a ≤ a1 − δ, we obtain .

||e−a(m−n) Am,n || ≤ D0 e(−λ0 −a1 +δ+a)(n−m)+ε|n| ≤ D0 e−λ0 (n−m)+ε|n|

for .m ≤ n. Therefore, the sequence .(e−a An )n∈Z has a tempered exponential expansion with constants .λ0 and .D0 . Analogously, one can show that for .a ≥ ak + δ the sequence .(e−a An )n∈Z has a tempered exponential contraction with some constants .λk and .Dk . Now let κ(δ) = min{λ0 , . . . , λk }

.

and

C(δ) = max{D0 , . . . , Dk }.

It follows from the former discussion that for each .a ∈ R \ Σδ the sequence (e−a An )n∈Z has a tempered exponential dichotomy with constants .κ(δ) and .C(δ), which thus are independent of a (the last constant may depend on .ε). ⨆ ⨅

.

Before proceeding, we recall the notion of exponential dichotomy with respect to a sequence of norms, which is convenient for what follows. Let .||·|| for .n ∈ Z be norms on .Rd . Definition 4.1 A sequence of invertible .d × d matrices .(An )n∈Z is said to have an exponential dichotomy with respect to the norms .||·||n if there are projections .Pn for d .n ∈ Z satisfying (1.1) and constants .λ, D > 0 such that for each .x ∈ R we have ||Am,n Pn x||m ≤ De−λ(m−n) ||x||n

.

for m ≥ n

(4.1)

and ||Am,n (Id − Pn )x||m ≤ De−λ(n−m) ||x||n

.

for m ≤ n.

(4.2)

4.1 Linear Perturbations

47

Proceeding as in the proof of Theorem 2.4, one can show that .(An )n∈Z has a tempered exponential dichotomy if and only if for each .ε > 0 the sequence .(An )n∈Z has an exponential dichotomy with respect to some norms .||·||n satisfying (2.25). In fact, given a sequence .(An )n∈Z with a tempered exponential dichotomy, one can take ( ( ) ) ||x||n = sup eλ(m−n) ||Am,n Pn x|| + sup e−λ(m−n) ||Am,n Qn x||

.

m≥n

(4.3)

m≤n

(which corresponds to let .ci = 0 in (2.24)). It turns out that for these norms one can take .D = 1 in (4.1) and (4.2). Given .l ∈ N, let .Bn (θ ) be .d × d matrices for .n ∈ Z and .θ ∈ Rl . We say that 1 .(Bn (θ ))n∈Z is a smooth family if each map .θ |→ Bn (θ ) is of class .C (that is, if 1 all entries of each matrix .Bn (θ ) are of class .C in .θ ). We denote its derivative by ' .Bn (θ ), taken entry by entry. Proposition 4.2 Let .(An )n∈Z be a sequence of matrices with a tempered exponential dichotomy and let .(Bn (θ ))n∈Z be a smooth family. If ⎧

||B ' (θ )x||n+1 ||Bn (θ )x||n+1 , sup n .c = sup sup sup ||x||n ||x||n x/=0 θ∈Rl n∈Z x/=0

⎫ (4.4)

is sufficiently small, then: 1. .(An + Bn (θ ))n∈Z has a tempered exponential dichotomy for each .θ ∈ Rl , with the constant .λ replaced by .λ − log(1 + 2c). 2. The corresponding projections .Pn (θ ) are of class .C 1 in .θ for each .n ∈ Z. Proof It follows from the discussion after Definition 4.1 that .(An )n∈Z has an exponential dichotomy with respect to the norms in (4.3), with constants .λ and .D = 1. Therefore, provided that c is sufficiently small, it follows from the robustness result in [11] that each sequence .(An + Bn (θ ))n∈Z has an exponential dichotomy with respect to the same norms, with the constant .λ replaced by λ − log(1 + 2cD) = λ − log(1 + 2c)

.

(although that paper considers only a constant sequence of norms, it is straightforward to modify the argument to consider an arbitrary sequence of norms). This yields property 1. To obtain property 2 we need to recall some material from [11]. Consider the stable and unstable spaces .En (θ ) and .Fn (θ ) for the sequence .(An + Bn (θ ))n∈Z , and the stable and unstable spaces .En and .Fn for .(An )n∈Z . Then there are linear maps ϕn (θ ) : En → Fn

.

and

ψn (θ ) : Fn → En

whose graphs are, respectively, .En (θ ) and .Fn (θ ). This means that

48

4 Parameter-Dependent Dynamics

En (θ ) = (IdEn + ϕn (θ ))En

.

(4.5)

and Fn (θ ) = (IdFn + ψn (θ ))Fn .

.

Moreover, the functions .ϕn (θ ) and .ψn (θ ) are of class .C 1 in .θ and by taking c sufficiently small in (4.4), the operator (IdEn + ϕn (θ ))Pn + (IdFn + ψn (θ ))Qn = Id + ϕn (θ )Pn + ψn (θ )Qn

.

becomes invertible (here .Pn and .Qn = Id − Pn are the projections associated to the splitting .Rd = En ⊕ Fn ). Let L be an invertible linear map such that LP0 L−1 = P := diag(IdRn1 , 0),

.

(4.6)

where .n1 = dim En , and define ┐−1 ┌ S(θ ) = L Id + ϕ0 (θ )P0 + ψ0 (θ )Q0 .

.

(4.7)

Note that .S(θ ) is invertible and both maps .S(θ ) and .S(θ )−1 are of class .C 1 in .θ . Moreover, by (4.5) we have P0 (θ )(IdE0 + ϕ0 (θ ))P0 = P0 + ϕ0 (θ )P0 ,

.

which gives ┐ ┌ P0 (θ )S(θ )−1 = P0 (θ ) Id + ϕ0 (θ )P0 + ψ0 (θ )Q0 L−1 = (P0 + ϕ0 (θ )P0 )L−1 ┐ ┌ = Id + ϕ0 (θ )P0 + ψ0 (θ )Q0 P0 L−1 ┐ ┌ = Id + ϕ0 (θ )P0 + ψ0 (θ )Q0 L−1 P

.

(4.8)

= S(θ )−1 P . Therefore, it follows from (1.1) that Pn (θ ) = Cn,0 (θ )S(θ )−1 P S(θ )C0,n (θ ),

.

(4.9)

where .Cm,n (θ ) is given by (1.18) with each matrix .An replaced by .An + Bn (θ ). It follows readily from (4.9) that the projections .Pn (θ ) are of class .C 1 in .θ . ⨆ ⨅ One can now establish the main result of the section. Essentially it says that the tempered spectrum varies little under sufficiently small linear perturbations.

4.1 Linear Perturbations

49

Theorem 4.1 Let .(An )n∈Z be a sequence Uof invertible .d ×d matrices with tempered growth and tempered spectrum .Σ = ki=1 [ai , bi ]. Moreover, let .(Bn (θ ))n∈Z be a smooth family. Given .δ > 0 such that .Σδ has k connected components, if the constant c in (4.4) is sufficiently small, then: 1. The spectrum .Σ(θ ) of each sequence .(An + Bn (θ ))n∈Z has at least k connected components and satisfies .Σ(θ ) ⊂ Σδ . 2. The projections .Pna (θ ) of the exponential dichotomy of .(e−a (An + Bn (θ )))n∈Z for each .a ∈ R \ Σ(θ ) satisfy .Pna (θ ) = Pnb (θ ) for a and b in the same connected component of .R \ Σδ and are of class .C 1 in .θ . Proof Take .a ∈ (R \ Σδ ) ∩ [a1 − δ, bk + δ] and write Cna (θ ) = e−a An + e−a Bn (θ ).

.

(4.10)

By Proposition 4.1, the sequence .(e−a An )n∈Z has a tempered exponential dichotomy with constants .κ(δ) and .C(δ). Hence, it follows from (2.26) and (2.27) (with .ci = 0) that .(e−a An )n∈Z has an exponential dichotomy with respect to the sequence of norms in (4.3), with constants .κ(δ) and 1. It follows from Proposition 4.2 that if the supremum in (4.4) is sufficiently small (depending only on .κ(δ) since the second constant is 1), then the sequence .(Cna (θ ))n∈Z also has an exponential dichotomy with respect to the same sequence of norms. Therefore, a l .(Cn (θ ))n∈Z has a tempered exponential dichotomy for all .θ ∈ R and a ∈ (R \ Σδ ) ∩ [a1 − δ, bk + δ].

.

Moreover, the corresponding projections .Pna (θ ) are of class .C 1 in .θ . Now take .a ∈ R \ (a1 − δ, bk + δ). Since the projections .Pna1 −δ (θ ) and .Pnbk +δ (θ ) are of class .C 1 in .θ , the dimensions of the corresponding stable spaces (and so also of the unstable spaces) are independent of .θ and coincide, respectively, with the dimensions of the stable spaces for the sequences (e−a1 +δ An )n∈Z

.

and

(e−bk −δ An )n∈Z .

Therefore, for .a = a1 − δ the sequence .(Cna (θ ))n∈Z has a tempered exponential expansion and so the same happens for .a < a1 − δ. Similarly, for .a = ar + δ the sequence .(Cna (θ ))n∈Z has a tempered exponential contraction and so the same happens for .a > ar + δ. This shows that .Σ(θ ) ⊂ Σδ . Moreover, .Pna (θ ) = Pnb (θ ) for a and b in the same connected component of .R \ Σ(θ ), and so also in the same connected component of .R \ Σδ . Since the stable and unstable spaces have dimensions that are independent of the parameter .θ , for any .a1 , a2 ∈ R \ Σ(θ ) in different connected components of .R \ Σδ the dimensions of the corresponding stable and unstable spaces are distinct. This shows that there is at least one point in .Σ(θ ) between .a1 and .a2 , and so this spectrum has at least k connected components. ⨆ ⨅

50

4 Parameter-Dependent Dynamics

4.2 Nonlinear Perturbations In this section we consider smooth parameter-dependent perturbations of a linear nonautonomous dynamics (that is independent of the parameter) and we construct normal forms that also depend smoothly on the parameter. See Section 4.3 for the general case when both the linear part and the perturbation may depend on the parameter. Let .(An )n∈Z be a sequence of invertible .d ×d matrices with tempered growth. Let also .(Bn )n∈Z be the cohomologous sequence constructed in Theorem 3.1. Moreover, let .fn : Rd × Rl → Rd be continuous maps of class .C p in the first variable, for some .p ≥ 1, and of class .C 1 in the second variable. We assume that .fnθ (0) = 0 and θ l θ .d0 fn = 0 for all .n ∈ Z and .θ ∈ R , where .fn = fn (·, θ ). In a similar manner to θ that in Section 3.2, we write the maps .fn in the form Σ

fnθ (x) =

.

r∈Nk0 ,2≤|r|≤p

1 r θ r ∂ f x + o(||x||p ). r! 0 n

Now we consider the nonlinear dynamics xn+1 = Bn xn + fnθ (xn )

.

for n ∈ Z.

(4.11)

Theorem 4.2 Let .(An )n∈Z be a sequence of invertible .d ×d matrices with tempered growth and let .(Bn )n∈Z be the cohomologous sequence constructed in Theorem 3.1. Moreover, let .fn : Rd × Rl → Rd be continuous maps of class .C p in the first variable, for some .p ≥ 1, and of class .C 1 in the second variable, with .fnθ (0) = 0 and .d0 fnθ = 0 for .n ∈ Z and .θ ∈ Rl , such that ⎫ ⎧ || r θ || || || 1 r θ || ∂(∂0 fn ) || ≤ 0 for 2 ≤ |r| ≤ p. log max ||∂0 fn ||, || . lim sup ∂θ || θ∈Rl n→±∞ |n|

(4.12)

Then: 1. For each .θ ∈ Rl there are polynomials .hθn : Rd → Rd with .hθn (0) = 0 and θ .d0 hn = 0 for .n ∈ Z satisfying .

lim sup n→±∞

1 log||hθn (x)|| ≤ 0. |n|

2. Letting .xn = yn + hθn (yn ) for .n ∈ Z in (4.11) we obtain yn+1 = Bn yn + gnθ (yn )

.

for n ∈ Z,

(4.13)

4.3 General Perturbations

51

where .gnθ = (gn1θ , . . . , gnkθ ) : Rd → Rd are maps of class .C p with .gnθ (0) = 0 and .d0 gnθ = 0. 3. .∂0r gniθ = 0 for all .n ∈ Z, .i = 1, . . . , k and .r ∈ Nk0 with .2 ≤ |r| ≤ p such that .(i, r) is not a resonance. 4. The maps .(x, θ ) |→ hθn (x) are continuous and are of class .C 1 in .θ . Proof One can repeat the construction in the proof of Theorem 3.2 to obtain parameter-dependent normal forms for the perturbed equation (4.11). We recall how the maps .hθn are constructed. Assume that .(i, r) is not a resonance and let .Blm,n be as in (1.18) with each matrix .An replaced by .Bnl . We define kθ hθn = (h1θ n , . . . , hn ),

.

jθ

where .hn = 0 for .j /= i, hiθ n (x) = −

+∞ Σ

Bin,l+1

1 r iθ 1 1 r1 ∂ f (Bl,n x ) · · · (Bkl,n x k )rk r! 0 l

(4.14)

Bin,l+1

1 r iθ 1 1 r1 ∂ f (Bl,n x ) · · · (Bkl,n x k )rk r! 0 l

(4.15)

.

l=n

whenever .ai > and hiθ n (x) =

n−1 Σ

.

l=−∞

whenever .bi < . Using condition (4.12), one can show as in the proof of Theorem 3.2 that these series converge absolutely and uniformly in .θ ∈ Rl . Moreover, by construction .hθn is a polynomial. Since each term in the series is continuous in .(x, θ ), the map .(x, θ ) |→ hθn (x) is automatically continuous. To show that .hθn is of class .C 1 in .θ one can proceed similarly, writing fnθ = (fn1θ , . . . , fnkθ )

.

∂ (∂0r fliθ ) in (4.14) and (4.15). Using condition (4.12), one and replacing .∂0r fliθ by . ∂θ can show that the new series converge absolutely and uniformly in .θ ∈ Rl (the estimates are identical to those in the proof of Theorem 3.2). This implies that .hθn is of class .C 1 in .θ . ⨆ ⨅

4.3 General Perturbations Finally, in this section we construct normal forms for a nonautonomous dynamics in which both the linear part and the perturbation may depend on a parameter. The

52

4 Parameter-Dependent Dynamics

construction uses an invariant splitting that is independent of the parameter with respect to which the dynamics has a block-diagonal form. We continue to write .ni = dim Hni with the spaces .Hni as in (2.1). Theorem 4.3 Let .(An )n∈Z be a sequence Uof invertible .d ×d matrices with tempered growth and tempered spectrum .Σ = ki=1 [ai , bi ]. Moreover, let .(Bn (θ ))n∈Z be a smooth family. Given .δ > 0 such that .Σδ has k connected components, if the constant c in (4.4) is sufficiently small, then: 1. Each sequence .(An + Bn (θ ))n∈Z is cohomologous to a sequence .(Dn (θ ))n∈Z with Dn (θ ) = diag(Dn1 (θ ), . . . , Dnk (θ ))

.

for n ∈ Z,

(4.16)

where each .Dni (θ ) is an .ni × ni matrix of class .C 1 in .θ . 2. There are tempered sequences .(Un (θ ))n∈Z composed of matrices of class .C 1 in .θ that satisfy Dn (θ ) = Un+1 (θ )−1 (An + Bn (θ ))Un (θ )

.

for n ∈ Z.

(4.17)

3. The spectrum of each sequence .(Dni (θ ))n∈Z is contained in .[ai − δ, bi + δ] and these intervals are pairwise disjoint. Proof For the matrices .Cna (θ ) in (4.10) and for a in a bounded connected component of .R \ Σδ , it follows from Theorem 4.1 that .(Cna (θ ))n∈Z has a tempered exponential dichotomy, for each .θ ∈ Rl . Let .Pna (θ ) be the corresponding projections. We define In (θ ) = (C0n,0 (θ )S(θ )−1 )∗ C0n,0 (θ )S(θ )−1

.

and Jn (θ ) = P In (θ )P + (Id − P )In (θ )(Id − P ),

.

with P as in (4.6) and .S(θ ) as in (4.7). Here .Cam,n (θ ) denotes the expression in (1.18) with each matrix .An replaced by .Cna (θ ). Moreover, let .Rn (θ ) be the unique positivedefinite square root of the positive-definite symmetric matrix .Jn (θ ). Finally, let Vn (θ ) = C0n,0 (θ )S(θ )−1 Rn (θ )−1 ,

.

(4.18)

and Dn (θ ) = Vn+1 (θ )−1 Cn0 (θ )Vn (θ ).

.

We know from the proof of Proposition 4.2 that the map .S(θ ) is of class .C 1 in .θ , and so the same will happen to .Vn (θ ) and .Dn (θ ) provided that .Rn (θ ) is shown to

4.3 General Perturbations

53

be of class .C 1 in .θ . One can verify that the map .F : A |→ A2 on the set of positivedefinite symmetric matrices has the continuous derivative dA F X = AX + XA.

.

It follows from work in [54] (see also [35]) that there is X satisfying .dJn (θ) F X = B if and only if the matrices ⎛ .

0 Jn (θ ) B −Jn (θ )

⎞ and

diag(Jn (θ ), −Jn (θ ))

are conjugate, which indeed holds because .Jn (θ ) is invertible. Therefore, F is a .C 1 diffeomorphism, and so the map .Rn (θ ) is of class .C 1 in .θ . Following closely the proof of Lemma 3.1, we find that there is .d > 0 such that .

||Vn (θ )x||n ≤d ||x||n x/=0 sup

and

||Vn (θ )−1 x||n ≤d ||x||n x/=0 sup

(4.19)

for all .n ∈ Z and .θ ∈ Rl . Moreover, the matrices .Dn (θ ) have a block-diagonal form Dn (θ ) = diag(Dn1 , Dn2 )

.

for n ∈ Z,

where .Dni is a .di × di matrix for .i = 1, 2 and .n ∈ Z, where .d1 and .d2 are the dimensions of the stable and unstable spaces of the tempered exponential dichotomy of .(Cna (θ ))n∈Z . In a similar manner to that in (4.8), one can show that S(θ )P0a (θ )S(θ )−1 = P

.

with .P = diag(IdRd1 , 0), and so also that Dn (θ )P = Vn+1 (θ )−1 Cn0 (θ )Pna (θ )Vn (θ )

.

and Dn (θ )(Id − P ) = Vn+1 (θ )−1 Cn0 (θ )(Id − Pna (θ ))Vn (θ )

.

(see (3.10) and (3.11)). In view of (4.19), for each .θ ∈ Rl the sequence (e−a Dn (θ ))n∈Z has a tempered exponential dichotomy. Proceeding as in the proof of Theorem 3.1, we find that for each .θ ∈ Rl the sequence .(An + Bn (θ ))n∈Z is cohomologous to a sequence .(Dn (θ ))n∈Z with .Dn (θ ) as in (4.16) and of class .C 1 in .θ . Finally, the matrices .Un (θ ) are obtained multiplying the various matrices .Vn (θ ) in (4.18) (one for each bounded connected component of the set .R \ Σδ ) and again are of class .C 1 in .θ . By (4.19) we also have

.

54

4 Parameter-Dependent Dynamics

.

||Un (θ )x||n < +∞ ||x||n θ∈Rl n∈Z x/=0 sup sup sup

and

||Un (θ )−1 x||n < +∞. ||x||n θ∈Rl n∈Z x/=0 sup sup sup

⨆ ⨅

This completes the proof of the theorem. 1 .C

Now we establish the regularity on the parameter of the coordinate change that gives the normal form, provided that a certain gap condition on the spectrum is satisfied. Let .fn : Rd × Rl → Rd be continuous maps of class .C p in the first variable, for some .p ≥ 1, and of class .C 1 in the second variable, with .fnθ (0) = 0 and .d0 fnθ = 0 for .n ∈ Z and .θ ∈ Rl . Under the assumptions of Theorem 4.3, we consider the dynamics xn+1 = Dn (θ )xn + fnθ (xn )

.

for n ∈ Z,

(4.20)

with the matrices .Dn (θ ) as in (4.16). We note that a dynamics of the form x¯n+1 = (An + Bn (θ ))x¯n + f¯nθ (x¯n ) for n ∈ Z

.

can always be transformed into (4.20). Indeed, by (4.17), letting .x¯n = Un (θ )xn for each .n ∈ Z, we obtain (4.20) with fnθ = Un+1 (θ )−1 f¯nθ ◦ Un (θ ).

.

Write .D¯ n (θ ) = Dn (θ )−1 and denote by .Dn' (θ ) and .D¯ n' (θ ), respectively, the derivatives of .Dn (θ ) and .D¯ n (θ ) with respect to .θ , taken entry by entry. We assume that ⎫ ⎧ ||Dn' (θ )x||n+1 ||Dn (θ )x||n+1 , sup N = sup sup sup ||x||n ||x||n x/=0 θ∈Rl n∈Z x/=0 (4.21) . ⎫ ⎧ ||D¯ n (θ )x||n ||D¯ n' (θ )x||n + sup sup sup < +∞. , sup x/=0 ||x||n+1 θ∈Rl n∈Z x/=0 ||x||n+1 Theorem 4.4 Let .(An )n∈Z be a sequence of invertible .d ×d matrices with tempered growth and let .(Bn (θ ))n∈Z be a smooth family. Moreover, let .fn : Rd × Rl → Rd be continuous maps of class .C p in the first variable, for some .p ≥ 1, and of class 1 θ θ l .C in the second variable, with .fn (0) = 0 and .d0 fn = 0 for .n ∈ Z and .θ ∈ R , satisfying (4.12) and (4.21). Then: 1. For each .θ ∈ Rl there are polynomials .hθn : Rd → Rd with .hθn (0) = 0 and θ .d0 hn = 0 for .n ∈ Z satisfying (4.13). 2. Letting .xn = yn + hθn (yn ) for .n ∈ Z in (4.20) we obtain yn+1 = Dn (θ )yn + gnθ (yn )

.

for n ∈ Z,

(4.22)

4.3 General Perturbations

55

where .gnθ = (gn1θ , . . . , gnkθ ) : Rd → Rd are maps of class .C p with .gnθ (0) = 0 and .d0 gnθ = 0. 3. .∂0r gniθ = 0 for all .n ∈ Z, .i = 1, . . . , k and .r ∈ Nk0 with .2 ≤ |r| ≤ p such that .(i, r) is not a resonance. 4. If whenever .(i, r) is not a resonance we have ai > + max (bj − aj )

.

j =1,...,k

or

bi < − max (bj − aj ), j =1,...,k

(4.23)

then the maps .(x, θ ) |→ hθn (x) are continuous and are of class .C 1 in .θ . Proof Again the maps .hθn are obtained as in the proof of Theorem 3.2. Assume that l .(i, r) is not a resonance and let .Dm,n (θ ) be as in (1.18) with each .An replaced by l .Dn (θ ). We define kθ hθn = (h1θ n , . . . , hn ),

.

jθ

where .hn = 0 for .j /= i, hiθ n (x) = −

+∞ Σ

.

l=n

1 Din,l+1 (θ ) ∂0r fliθ (D1l,n (θ )x 1 )r1 · · · (Dkl,n (θ )x k )rk r!

whenever .ai > and iθ .hn (x)

=

n−1 Σ l=−∞

1 Din,l+1 (θ ) ∂0r fliθ (D1l,n (θ )x 1 )r1 · · · (Bkl,n x k )rk r!

whenever .bi < . Again one can show as in the proof of Theorem 3.2 that these series converge absolutely and uniformly in .θ ∈ Rl , and by construction .hθn is a polynomial. Since each term in the series is continuous in .(x, θ ), each map θ .(x, θ ) |→ hn (x) is automatically continuous. It remains to show that these maps are 1 of class .C in .θ . Take .ci < ai and .ci > bi for .i = 1, . . . , k such that ci > + max (cj − cj )

(4.24)

ci < − max (cj − cj )

(4.25)

.

j =1,...,k

whenever .ai > and .

j =1,...,k

whenever .bi < , with .c and .c as in (3.22). Assume that (4.24) holds. For .l ≥ n let

56

4 Parameter-Dependent Dynamics

Gl (θ ) = Din,l+1 (θ )Fl (θ )(D1l,n (θ )x 1 )r1 · · · (Dkl,n (θ )x k )rk ,

.

where .Fl (θ ) = ∂0r fliθ . Then G'l (θ ) = S1 + S2 + S3 ,

.

where S1 = (Din,l+1 )' (θ )Fl (θ )(D1l,n (θ )x 1 )r1 · · · (Dkl,n (θ )x k )rk , .

S2 = Din,l+1 (θ )Fl' (θ )(D1l,n (θ )x 1 )r1 · · · (Dkl,n (θ )x k )rk , Σ j S3 = rj Din,l+1 (θ )Fl (θ )(D1l,n (θ )x 1 )r1 · · · El (θ ) · · · (Dkl,n (θ )x k )rk j :rj ≥1

with El (θ ) = (Dl,n (θ )x j )rj −1 (Dl,n )' (θ )x j . j

.

j

j

Since Din,l+1 (θ ) = Dni (θ )−1 · · · Dli (θ )−1 ,

.

i (θ ) = D i (θ )−1 we obtain writing .D¯ m m

(Din,l+1 )' (θ ) =

l Σ

.

i ' Din,m (θ )(D¯ m ) (θ )Dim+1,l+1 (θ ).

m=n

In a similar manner to that in the proof of Theorem 3.2, for .m = n, . . . , l we have ||Din,m (θ )|| ≤ Deci (n−m)+ε|m|

.

and

||Dim+1,n (θ )|| ≤ Deci (m−n)+ε|n|

for some constant .D > 0. Therefore, l || Σ || || i ' i || ≤ || ||D (θ )(D¯ i )' (θ )Di ) (θ )D (θ ) n,m m n,l+1 l+1,n m+1,n (θ )

|| ||(Di

m=n

≤ .

l Σ

Deci (n−m)+ε|m| N Deci (m−n)+ε|n| (4.26)

m=n

= N1 eci

l Σ

e(ci −ci +ε)(m−n)+2ε|n|

m=n

≤ N¯ 1 e

(ci −ci +ε)(l−n)+2ε|n|

4.3 General Perturbations

57

for some constants .N1 , N¯ 1 > 0. Proceeding analogously we also obtain l−1 Σ

(Dil,n )' (θ )Din,l (θ ) =

i ' Dil,m+1 (θ )(Dm ) (θ )Dim,l (θ ).

.

m=n

Moreover, for .m = n, . . . , l − 1 we have ||Dim,l (θ )|| ≤ Deci (m−l)+ε|l|

and

.

||Dil,m+1 (θ )|| ≤ Deci (l−m)+ε|m|

for some constant .D > 0. Therefore, l−1 || Σ || || || i i ' i || ||(Di )' (θ )Di (θ )|| ≤ ||D l,n n,l l,m+1 (θ )(Dm ) (θ )Dm,l (θ ) m=n

≤

l−1 Σ

Deci (l−m)+ε|m| NDeci (m−l)+ε|l|

m=n .

= N2

l−1 Σ

(4.27) e

(ci −ci +ε)(l−m)+2ε|l|

m=n

≤ N¯ 2 e(ci −ci +ε)(l−n)+2ε|l| ≤ N¯ 2 e(ci −ci +3ε)(l−n)+2ε|n| for some constants .N2 , N¯ 2 > 0. Writing (Din,l+1 )' (θ ) = (Din,l+1 )' (θ )Dil+1,n (θ )Din,l+1 (θ ),

.

by (4.12) and (4.26) we obtain

.

Σk ||S1 || ≤ N3 e(ci −ci +ε)(l−n)+2ε|n| e2ε|l| e(ci −)(n−l)+ε|n| i=1 ri k r k · · · ||x ||

||x 1 ||r1

≤ N3 e(ci −(ci −ci )−−2ε)(n−l)+ε|n|(4+|r|) for some constant .N3 > 0. Proceeding as in (3.25) we also obtain

.

Σk ||S2 || ≤ N4 e(ci −)(n−l)+2ε|l|+ε|n| i=1 ri k r · · · ||x || k

||x 1 ||r1

≤ N4 e(ci −−2ε)(n−l)+ε|n|(2+|r|)

58

4 Parameter-Dependent Dynamics

for some constant .N4 > 0. Similarly, we have (Dl,n )' (θ )x j = (Dl,n )' (θ )Dn,l (θ )Dl,n (θ )x j ,

.

j

j

j

j

and so it follows from (4.27) that ||S3 || · · · ||x k ||rk Σ ≤ N5 rj eci (n−l)+2ε|l|−(n−l)+ε|n|·|r|+(cj −cj +3ε)(l−n)+2ε|n|

||x 1 ||r1 .

j :rj ≥1

≤ N¯ 5 e[ci −−maxj =1,...,k (cj −cj )−5ε](n−l)+ε|n|(4+|r|) for some constants .N5 , N¯ 5 > 0. From these estimates we finally obtain .

||G'l (θ )|| ≤ N6 e[ci −−maxj =1,...,k (cj −cj )−5ε](n−l)+ε|n|(4+|r|) ||x 1 ||r1 · · · ||x k ||rk

(4.28)

Σ ' for some constant .N6 > 0. By (4.24) and (4.28) the series . +∞ l=n Gl (θ ) is absolutely and uniformly convergent in .θ provided that .ε is sufficiently small. This implies that iθ θ 1 .hn and so also .hn are of class .C in .θ . The case when (4.25) holds can be treated analogously. ⨆ ⨅ We note that when all intervals of the tempered spectrum of the sequence (An )n∈Z reduce to points, condition (4.23) becomes condition (3.15). In other words, there is no need for a spectral gap condition. This leads to the following special case of Theorem 4.4.

.

Theorem 4.5 Let .(An )n∈Z be a sequence of invertible .d ×d matrices with tempered growth and tempered spectrum .Σ = {a1 , . . . , ak }, and let .(Bn (θ ))n∈Z be a smooth family. Moreover, let .fn : Rd ×Rl → Rd be continuous maps of class .C p in the first variable, for some .p ≥ 1, and of class .C 1 in the second variable, with .fnθ (0) = 0 and .d0 fnθ = 0 for .n ∈ Z and .θ ∈ Rl , satisfying (4.12) and (4.21). Then: 1. For each .θ ∈ Rl there are polynomials .hθn : Rd → Rd with .hθn (0) = 0 and θ .d0 hn = 0 for .n ∈ Z satisfying (4.13). 2. Letting .xn = yn + hθn (yn ) for .n ∈ Z in (4.20) we obtain (4.22), where .gnθ = (gn1θ , . . . , gnkθ ) : Rd → Rd are maps of class .C p with .gnθ (0) = 0 and .d0 gnθ = 0. 3. .∂0r gniθ = 0 for all .n ∈ Z, .i = 1, . . . , k and .r ∈ Nk0 with .2 ≤ |r| ≤ p and .ai /= . 4. The maps .(x, θ ) |→ hθn (x) are continuous and are of class .C 1 in .θ . Notes Proposition 4.1 and its proof are taken from [17]. The notion of exponential dichotomy with respect to a sequence of norms in Definition 4.1, which is crucial for some of the arguments, follows the approach in the nonuniform hyperbolicity theory

4.3 General Perturbations

59

(see [10]) which considers sequences of Lyapunov norms, or even sequences of Lyapunov inner products. With respect to these norms the nonuniform or tempered behavior becomes uniform (see (2.26) and (2.27)). More precisely, the nonuniform behavior is transferred from the contraction and expansion of the exponential dichotomy to the norms in (2.25). Proposition 4.2 is a consequence of a robustness result in [11] for exponential dichotomies depending on a parameter. Essentially it says that sufficiently small smooth linear perturbations of a tempered exponential dichotomy still have tempered exponential dichotomies, with stable and unstable spaces that depend smoothly on the parameter. The case of higher smoothness can also be considered, although it would require more elaborate computations for the higher derivatives. The problem of robustness of exponential dichotomies was first discussed by Massera and Schäffer [63] (see also earlier work of Perron in [76]), Coffman and Schäffer [25], Coppel [29], and in the infinite-dimensional setting by Dalec’ki˘ı and Kre˘ın [32]. For more recent works we refer the reader to [22, 69, 82] and the references therein. Moreover, for the case of continuous time, see [50, 106] for exponential dichotomies on .R, and [73] for exponential dichotomies on .R+ . The related work in [110] considers infinite-dimensional random difference equations. Using the former preparatory notions and results, Theorem 4.1 shows that the tempered spectrum varies little under sufficiently small linear parameter-dependent perturbations, also with a smooth dependence of the projections of the perturbed dynamics on the parameter. Theorem 4.1 and its proof are adapted from [16]. Theorem 4.2 is a simple modification of Theorem 3.2. The only novelty is the smooth dependence of the coordinate changes on the parameter. The result is adapted from [16], which considers the more general case of perturbations of class p with .p ≥ 1. While Theorem 4.1 considers linear perturbations, Theorem 4.2 .C considers nonlinear perturbations, both possibly depending on a parameter. We emphasize that the techniques are distinct in the two cases. In Section 4.3 we consider simultaneously linear and nonlinear parameterdependent perturbations. Theorem 4.3 and its proof are adapted from [16], which again considers the more general case of perturbations of class .C p with .p ≥ 1. The argument is an elaboration of the proof of Theorem 3.1, which essentially gives the block-diagonalization in (4.16), but not the smooth dependence of the coordinate change on the parameter. However, we emphasize that even the blockdiagonalization is nontrivial since the blocks are written for all values of the parameter with respect to the same splitting of the ambient space, which thus is independent of the parameter. The former developments are then used to prove Theorem 4.4, which establishes the smooth dependence on the parameter of the coordinate change that gives the normal form when we consider simultaneously linear and nonlinear parameterdependent perturbations. The theorem and its proof are adapted from [16].

Chapter 5

The Case of Differential Equations

This chapter presents a version of some results in the former chapters for differential equations, that is, for a dynamics with continuous time. In particular, we introduce the notions of tempered exponential dichotomy and of tempered spectrum for a nonautonomous linear equation .x ' = A(t)x on .Rd . Then we describe all possible forms of the tempered spectrum and its relation to the lower and upper Lyapunov exponents. We also give explicit examples of differential equations for all possible forms of the spectrum. Finally, we detail the construction of normal forms for the perturbations of a nonautonomous linear equation. We avoid repeating the details that were already given before.

5.1 Tempered Spectrum and Lyapunov Exponents We first introduce the notion of tempered exponential dichotomy for a linear differential equation and, more generally, for an invertible evolution family (for simplicity of the exposition, we avoid considering noninvertible evolution families here). Let .A(t) be .d × d matrices varying continuously with .t ∈ R. Then all solutions of the linear equation x ' = A(t)x

.

(5.1)

are global (that is, are defined for all .t ∈ R). We define .d × d matrices .T (t, s) for t, s ∈ R by requiring that

.

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

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 L. Barreira, C. Valls, Spectra and Normal Forms, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-51897-3_5

61

62

5 The Case of Differential Equations

for any solution x of Equation (5.1) and all .t, s ∈ R. Note that T (t, t) = Id and

T (t, τ )T (τ, s) = T (t, s)

.

for t, τ, s ∈ R.

(5.2)

Any family .(T (t, s))t,s∈R of invertible .d × d matrices satisfying (5.2) is called an invertible evolution family. Definition 5.1 An invertible evolution family .(T (t, s))t,s∈R is said to have a tempered exponential dichotomy if: 1. There are projections .P (s) : Rd → Rd for .s ∈ R satisfying T (t, s)P (s) = P (t)T (t, s) for t, s ∈ R.

.

(5.3)

2. There are a constant .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that for each .t, s ∈ R we have .

||T (t, s)P (s)|| ≤ De−λ(t−s)+ε|s|

for t ≥ s

||T (t, s)Q(s)|| ≤ De−λ(s−t)+ε|s|

for t ≤ s,

and .

where .Q(s) = Id − P (s). We shall say that the linear equation .x ' = A(t)x has a tempered exponential dichotomy if its evolution family has a tempered exponential dichotomy. One can also consider the more general case when the matrices .T (t, s) need not be invertible (in a similar manner to that in Chapter 1 where the matrices .An need not be invertible). But for simplicity of the exposition, we consider only invertible matrices (obtained, for example, from the solutions of Equation (5.1)). A function .D : R → R+ is said to be upper tempered if .

lim sup s→±∞

1 log D(s) ≤ 0. |s|

Note that this happens if and only if given .ε > 0, there is .D = D(ε) > 0 such that D(s) ≤ Deε|s|

.

for all s ∈ R.

Thus, the equation .x ' = A(t)x has a tempered exponential dichotomy if and only if there are projections .P (s) for .s ∈ R satisfying (5.3), and there are a constant .λ > 0 and an upper tempered function D such that ||T (t, s)P (s)|| ≤ D(s)e−λ(t−s)

.

for t ≥ s

5.1 Tempered Spectrum and Lyapunov Exponents

63

and ||T (t, s)Q(s)|| ≤ D(s)e−λ(s−t)

.

for t ≤ s.

We shall also say that .(T (t, s))t,s∈R has a tempered exponential contraction if it has a tempered exponential dichotomy with .P (t) = Id for all .t ∈ R, and that .(T (t, s))t,s∈R has a tempered exponential expansion if it has a tempered exponential dichotomy with .P (t) = 0 for all .t ∈ R. For any tempered exponential dichotomy, the sets E(s) = P (s)(Rd )

.

and

F (s) = Q(s)(Rd )

are called, respectively, the stable and unstable spaces at time s. They satisfy Rd = E(s) ⊕ F (s) for s ∈ R

.

and can be univocally characterized as follows. Proposition 5.1 Assume that the equation .x ' = A(t)x has a tempered exponential dichotomy. For each .s ∈ R, we have ⎧ ⎫ d .E(s) = v ∈ R : sup||T (t, s)v|| < +∞ t≥s

and ⎧ ⎫ d .F (s) = v ∈ R : sup||T (t, s)v|| < +∞ . t≤s

The proof of Proposition 5.1 is analogous to the proof of Proposition 1.1, and so we avoid repeating the details (in fact the argument is now somewhat simpler since we are considering invertible evolution families). The notion of tempered spectrum is defined in terms of the notion of tempered exponential dichotomy. Definition 5.2 The tempered spectrum (or, simply, the spectrum) of a linear equation .x ' = A(t)x is the set .Σ = Σ(A) of all numbers .a ∈ R such that the equation x ' = (A(t) − aId)x

.

(5.4)

does not have a tempered exponential dichotomy. Instead of considering the differential equation in (5.4), one can consider, equivalently, its evolution family .(T a (t, s))t,s∈R , which is given by

64

5 The Case of Differential Equations

T a (t, s) = e−a(t−s) T (t, s)

for t, s ∈ R.

.

(5.5)

Given .a ∈ R and .s ∈ R, let ⎧ ⎫ d a .E (s) = v ∈ R : sup||T (t, s)v|| < +∞ a

t≥s

and ⎧ ⎫ d a .F (s) = v ∈ R : sup||T (t, s)v|| < +∞ . a

t≤s

The following result describes all possible forms of the tempered spectrum for a nonautonomous linear equation. Theorem 5.1 For a linear equation .x ' = A(t)x, either .Σ = ∅ or .Σ = Uk i=1 |ai , bi | for some numbers as in (1.12) and some positive integer .k ≤ d. The proof of Theorem 5.1 is similar to the proof of Theorem 1.1 and so it is omitted. The reason for this similarity is that the necessary changes essentially amount to replace .Am,n by .T (t, s) everywhere. Let .x ' = A(t)x be a linear equation with nonempty tempered spectrum. Then Σ=

k | |

.

|ai , bi |

i=1

for some numbers .ai and .bi as in (1.12) and some positive integer .k ≤ d. Take ci ∈ (bi , ai+1 ) for .i = 1, . . . , k − 1. Moreover, take .c0 < a1 if .a1 /= −∞ and .c0 = −∞ otherwise, and take .ck > bk if .bk /= +∞ and .ck = +∞ otherwise. We define .

H i (s) = E ci (s) ∩ F ci−1 (s)

.

for i = 1, . . . , k,

with the convention that E +∞ (s) = F −∞ (s) = Rd .

.

In addition, we define H 0 (s) = E c0 (s) and

.

H k+1 (s) = F ck (s),

with the convention that E −∞ (s) = F +∞ (s) = {0}.

.

(5.6)

5.1 Tempered Spectrum and Lyapunov Exponents

65

One can easily verify that the subspaces .H i (s) are independent of the numbers .ci . Moreover, T (t, s)H i (s) = H i (t) for t, s ∈ R.

.

Theorem 5.2 Let .x ' = A(t)x be a linear equation with nonempty tempered spectrum. Then for each .s ∈ R the following properties hold: ⊕ i 1. .Rd = k+1 i=0 H (s). i 2. Given .v ∈ H (s) \ {0} with .i ∈ {1, . . . , k}, we have .

lim inf t→±∞

1 1 log||T (t, s)v||, lim sup log||T (t, s)v|| ∈ |ai , bi |. t t→±∞ t

3. Given .v ∈ H 0 (s) \ {0}, we have .

lim inf t→±∞

1 1 log||T (t, s)v|| = lim sup log||T (t, s)v|| = −∞, t t→±∞ t

and given .v ∈ H k+1 (s) \ {0}, we have .

lim inf t→±∞

1 1 log||T (t, s)v|| = lim sup log||T (t, s)v|| = +∞. t t→±∞ t

The proof of Theorem 5.2 can be obtained following the proof of Theorem 2.1. Now we recall the notion of tempered growth. Definition 5.3 We say that an invertible evolution family .(T (t, s))t,s∈R has tempered growth if there are a constant .γ ≥ 0 and for each .ε > 0 a constant .K = K(ε) > 0 such that ||T (t, s)|| ≤ Keγ |t−s|+ε|s|

.

for t, s ∈ R.

We shall also say that Equation (5.1) has tempered growth if its evolution family has tempered growth. For example, any such equation with bounded coefficient matrix has tempered growth. This follows from Gronwall’s lemma. Proposition 5.2 If .supt∈R ||A(t)|| < +∞, then the equation .x ' = A(t)x has tempered growth. Proof Take .t ≥ s. Since ⌠ x(t) = x(s) +

t

A(τ )x(τ ) dτ,

.

s

66

5 The Case of Differential Equations

we obtain ⌠ ||x(t)|| ≤ ||x(s)|| +

t

.

||A(τ )|| · ||x(τ )|| dτ.

s

It follows from Gronwall’s lemma that ||x(t)|| ≤ ||x(s)||e

⌠t

.

s

||A(τ )|| dτ

and so ||T (t, s)|| ≤ e

.

⌠t s

||A(τ )|| dτ

.

Now we consider the adjoint equation .y ' = −A(t)∗ y. Since .

∂ T (s, t) = −T (s, t)A(t), ∂t

its evolution family is given by .(t, s) |→ T (s, t)∗ . It follows as before that ||y(t)|| ≤ ||y(s)||e

.

⌠t s

||−A(τ )∗ || dτ

= ||y(s)||e

⌠t s

||A(τ )|| dτ

and so ||T (s, t)|| = ||T (s, t)∗ || ≤ e

.

⌠t s

||A(τ )|| dτ

.

This gives the desired statement.

⨆ ⨅

The spectrum of any linear equation with tempered growth is automatically bounded and nonempty. Proposition 5.3 Let .x ' = A(t)x be a linear equation with tempered growth. Then Σ is bounded and nonempty.

.

This follows as in the proof of Proposition 2.1. Under the assumption of Proposition 5.3, it follows readily from Theorem 5.1 that (2.18) holds for some real numbers .ai and .bi as in (1.12) and some positive integer .k ≤ d. The following result is analogous to Theorem 2.3. Theorem 5.3 Let .x ' = UA(t)x be a linear equation with tempered growth and tempered spectrum .Σ = ki=1 [ai , bi ]. Then for each .s ∈ R the following properties hold: ⊕ 1. .Rd = ki=1 H i (s). 2. Given .v ∈ Rd \ {0}, there is .i ∈ {1, . . . , k} such that .

lim inf t→±∞

1 1 log||T (t, s)v||, lim sup log||T (t, s)v|| ∈ [ai , bi ]. t t→±∞ t

5.2 Examples of Tempered Spectra

67

5.2 Examples of Tempered Spectra In this section we give explicit examples of linear differential equations for all possible forms of the nonuniform spectrum .Σ described in Theorem 5.1. Example 5.1 (.Σ = ∅) The linear equation .x ' = A(t)x with .A(t) = 3t 2 Id originates the evolution family T (t, s) = et

.

3 −s 3

Id.

We claim that for each .a ∈ R Equation (5.4) has a tempered exponential expansion. Given .λ > 0, we define a function .f : R → R by f (t) = −at + t 3 − λt.

.

One can easily verify that there is .c > 0 such that f (t) ≤ f (s) + c

.

for any t ≤ s.

Therefore, using the notation in (5.5), we obtain ||T a (t, s)|| = ||e−a(t−s) T (t, s)|| ≤ ec−λ(s−t)

.

for t ≤ s,

and so indeed Equation (5.4) has a tempered exponential expansion. Hence, .Σ = ∅. Example 5.2 (.Σ = R) The linear equation .x ' = A(t)x with .A(t) = −ct sin tId for some .c > 0 originates the evolution family T (t, s) = ect cos t−cs cos s−c sin t+c sin s Id.

.

Take .a ∈ R and assume that the evolution family .T a (t, s) has a tempered exponential dichotomy with projections .P (t). Note that either .P (t) = Id for all .t ∈ R or .P (t) = 0 for all .t ∈ R. In the first case, there are a constant .λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that e−a(t−s) ||T (t, s)|| ≤ De−λ(t−s)+ε|s|

.

for t ≥ s.

Taking .t = 2π l and .s = 2π l − π with .l ∈ N, we obtain ec(4π l−π ) ≤ De(a−λ)π +ε(2π l−π ) ,

.

which fails for .ε < 2c and l sufficiently large. In the second case there are a constant λ > 0 and for each .ε > 0 a constant .D = D(ε) > 0 such that

.

e−a(t−s) ||T (t, s)|| ≤ De−λ(s−t)+ε|s|

.

for t ≤ s.

68

5 The Case of Differential Equations

Taking .t = 2π l and .s = 2π l + π with .l ∈ N, we obtain ec(4π l+π ) ≤ De(−a−λ)π +ε(2π l+π ) ,

.

which fails for .ε < 2c and l sufficiently large. Therefore, .Σ = R. Example 5.3 (.Σ nonempty and bounded) Take real numbers .ai and .bi as in (1.12) for some positive integer .k ≤ d. For .i = 1, . . . , k, let .φi : R → R be a continuous function with .φi (t) = ai for .t ≤ −1 and .φi (t) = bi for .t ≥ 1. We consider the linear equation .x ' = A(t)x on .Rd , where ( ) A(t) = diag c1 (t), . . . , cd (t)

(5.7)

√ 1 ci (t) = φi (t) + √ sin |t| + 1 + |t| cos t 2 1 + |t|

(5.8)

.

with .

for .i = 1, . . . , k, and .ci (t) = ck (t) for .i > k. Its evolution family is given by ( ) T (t, s) = diag T1 (t, s), . . . , Tk (t, s) ,

.

(5.9)

with Ti (t, s) = e

⌠t

.

s

√ √ φi (u) du+ 1+|t| sin t− 1+|s| sin s

(5.10)

for .i = 1, . . . , k, and .Ti (t, s) = Tk (t, s) for .i > k. Take .a > bi . Since .ai ≤ bi , for .t ≥ s, we have √

e−a(t−s) Ti (t, s) ≤ Ce−(a−bi )(t−s)+

.

√ 1+|t|+ 1+|s|

(5.11)

for some constant .C > 0. On the other hand, given .ε > 0, there is .D = D(ε) > 0 such that √

e

.

1+|t|

≤ Deε|t|

for t ∈ R.

(5.12)

Thus, it follows from (5.11) that e−a(t−s) Ti (t, s) ≤ CD 2 e−(a−bi )(t−s)+ε|t|+ε|s| .

≤ CD 2 e−(a−bi −ε)(t−s)+2ε|s|

for .t ≥ s. Since .a − bi > 0 and .ε is arbitrary, the evolution family Tia (t, s) = e−a(t−s) Ti (t, s)

.

(5.13)

5.2 Examples of Tempered Spectra

69

has a tempered exponential contraction. On the other hand, for .a < ai and .t ≤ s, we have ¯ 2 e(a−ai +ε)(s−t)+2ε|s| e−a(t−s) Ti (t, s) ≤ CD

.

for some constant .C¯ > 0, and so the evolution family .Tia (t, s) has a tempered exponential expansion. We also consider the case when .a ∈ [ai , bi ]. Note that e

.

−a(t−s)

⌠ √ √ e(bi −a)(t−s)+ 1+t sin t− 1+s sin s

Ti (t, s) =

√ √ e(ai −a)(t−s)+ 1+|t| sin t− 1+|s| sin s

if t, s ≥ 1, if t, s ≤ −1.

Since .bi − a ≥ 0, the formula for .t, s ≥ 1 shows that .Tia (t, s) does not have a tempered exponential contraction, and since .ai − a ≤ 0, the formula for .t, s ≤ −1 shows that .Tia (t, s) does not have a tempered exponential expansion. Hence, this evolution family has a tempered exponential dichotomy if and only if .a ∈ R \ [ai , bi ]. U Now take .a ∈ R \ ki=1 [ai , bi ]. When .a > bk , the evolution family .T a (t, s) has a tempered exponential contraction. Moreover, when .a < a1 , it has a tempered exponential expansion. Finally, take .i ∈ {1, . . . , k − 1} and .bi < a < ai+1 . Then a .T (t, s) has a tempered exponential dichotomy with projections .P (t) given by P (t)x = (x1 , . . . , xi , 0, . . . , 0).

.

U U Therefore, .Σ ⊂ ki=1 [ai , bi ]. To show that . ki=1 [ai , bi ] ⊂ Σ, take .a ∈ [ai , bi ] for some i and assume that .a ∈ / Σ. Then each evolution family in (5.13) should have a tempered exponential dichotomy, but this cannot happen because .a ∈ [ai , bi ]. Therefore, .a ∈ Σ. Example 5.4 (.Σ unbounded from above) Take .2 ≤ k ≤ d. Let .A(t) be as in (5.7), with .ci (t) as in (5.8) for .i = 1, . . . , k − 1, and √ 1 ci (t) = ψ(t) + √ sin |t| + 1 + |t| cos t 2 1 + |t|

.

(5.14)

for .i ≥ k, taking a continuous function .ψ : R → R with .ψ(t) = ak for .t ≤ −1 and ψ(t) = 2t for .t ≥ 1. Its evolution family is given by (5.9), with .Ti (t, s) as in (5.10) for .i = 1, . . . , k − 1, and

.

Ti (t, s) = e

.

⌠t s

√ √ ψ(u) du+ 1+|t| sin t− 1+|s| sin s

for .i ≥ k. Using Example 5.3, it suffices to study .Tk (t, s). Take .a < ak . By (5.12), for .t ≤ s, we have

70

5 The Case of Differential Equations √

e−a(t−s) Tk (t, s) ≥ Ce−(a−ak )(s−t)−

√ 1+|t|− 1+|s|

≥ CD 2 e−(a−ak )(s−t)−ε|t|−ε|s|

.

≥ CD 2 e−(a−ak +ε)(s−t)−2ε|s| for some constant .C > 0. Since .a − ak < 0 and .ε is arbitrary, the evolution family Tka (t, s) has a tempered exponential expansion. Proceeding as in Example 5.3, one can show that

.

Σ=

k−1 | |

.

[ai , bi ] ∪ [ak , +∞).

i=1

Similarly, one can take .A(t) = ck (t)Id, with .ck (t) as in (5.14) with .i = k. Proceeding as before, we find that .Σ = [ak , +∞). Example 5.5 (.Σ unbounded from below) Take .2 ≤ k ≤ d. Let .A(t) be as in (5.7), with √ 1 ¯ c1 (t) = ψ(t) + √ sin |t| + 1 + |t| cos t, 2 1 + |t|

.

(5.15)

ci (t) as in (5.8) for .i = 2, . . . , k, and .ci (t) = ck (t) for .i > k, taking a continuous ¯ ¯ = 2t for .t ≤ −1 and .ψ(t) = b1 for .t ≥ 1. Its function .ψ¯ : R → R with .ψ(t) evolution family is given by (5.9), with

.

T1 (t, s) = e

.

⌠t s

√ √ ¯ ψ(u) du+ 1+|t| sin t− 1+|s| sin s

Ti (t, s) as in (5.10) for .i = 2, . . . , k, and .Ti (t, s) = Tk (t, s) for .i > k. Take .a > b1 . By (5.12), for .t ≥ s, we have

.

√

e−a(t−s) T1 (t, s) ≤ Ce−(a−b1 )(t−s)+

√ 1+|t|+ 1+|s|

≤ CD 2 e−(a−b1 )(t−s)+ε|t|+ε|s|

.

≤ CD 2 e−(a−b1 −ε)(t−s)+2ε|s| for some constant .C > 0. Since .a − b1 > 0 and .ε is arbitrary, the evolution family T1a (t, s) has a tempered exponential contraction. Proceeding as in Example 5.3, one can show that

.

Σ = (−∞, b1 ] ∪

k | |

.

i=2

[ai , bi ].

5.3 Block-Diagonalization of a Linear Dynamics

71

Similarly, one can take .A(t) = c1 (t)Id, with .c1 (t) as in (5.15). Proceeding as before, we find that .Σ = (−∞, b1 ]. Example 5.6 (.Σ /= R unbounded from above and below) Take .2 ≤ k ≤ d. Let A(t) be as in (5.7), with .c1 (t) as in (5.15), .ci (t) as in (5.8) for .i = 2, . . . , k − 1, and .ci (t) as in (5.14) for .i ≥ k. Proceeding as in Example 5.3, one can show that .

Σ = (−∞, b1 ] ∪

k−1 | |

.

[ai , bi ] ∪ [ak , +∞).

i=2

Finally, let .A(t) be as in (5.7), with .c1 (t) as in (5.15), and .ci (t) as in (5.14) for i ≥ 2. Proceeding as in Example 5.3, we find that

.

Σ = (−∞, b1 ] ∪ [ak , +∞).

.

5.3 Block-Diagonalization of a Linear Dynamics Mainly as a preparation for the construction of normal forms, we show that any linear equation can be reduced to an equation in block-diagonal form. For simplicity of the exposition, we consider only equations with tempered growth (which also suffices for the construction of normal forms in Section 3.2). Consider invertible .d × d matrices .U (t) with entries of class .C 1 in .t ∈ R. For the nonautonomous coordinate change x(t) = U (t)y(t) for t ∈ R,

.

one can easily verify that y ' (t) = B(t)y,

.

where B(t) = U (t)−1 A(t)U (t) − U (t)−1 U ' (t).

.

(5.16)

We note that the entries of .B(t) vary continuously with .t ∈ R. Now let .T (t, s) be the evolution family associated to the equation .x ' = A(t)x. Then the evolution family ' .T¯ (t, s) associated to the equation .y = B(t)y is given by T¯ (t, s) = U (t)−1 T (t, s)U (s).

.

(5.17)

72

5 The Case of Differential Equations

Definition 5.4 A family .(U (t))t∈R of invertible .d × d matrices .U (t) is said to be tempered if .

1 log||U (t)|| = 0 t→±∞ t lim

1 log||U (t)−1 || = 0. t→±∞ t

and

lim

Two linear equations x ' = A(t)x

.

and

y ' = B(t)y

are said to be cohomologous if there is a tempered family of matrices .(U (t))t∈R with .U (t) of class .C 1 in t that satisfies (5.16). The following result can be obtained as in the proof of Proposition 3.1. Proposition 5.4 Two cohomologous equations have the same tempered spectrum. Moreover, it follows readily from the definitions that if one equation has tempered growth, then the other too. One can now perform the block-diagonalization of any linear Equation (5.1) with tempered growth. ' Theorem 5.4 Any and tempered Uk linear equation .x = A(t)x with tempered growth spectrum .Σ = i=1 [ai , bi ] is cohomologous to an equation .y ' = B(t)y with

( ) B(t) = diag B 1 (t), . . . , B k (t) ,

.

where each .B i (t) is an .ni × ni matrix varying continuously with .t ∈ R such that Σ(B i ) = [ai , bi ] for i = 1, . . . , k.

.

Proof We first establish an auxiliary result. Lemma 5.1 If the equation .x ' = A(t)x has a tempered exponential dichotomy with projections .P (t), then it is cohomologous to an equation .y ' = B(t)y with B(t) = diag(B 1 (t), B 2 (t)),

.

(5.18)

where each .B i (t) is an .mi × mi matrix varying continuously with .t ∈ R taking m1 = dim P (t)(Rd )

.

and

m 2 = d − m1 .

Moreover, the equation .y ' = B(t)y has a tempered exponential dichotomy with projections .P = diag(IdRm1 , 0) independent of t. Proof of the lemma The argument builds on the proof of Lemma 3.1. Let .N(t) = T (t, 0)S −1 , where S is an invertible .d × d matrix such that .SP (0)S −1 = P , and define

5.3 Block-Diagonalization of a Linear Dynamics

I (t) = P N(t)∗ N(t)P + (Id − P )N(t)∗ N(t)(Id − P ).

.

73

(5.19)

The matrix .I (t) is symmetric and positive-definite, and so it has a unique positivedefinite square root .J (t). Moreover, it follows from (5.19) that .I (t) is a block matrix (with blocks of dimensions .m1 and .m2 ), and thus the same happens to .J (t). Finally, we define U (t) = N(t)J (t)−1 = T (t, 0)S −1 J (t)−1 .

.

Clearly, the map .t |→ T (t, 0) is of class .C 1 . Moreover, it is shown in the proof of Theorem 4.3 that the map .A |→ A2 on the set of positive-definite symmetric matrices is a .C 1 diffeomorphism, and so .J (t) and thus also .J (t)−1 are of class .C 1 in t. This shows that .U (t) is of class .C 1 in t and so .B(t) is continuous in t. One can proceed as in the proof of Lemma 3.1 to verify that Id = P U (t)∗ U (t)P + (Id − P )U (t)∗ U (t)(Id − P )

.

and thus that for each .ε > 0 there is .D = D(ε) > 0 such that .

√ 1 −ε|t| ≤ ||U (t)−1 ||−1 ≤ ||U (t)|| ≤ 2. e D(ε)

Therefore, the family .(U (t))t∈R is tempered. Now observe that U (t)−1 P (t)U (t) = J (t)N(t)−1 P (t)N(t)J (t)−1 .

= J (t)P J (t)−1 = P

(5.20)

(since .J (t) is a block matrix, we have .P J (t) = J (t)P ). For .T¯ (t, s) as in (5.17), it follows from (5.20) that P T¯ (t, s) = U (t)−1 P (t)U (t)T¯ (t, s) = U (t)−1 P (t)T (t, s)U (s) .

= U (t)−1 T (t, s)P (s)U (s) = U (t)−1 T (t, s)U (s)U (s)−1 P (s)U (s) = T¯ (t, s)P .

Therefore, .T¯ (t, s) has a block-diagonal form. Since it is the evolution family associated to the equation .y ' = B(t)y, we conclude that .B(t) has the block-diagonal form in (5.18). Moreover, since .J (t)−1 has a block-diagonal form, we obtain

74

5 The Case of Differential Equations

T¯ (t, s)P = U (t)−1 T (t, s)U (s)P = U (t)−1 T (t, s)T (s, 0)S −1 J (s)−1 P = U (t)−1 T (t, s)T (s, 0)S −1 P J (s)−1 .

= U (t)−1 T (t, s)T (s, 0)P (0)S −1 J (s)−1 = U (t)−1 T (t, s)T (s, 0)P (0)T (0, s)U (s) = U (t)−1 T (t, s)P (s)U (s)

and so also T¯ (t, s)(Id − P ) = U (t)−1 T (t, s)(Id − P (s))U (s).

.

Since the family .(U (t))t∈R is tempered, this implies that the equation .y ' = B(t)y has a tempered exponential dichotomy with projections P independent of n, with 1 1 .B (t) and .B (t) in (5.18) corresponding, respectively, to contraction and expansion. ⨆ ⨅ Using the block-diagonalization in Lemma 5.1, the remainder of the argument can be obtained following closely the proof of Theorem 3.1. ⨆ ⨅

5.4 Construction of Normal Forms In this section we construct normal forms, using the tempered spectrum and the block-diagonalization of a linear equation described in Section 5.3. Let .x ' = A(t)x be a linear equation on .Rd with tempered growth and let .B(t) be the matrices constructed in Theorem 5.4. We consider the nonlinear dynamics x ' = B(t)x + f (t, x)

.

(5.21)

for some continuous map .f : R × Rd → Rd of class .C p in the second variable, for some .p ≥ 1, with .ft (0) = 0 and .d0 ft = 0 for all .t ∈ R, where .ft = f (t, ·). Given .x ∈ Rd , we write .x = (x 1 , . . . , x k ), where .x i ∈ Rni for .i = 1, . . . , k with i .ni = dim H (s) (see (5.6)). One can write each map .ft as a Taylor series Σ

ft (x) =

.

r∈Nk0 ,2≤|r|≤p

1 r ∂ ft x r + o(||x||p ), r! 0

using the same notation as in Section 3.2. Writing ft = (ft1 , . . . , ftk ),

.

(5.22)

5.4 Construction of Normal Forms

75

with .fti : Rd → Rni for .i = 1, . . . , k, the term .(1/r!)∂0r fti x r in (5.22) is said to be resonant if the pair .(i, r) is a resonance (in the sense of Definition 3.2). Finally, we construct normal forms for the perturbations of a linear equation in block-diagonal form. Again this corresponds to eliminate all nonresonant terms in (5.22) up to order p by making an appropriate coordinate change. Theorem 5.5 Let .x ' = A(t)x be a linear equation on .Rd with tempered growth and let .B(t) be the matrices constructed in Theorem 5.4. Moreover, let .f : R×Rd → Rd be a continuous map of class .C p in the second variable, for some .p ≥ 1, with .ft (0) = 0 and .d0 ft = 0 for .t ∈ R, such that .

lim sup t→±∞

1 log||∂0r ft || ≤ 0 |t|

for 2 ≤ |r| ≤ p.

Then: 1. There are polynomials .ht : Rd → Rd with .ht (0) = 0 and .d0 ht = 0 for .t ∈ R satisfying .

lim sup t→±∞

1 log||ht (x)|| ≤ 0. |t|

2. Letting .x(t) = y(t) + ht (y(t)) for .t ∈ R in (5.21), we obtain y ' = B(t)y + g(t, y)

.

(5.23)

for some continuous map .g : R × Rd → Rd of class .C p in the second variable, with .gt (0) = 0 and .d0 gt = 0 for all .t ∈ R. 3. Writing .gt = (gt1 , . . . , gtk ), we have .∂0r gti = 0 for all .t ∈ R, .i = 1, . . . , k, and k .r ∈ N with .2 ≤ |r| ≤ p such that .(i, r) is not a resonance. 0 Proof Assume that .(i, r) is not a resonance. It follows from (3.15) that either .ai > or .bi < . Let .Tl (t, s) be the evolution family associated to the linear equation .x ' = B l (t)x. We define maps .ht : Rd → Rd by ht = (h1t , . . . , hkt ),

.

j

where .ht = 0 for .j /= i, ⌠ hit (x) = −

+∞

.

t

1 Ti (t, τ ) ∂0r fτi (T1 (τ, t)x 1 )r1 · · · (Tk (τ, t)x k )rk dτ r!

(5.24)

whenever .ai > and ⌠ hit (x) =

.

t

1 Ti (t, τ ) ∂0r fτi (T1 (τ, t)x 1 )r1 · · · (Tk (τ, t)x k )rk dτ r! −∞

(5.25)

76

5 The Case of Differential Equations

whenever .bi < . One can verify in a similar manner to that in the proof of Theorem 3.2 that the integrals are well defined. Moreover, one can show as in the proof of Theorem 4.4 that the derivatives of the polynomial .ht can be obtained passing them inside the integrals in (5.24) and (5.25) (which converge absolutely and uniformly for t in any compact interval). Finally, it follows from the explicit form of the integrals that the map .(t, x) |→ ht (x) is of class .C 1 . Indeed, since the integrand .F = F (t, τ, x) in (5.24) is of class .C 1 and the integrals ⌠

+∞

.

t

∂ F (t, τ, x) dτ ∂t

⌠ and t

+∞

∂ F (t, τ, x) dτ ∂x

converge absolutely and uniformly for .(t, x) in any compact set, the function (t, x) |→ hit (x) is of class .C 1 . A similar argument applies to the integral in (5.25). The coordinate change .x(t) = y(t) + ht (y(t)) takes (5.21) to (5.23), where

.

⎞ ⎞ ⎛ ⎛ ∂ht ∂ht −1 (B(t) + ft )(y + ht (y)) − − B(t)y. g(t, y) = Id + ∂t ∂y

.

Therefore, ⎞ ⎛ ∂ht ∂ht (B(t)y + g(t, y)) = B(t)y + B(t)ht (y) + ft (y + ht (y)) − . Id + ∂t ∂y and so g(t, y) = −

.

∂ht ∂ht B(t)y + B(t)ht (y) + ft (y) − + o(||y|||r| ). ∂y ∂t

We show that .

1 ∂ht ∂ht = B(t)ht (y) − B(t)y + ∂0r fti y r . ∂t ∂y r!

(5.26)

For each .ξ ∈ Rd , when .ai > , we obtain .

⎞ ⎛ 1 ∂y Ti (t, τ ) ∂0r fτi (T1 (τ, t)y 1 )r1 · · · (Tk (τ, t)y k )rk ξ dτ r! t ⌠ +∞ Σ 1 =− rj B i (t)Ti (t, τ ) ∂0r fτi (T1 (τ, t)y 1 )r1 · · · r! t

∂hit ξ =− ∂y

⌠

+∞

j :rj ≥1

· · · (Tj (τ, t)y j )rj −1 (Tj (τ, t)ξj ) · · · (Tk (τ, t)y k )rk dτ and, similarly, when .bi < ,

5.4 Construction of Normal Forms

.

77

⌠ t Σ ∂hit 1 rj B i (t)Ti (t, τ ) ∂0r fτi (T1 (τ, t)y 1 )r1 · · · ξ= ∂y r! −∞ j :rj ≥1

· · · (Tj (τ, t)y j )rj −1 (Tj (τ, t)ξj ) · · · (Tk (τ, t)y k )rk dτ. On the other hand, taking into account that .

∂ T (t, τ ) = B(t)T (t, τ ) ∂t

and

∂ T (τ, t) = −T (τ, t)B(t), ∂t

when .ai > , we have 1 ∂hit = ∂0r fti y r − . ∂t r! ⌠ Σ rj + j :rj ≥1

⌠

+∞

t +∞

t

1 Ti (t, τ ) ∂0r fτi (T1 (τ, t)y 1 )r1 · · · (Tk (τ, t)y k )rk dτ r!

1 B i (t)Ti (t, τ ) ∂0r fτi (T1 (τ, t)y 1 )r1 · · · r!

· · · (Tj (τ, t)y j )rj −1 (Tj (τ, t)B j (t)y j ) · · · (Tk (τ, t)y k )rk dτ =

1 r i r ∂hi ∂0 ft y + B i (t)hit (y) − t B(t)y r! ∂y

and, similarly, when .bi < , 1 ∂hit = ∂0r fti y r + . r! ∂t ⌠ Σ rj − j :rj ≥1

⌠

t

1 Ti (t, τ ) ∂0r fτi (T1 (τ, t)y 1 )r1 · · · (Tk (τ, t)y k )rk dτ r! −∞

t

1 B i (t)Ti (t, τ ) ∂0r fτi (T1 (τ, t)y 1 )r1 · · · r! −∞

· · · (Tj (τ, t)y j )rj −1 (Tj (τ, t)B j (t)y j ) · · · (Tk (τ, t)y k )rk dτ =

1 r i r ∂hi ∂0 ft y + B i (t)hit (y) − t B(t)y. ∂y r! j

Hence, identity (5.26) holds (since .ht = 0 for .j /= i). Therefore, ⎞ ⎛ 1 r i r ∂ f y , 0, . . . , 0 + o(||y|||r| ), .g(t, y) = f (t, y) − 0, . . . , 0, r! 0 t which thus eliminates the term .(1/r!)∂0r fti y r . This completes the proof.

⨆ ⨅

Notes Most of the notions and results in this chapter are close to those in Chapters 1, 2, and 3 and thus why we present them somewhat briefly. We also avoided duplicating

78

5 The Case of Differential Equations

the proofs, and whenever possible, we referred instead to the appropriate places in those chapters while highlighting the differences. The notion of tempered spectrum in Definition 5.2 goes back to work of Sacker and Sell in [92] (especially in the case of continuous time), who introduced a spectrum for linear cocycles over a flow using uniform exponential dichotomies. The results in Theorems 5.1 and 5.2, respectively, on the structure of the tempered spectrum and on its relation to the Lyapunov exponents, follow work in [13] in the infinite-dimensional setting. Siegmund [97] and then Zhang [109] obtained earlier versions of Theorem 5.1 for spectra defined, respectively, in terms of uniform exponential dichotomies and nonuniform exponential dichotomies (the latter using the spectrum introduced in [23]). The examples in Section 5.2 are adapted from examples in [13] in the infinitedimensional setting. We tried as much as possible to follow the approach of Chapter 1 (notice that the possible forms of the spectrum are the same in the two settings). The result in Theorem 5.4 and its proof are based on Theorem 3.1. The decomposition obtained in Lemma 5.1 is a slight modification of a construction described by Coppel in [31] in terms of uniform exponential dichotomies. His work was generalized by Siegmund in [99] to consider coefficient matrices that are locally integrable (but not necessarily continuous) and a block-diagonalization into further blocks related to the spectrum introduced in [97] in the same manner as in Theorem 5.4. Finally, Theorem 5.5 on the construction of normal forms is based on work of Siegmund in [98], up to the point that he uses a spectrum defined in terms of uniform exponential dichotomies. See also [109] for the construction of normal forms using a spectrum defined in terms of nonuniform exponential dichotomies.

Chapter 6

Infinite-Dimensional Dynamics

This chapter is dedicated to some generalizations of the notions and results in the former chapters for an infinite-dimensional dynamics. This includes the description of all possible forms of the tempered spectrum for a sequence of compact linear operators, which leads to new forms of the spectrum. We also consider the construction of normal forms. Finally, we give examples of sequences of compact linear operators for all possible forms of the tempered spectrum.

6.1 Structure of the Spectrum Let .(An )n∈Z be a sequence of bounded linear operators acting on an infinitedimensional Banach space .X = (X, ||·||). The notion of tempered exponential dichotomy can again be introduced as in Section 1.1. We continue to denote the stable and unstable spaces of a tempered exponential dichotomy by .En and .Fn for each .n ∈ Z. For simplicity of the exposition, we consider only sequences of compact linear operators. We recall that a linear operator A between normed spaces is said to be compact if it maps bounded sets to sets with compact closure. Any compact operator is necessarily bounded. It turns out that for any sequence of compact operators with a tempered exponential dichotomy all unstable spaces are finite-dimensional. Proposition 6.1 For any sequence of compact linear operators .(An )n∈Z with a tempered exponential dichotomy, we have .dim Fn < ∞ for all .n ∈ Z. Proof Since compact operators are never invertible on infinite-dimensional spaces, it follows readily from the invertibility assumption in the notion of tempered exponential dichotomy that all unstable spaces are finite-dimensional. ⨆ ⨅

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 L. Barreira, C. Valls, Spectra and Normal Forms, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-51897-3_6

79

80

6 Infinite-Dimensional Dynamics

We also define tempered spectrum as before (see Definition 1.2). The following result builds on Theorem 1.1 to give a description of all possible forms of the tempered spectrum for a sequence of compact operators. Theorem 6.1 For a sequence of compact linear operators .(An )n∈Z , one of the following alternatives holds: 1. .Σ = U ∅. 2. .Σ = ki=1 |ai , bi | for some numbers .

+ ∞ ≥ b1 ≥ a1 > b2 ≥ a2 > · · · > bk ≥ ak ≥ −∞

(6.1)

and some U integer .k ∈ N. 3. .Σ = ∞ i=1 |ai , bi | for some numbers .

+ ∞ ≥ b1 ≥ a1 > b2 ≥ a2 > · · ·

(6.2)

such that U .ai → −∞ when .i → ∞. 4. .Σ = ∞ i=1 |ai , bi | ∪ (−∞, b∞ ] for some numbers as in (6.2) such that .ai → b∞ ∈ R when .i → ∞. Proof To the possible extent, we follow the proof of Theorem 1.1. First observe that Lemmas 1.1 and 1.2 also hold in the present context (it is important to use the dimensions of the unstable spaces in Lemma 1.2 since these are the ones that are finite-dimensional). We also need an additional property. Lemma 6.1 For each .c ∈ R \ Σ , the set .Σ ∩ [c, +∞) is the union of finitely many closed intervals. Proof of the lemma We proceed in a similar manner to that in the proof of Theorem 1.1. Write .dim F c = d and assume that .Σ ∩ [c, +∞) had at least .d + 2 connected components. Then there would be points .c1 , . . . , cd+2 ∈ R \ Σ , say with c = cd+2 < cd+1 < · · · < c2 < c1 ,

.

such that .[ci+1 , ci ] ∩ Σ = / ∅ for .i = 1, . . . , d + 1. It follows from Lemma 1.2 that dim F c ≥ d + 1, which contradicts the definition of d. Therefore, .Σ ∩ [c, +∞) is ⨆ ⨅ the union of at most .d + 1 closed intervals.

.

We proceed with the proof of the theorem. Since .Σ is closed, if it has finitely many connected components, then either it is empty or the second alternative in the theorem holds. Now assume that .Σ has infinitely many connected components. Take .c1 ∈ R \ Σ . By Lemma 6.1 the set .Σ ∩ [c1 , +∞) is the union of finitely many closed intervals. Note that .Σ ∩ (−∞, c1 ) is nonempty. Indeed, otherwise .Σ = Σ ∩ [c1 , +∞) and the spectrum would have finitely many connected components. Now observe that there is .c2 < c1 with .c2 ∈ R \ Σ such that

6.2 Examples of Tempered Spectra

81

(c2 , c1 ) ∩ Σ = / ∅.

.

Indeed, otherwise .Σ ∩ (−∞, c1 ) would have a single connected component, and again the spectrum would have finitely many connected components. One can proceed inductively to find a strictly decreasing sequence .(cn )n∈N in .R \ Σ such that (cn+1 , cn ) ∩ Σ = / ∅ for n ∈ N.

.

There are two possibilities: 1. If .cn → −∞ when .n → ∞, then it follows from Lemma 6.1 that the third alternative in the theorem holds. 2. If .cn → b∞ when .n → ∞ for some .b∞ ∈ R, then it follows from Lemma 6.1 that (b∞ , +∞) ∩ Σ =

∞ | |

.

|ai , bi |

i=1

for some constants as in (6.2) with .an , bn → b∞ when .n → ∞. It also follows from the lemma that .(−∞, b∞ ] ⊂ Σ , and so the last alternative in the theorem holds. ⨆ ⨅

This completes the proof of the theorem.

6.2 Examples of Tempered Spectra We can also give examples of sequences of compact linear operators for all possible forms of the tempered spectrum .Σ described in Theorem 6.1. We shall always consider the Σ Banach space .X = l2 (N) of sequences .x = (xn )n∈N of real numbers such that . n∈N |xn |2 < +∞ equipped with the norm ||x|| =

.

⎛Σ

⎞1/2 |xn |

2

.

n∈N

We recall (see, for example, [27]) that in a separable Hilbert space (such as .l2 (N)) with a basis .(en )n∈N , if .A : l2 (N) → l2 (N) is a linear operator such that .Aen = λn en for all .n ∈ N, then A is compact if and only if .λn → 0 when .n → ∞. Examples corresponding to the first two alternatives in Theorem 6.1 can be obtained closely imitating the examples given in Section 1.2. For completeness, we briefly describe the constructions.

82

6 Infinite-Dimensional Dynamics

Example 6.1 (.Σ = ∅) For each .n ∈ Z, let .An be the linear operator defined by ( ) 3 3 An x = e(n+1) −n x1 , 0, . . . .

.

One can proceed as in Example 1.1 to show that .Σ = ∅. Example 6.2 (.Σ = R) For each .n ∈ Z, let .An be the linear operator defined by ( ) An x = e(n+1) cos(n+1)−n cos n x1 , 0, . . . .

.

Similarly, one can proceed as in Example 1.2 to show that .Σ = R. Example 6.3 (.Σ nonempty and bounded, with finitely many connected components) Assume that all numbers .ai and .bi in (6.1) are finite. For each .n ∈ Z, let .An be the linear operator defined by ( ) An x = cn1 x1 , . . . , cnk xk , 0, 0, . . .

.

(6.3)

with .cni as in (1.22) for .i = 1, . . . , k. One can show as in Example 1.3 that Σ =

k | |

.

[ai , bi ].

i=1

Example 6.4 (.Σ unbounded from above or below, with finitely many connected components) Again assume that all numbers .ai and .bi in (6.1) are finite. For each .n ∈ Z, let .An be the linear operator defined by (6.3), with ⎧ 1 .cn

=

√

√ n+1 cos(n+1)− n cos n √ √ ea1 + |n+1| cos(n+1)− |n| cos n

e2n+1+

if n ≥ 0, if n < 0,

(6.4)

and .cni as in (1.22) for .i = 2, . . . , k. One can show as in Example 1.4 that Σ =

k | |

.

[ai , bi ] ∪ [a1 , +∞).

i=2

Similarly, for each .n ∈ Z, let .An be the linear operator defined by (6.3), with .cni as in (1.22) for .i = 1, . . . , k − 1, and k .cn

=

⎧ √ √ ebk + n+1 cos(n+1)− n cos n

√ √ e2n+1+ |n+1| cos(n+1)− |n| cos n

One can show as in Example 1.5 that

if n ≥ 0, if n < 0.

(6.5)

6.2 Examples of Tempered Spectra

83

Σ = (−∞, bk ] ∪

k−1 | |

.

[ai , bi ].

i=1

Finally, for each .n ∈ Z, let .An be the linear operator defined by (6.3), with .cn1 as in (6.4), .cni as in (1.22) for .i = 2, . . . , k − 1, and .cnk as in (6.5). Then Σ = (−∞, bk ] ∪

k−1 | |

[ai , bi ] ∪ [a1 , +∞).

.

i=2

Example 6.5 (Case 3 in Theorem 6.1) Take numbers .ai and .bi as in (6.2) with ai → −∞

.

when i → ∞.

(6.6)

For each .n ∈ Z, let .An be the linear operator defined by ( ) An x = cn1 x1 , cn2 x2 , cn3 x3 , . . . ,

.

with .cni as in (1.22) for .i ≥ 2, and either as in (1.22) or in (6.4) for .i = 1. By (6.6), for each .n ∈ Z we have .cni → 0 when .i → ∞, and so each operator .An is compact. To determine the tempered spectrum, U we proceed in a similar manner to that in Examples 1.3 and 1.4. Take .a ∈ R \ ∞ i=1 |ai , bi |. When .bi < a < ai−1 for some .i > 1, let .Pn = Id − Qn with Qn x = (x1 , . . . , xi−1 , 0, 0, . . .),

.

and when .a > b1 , let .Pn = Id. Proceeding as in those examples, we find that the sequence .(e−a An )n∈Z has a tempered exponential dichotomy, and that in fact Σ =

∞ | |

.

|ai , bi |.

i=1

Example 6.6 (Case 4 in Theorem 6.1) Take numbers .ai and .bi as in (6.2) with ai → b∞ ∈ R when .i → ∞. For each .n ∈ Z, let .An be the linear operator defined by

.

( ) An x = cn∞ x1 , cn1 x2 , cn2 x3 , . . . ,

(6.7)

.

with ⎧ ∞ .cn

=

√

√ n+1 cos(n+1)− n cos n √ √ e2n+1+ |n+1| cos(n+1)− |n| cos n

e b∞ +

if n ≥ 0, if n < 0

(6.8)

84

6 Infinite-Dimensional Dynamics

and ⎧ cni =

.

√ √ n+1 cos(n+1)− n cos n /c ni √ √ |n+1| cos(n+1)− a + |n| cos n i /cni e

e bi +

if n ≥ 0, if n < 0

(6.9)

for .i ≥ 1, where .cni = 1 + ie−|n| . For each .n ∈ Z we have .cni → 0 when .i → ∞, and so each operator .An is compact. Now observe that ⎧ √ √ b∞ (m−n)+ m cos m− n cos n ⎪ ⎪ ⎨e √ 2 √ = eb∞ m−n + m cos m− |n| cos n ⎪ √ √ ⎪ ⎩e(m+n)(m−n)+ |m| cos m− |n| cos n

∞ .cm,n

if m, n ≥ 0, if m ≥ 0, n < 0, if m, n < 0

and

i .cm,n

⎧ √ √ | | bi (m−n)+ m cos m− n cos n / m c ⎪ ⎪ p=n pi ⎨e √ √ | |m m cos m− b m−a n+ |n| cos n i i = e / p=n cpi ⎪ √ √ ⎪ ⎩eai (m−n)+ |m| cos m− |n| cos n / | |m c p=n pi

if m, n ≥ 0, if m ≥ 0, n < 0, if m, n < 0

for .i ≥ 1. Take .a > bi . Then √ √ |m|+ |n|

i e−a(m−n) cm,n ≤ e−(a−bi )(m−n)+

.

/

m | |

cpi

p=n

for .m ≥ n. Now take .ε > 0. By (1.25), for .m ≥ n we obtain i e−a(m−n) cm,n ≤ De−(a−bi )(m−n)+ε|m|+ε|n| .

≤ De−(a−bi −ε)(m−n)+2ε|n|

for some constant .D = D(ε) > 0. Since .a − bi > 0 and .ε is arbitrary, the sequence (e−a cni )n∈Z has a tempered exponential contraction. Similarly, for .a < ai the sequence .(e−a cni )n∈Z has a tempered exponential expansion. Indeed, by (1.25), for .m ≤ n we obtain

.

√ √ |m|− |n|

i e−a(n−m) cn,m ≥ e−(a−ai )(n−m)−

/

n | |

cpi

p=m

.

¯ −(a−ai +ε)(n−m)−2ε|n| ≥ De ¯ ε) > 0. Since .a − ai < 0 and .ε is arbitrary, the for some constant .D¯ = D(i, sequence .(e−a cni )n∈Z has a tempered exponential expansion. Finally, for .a > b∞ and .m ≥ n, we obtain

6.3 Normal Forms

85 √ √ |m|+ |n|

∞ e−a(m−n) cm,n ≤ e−(a−b∞ )(m−n)+ .

≤ D 2 e−(a−b∞ −ε)(m−n)+2ε|n| .

Since .a − b∞ > 0 and .ε is arbitrary, the sequence .(e−a cn∞ )n∈Z has a tempered exponential contraction. Now assume that .bi < a < ai−1 . For .j ≥ i and .m ≥ n we have e−a(m−n) cm,n ≤ De−(a−bj −ε)(m−n)+2ε|n| / j

m | |

cpj

p=n

.

≤ De−(a−bi −ε)(m−n)+2ε|n| , since .bj ≤ bi and .cpj ≥ 1. Analogously, for .j < i and .m ≤ n, we have j e−a(m−n) cm,n

2 −(a−aj +ε)(m−n)+2ε|n|

≤D e

n | |

cpj

p=m .

≤ D 2 e−(a−ai +ε)(m−n)+2ε|n| max

1≤j b1 . Given integers .l1 < l2 < · · · < ls , we have

86

6 Infinite-Dimensional Dynamics

X = Gn ⊕

s ⊕

.

Hnli

i=1

for .n ∈ Z, where cl

c

Gn = Enls +1 ⊕ Fn 1 ⊕

⊕

.

Hn1

l∈I

and I =N∩

s−1 | |

.

(li , li+1 ).

i=1

The following result reduces the sequence .(An )n∈Z to one in block-diagonal form. Let .mi = dim Hnli for each i and define .d = m1 + · · · + ms . Theorem 6.2 Any sequence of compact linear operators .(An )n∈Z whose tempered spectrum has infinitely many connected components and is bounded from above is cohomologous to a sequence .(Bn )n∈Z of linear operators Bn = diag(Bn1 , . . . , Bns+1 ) : Rd × Gn → Rd × Gn+1 ,

.

where .B i = (Bni )n∈Z for each .i = 1, . . . , s is a sequence of invertible .mi × mi matrices with Σ (B i ) = [ali , bli ]

.

for i = 1, . . . , s

and Bns+1 = An |Gn : Gn → Gn+1 .

.

Proof We proceed in a similar manner to that in the proof of Theorem 3.1. Write ¯n = G

s ⊕

.

Hnlk .

k=1

¯n → G ¯ n+1 is invertible. Moreover, for each .i = We note that each map .An |G¯ n : G ¯ n → Rd satisfying 1, . . . , s − 1 there is an invertible linear map .Li : G Li P0i Li = P ,

.

where .P0i is the projection of the tempered exponential dichotomy with splitting

6.3 Normal Forms

87

⎛ i ⊕ .

⎞ Hnlk

⎛ ⊕

k=1

s ⊕

⎞ Hnlk

.

k=i+1

One can now proceed as in the proof of Theorem 3.1 to obtain a sequence .(A¯ in )n∈Z of .d × d matrices of the form i )−1 An |G¯ n Uni = diag(Cni , Dni ) A¯ in = (Un+1

.

(6.10)

¯ n satisfying for some invertible linear maps .Uni : Rd → G .

lim

n→±∞

1 1 log||Uni || = lim log||(Uni )−1 || = 0 n→±∞ n n

and for some matrices .Cni and .Dni of dimensions .qi × qi and .(d − qi ) × (d − qi ), where .qi = m1 +· · · +mi . Proceeding inductively, one can further split the matrices in (6.10) to obtain a sequence .Bn as in the statement of the theorem. ⨆ ⨅ We use the tempered spectrum and the block-diagonalization in Theorem 6.2 to obtain normal forms for a nonlinear dynamics xn+1 = Bn xn + fn (xn )

.

for n ∈ Z,

for some maps .fn : X → X of class .C p , for some .p ≥ 1, with .fn (0) = 0 and .d0 fn = 0 for .n ∈ Z. We write Σ

fn (x) =

.

r∈Ns0 ,2≤|r|≤p

1 r ∂ fn x r + o(||x||p ). r! 0

(6.11)

Given .x ∈ X, let .x = (x 1 , . . . , x s , x s+1 ), with .x i ∈ Rmi for .i = 1, . . . , s and s+1 ∈ G . Moreover, given a vector .r = (r , . . . , r ) ∈ Ns , we write .x n 1 s 0 |r| = r1 + · · · + rs

.

and

∂ r fn = ∂xr11 · · · ∂xrss fn .

We also consider a corresponding notion of resonance. Given .i ∈ {1, . . . , s} and r ∈ Ns0 with .|r| ≥ 2, the pair .(i, r) is said to be a resonance if

.

ali ≤

s Σ

.

j =1

rj alj

and

bli ≥

s Σ j =1

Writing fn = (fn1 , . . . , fns , fns+1 ),

.

rj blj .

88

6 Infinite-Dimensional Dynamics

the component .(1/r!)∂0r fni x r in (6.11) is said to be resonant if the pair .(i, r) is a resonance. The following result gives the normal forms. Theorem 6.3 Let .(An )n∈Z be a sequence of compact linear operators and let (Bn )n∈Z be the sequence constructed in Theorem 6.2. Moreover, let .fn : X → X be maps of class .C p , for some .p ≥ 1, with .fn (0) = 0 and .d0 fn = 0 for .n ∈ Z satisfying property (3.16). Then:

.

1. There are polynomials .hn : Rd → Rd with .hn (0) = 0 and .d0 hn = 0 for .n ∈ Z satisfying (3.17). 2. Letting xni = yni + hin (yn1 , . . . , yns )

.

for n ∈ Z, i = 1, . . . , s,

and .xns+1 = yns+1 , we obtain yn+1 = Bn yn + gn (yn )

.

for n ∈ Z,

where .gn = (gn1 , . . . , gns+1 ) : X → X are maps of class .C p with .gn (0) = 0 and .d0 gn = 0. 3. .∂0r gni = 0 for all .n ∈ Z, .i = 1, . . . , s and .r ∈ Ns0 with .2 ≤ |r| ≤ p such that .(i, r) is not a resonance. Proof Assume that .(i, r) is not a resonance. Writing .x¯ = (x 1 , . . . , x s ), we define hn (x) ¯ = (h1n (x), ¯ . . . , hsn (x)), ¯

.

Σ j where .hn = 0 for .j /= i, with .hin given by (3.20) whenever .ali > sj =1 rj blj and Σ by (3.21) whenever .bli < sj =1 rj alj (with k replaced by s). Proceeding as in the proof of Theorem 3.2, one can show that each map .hn is a well-defined polynomial of degree at most p (depending only on the first s components). The remainder of ⨆ ⨅ the argument is identical to that in the proof of Theorem 3.2. Notes The notion of exponential dichotomy has a natural and straightforward generalization to the infinite-dimensional setting. On this respect, an important role was played by the books of Hale [40] (see also [41]) on delay equations and of Henry [46] on parabolic partial differential equations. Moreover, many linear delay equations (see, for example, [33, 41]) and many linear parabolic partial differential equations (see, for example, [46, 94]) generate evolution families .T (t, s) that are compact for .t > s+r with some .r ≥ 0 (in the case of delay equations r is the delay). Incidentally, it is easy to show that a dynamics with continuous time having bounded growth has an exponential dichotomy if and only if any of its discretizations has an exponential dichotomy. Here we consider the case of discrete time, with the study of sequences of compact linear operators. For details and references on exponential dichotomies

6.3 Normal Forms

89

and their applications in the infinite-dimensional setting, we refer the reader to the books [20, 42, 46, 94]. In a similar manner to that in Chapter 1, the notion of tempered exponential dichotomy can also be seen as a simplification of the notion of nonuniform hyperbolicity for cocycles over an invertible map. In particular, for a strongly measurable cocycle with values in the bounded linear operators on a separable Banach space over a measure-preserving map, Lian and Lu [59] showed that if all Lyapunov exponents are nonzero almost everywhere, then the dynamics has a tempered exponential dichotomy. Theorem 6.1 is taken from [8], which also considers the more general case of sequences of bounded linear operators satisfying a certain asymptotic compactness. The proof of the theorem builds on the proof of Theorem 1.1, and so we avoid repeating the arguments that are already there. The only new element consists of showing that for each .c ∈ R \ Σ , the set .Σ ∩ [c, +∞) is the union of finitely many closed intervals (see Lemma 6.1). This implies that the intervals of the spectrum may accumulate from the right at .−∞ or at a real number, but not from the left. For the most part, the examples in Section 6.2 follow those given in [8] and consist of sequences of compact linear operators on .l2 (N). One could also give examples of sequences of compact linear operators on the Banach space .L2 (S 1 ) with the Lebesgue measure on .S 1 . Theorems 6.2 and 6.3 are elaborations, respectively, of Theorems 3.1 and 3.2 and are taken from [14].

Chapter 7

Stable and Unstable Foliations

In this chapter we construct an unstable foliation building on the construction of unstable invariant manifolds for any sufficiently small perturbation of a tempered exponential dichotomy. One can also construct a stable foliation, simply by reversing time and so the corresponding details are omitted. The two foliations are crucial for the construction of smooth conjugacies in Chapter 8.

7.1 Preliminaries We recall that a sequence of positive numbers .(cn )n∈Z is said to be, respectively, lower tempered or upper tempered if .

1 log cn ≥ 0 or |n|

lim inf

n→±∞

lim sup n→±∞

1 log cn ≤ 0. |n|

In particular we have the following property. Proposition 7.1 Given an upper tempered sequence .(Dn )n∈Z and a constant .ε > 0, there is an upper tempered sequence .(D¯ n )n∈Z with .D¯ n ≥ max{1, Dn } and D¯ n e−ε|m| ≤ D¯ n+m ≤ D¯ n eε|m|

.

for n, m ∈ Z.

Proof We show that one can take { } D¯ n = sup 1, Dn+m e−ε|m| : m ∈ Z .

.

(7.1)

Since .(Dn )n∈Z is an upper tempered sequence, it follows from (1.6) that Dn+m e−ε|m| ≤ Deε|n+m|−ε|m| ≤ Deε|n|

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 L. Barreira, C. Valls, Spectra and Normal Forms, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-51897-3_7

91

92

7 Stable and Unstable Foliations

and so in particular .D¯ n is finite. Moreover, clearly, .D¯ n ≥ max{1, Dn }. We also have { } D¯ n+m = sup 1, Dn+m+l e−ε|l| : l ∈ Z . { } ≤ sup 1, Dn+m+l e−ε|m+l| : l ∈ Z eε|m| = D¯ n eε|m| and, similarly, .

{ } D¯ n+m ≥ sup 1, Dn+m+l e−ε|m+l| : l ∈ Z e−ε|m| = D¯ n e−ε|m| .

Now take .δ ∈ (0, ε). Since .(Dn )n∈Z is upper tempered, it follows from (1.6) with ε replaced by .δ that

.

Dn+m e−ε|m| ≤ Deδ|n+m|−ε|m| ≤ Deδ|n| .

.

By (7.1) we conclude that lim sup

.

n→±∞

1 log D¯ n ≤ δ, |n|

and it follows from the arbitrariness of .δ that .(D¯ n )n∈Z is upper tempered.

⨆ ⨅

Now let .(An )n∈Z be a sequence of invertible .d × d matrices with a tempered exponential dichotomy, say with constant .λ and upper tempered sequence .(Dn )n∈Z . Moreover, let .fn : Rd → Rd for .n ∈ Z be maps of class .C p , for some .p ≥ 2, with .fn (0) = 0 and .d0 fn = 0 for .n ∈ Z. We also consider the upper tempered sequence ¯ n )n∈Z given by Proposition 7.1 with .ε = λ/2. Finally, we assume that there are .(D upper tempered sequences .(cn,i )n∈Z for .0 ≤ i ≤ p and a lower tempered sequence .(rn )n∈Z in .(0, 1] satisfying D¯ n2 cn,1 ≤ a

.

for n ∈ Z

(7.2)

and some .a > 0 such that for each .n ∈ Z: 1. .fn (x) = 0 whenever .||x|| ≥ 2rn /3. 2. .||dxi fn || ≤ cn,i for .x ∈ Rd and .0 ≤ i ≤ p. Before proceeding, we illustrate how to give examples of maps .fn as above. Consider maps .f¯n : B(0, rn ) → Rd for .n ∈ Z of class .C p , for some .p ≥ 2, with .f¯n (0) = 0 and .d0 f¯n = 0 for .n ∈ Z, and a lower tempered sequence .(rn )n∈Z in .(0, 1]. We assume that for each .0 ≤ i ≤ p there are upper tempered sequences .(c ¯n,i )n∈Z with D¯ n2 c¯n,2 rn ≤ a¯

.

for n ∈ Z

(7.3)

7.1 Preliminaries

93

(taking .ε = λ/2 in Proposition 7.1) and some .a¯ > 0, such that ||dxi f¯n || ≤ c¯n,i

.

for n ∈ Z, x ∈ B(0, rn ) and 0 ≤ i ≤ p.

(7.4)

We also consider a smooth cutoff. Let .ρ : R → [0, 1] be a .C ∞ function such that ρ|[−1,1] = 1,

.

ρ|R\[−2,2] = 0 and

sup |ρ ' (x)| ≤ 2.

(7.5)

x∈R

Finally, let .sn = rn /3 and consider the maps .gn : Rd → Rd defined by ⎛ gn (x) = ρ

.

⎞ ||x|| ¯ fn (x). sn

Proposition 7.2 The maps .gn satisfy the same hypotheses as the maps .fn . Proof Let .ρn (x) = ρ(||x||/sn ). We have dx ρn u = ρ '

⎛

.

||x|| sn

⎞

1 sn ||x||

and so | ⎛ ⎞| | ' ||x|| | 1 | ≤ 2. | .||dx ρn || ≤ ρ | sn | sn sn Moreover, .

dx2 ρn (u, v) = ρ ''

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ||x|| 1 1 ' ||x|| , − +ρ sn sn ||x|| sn sn2 ||x||2 ||x||3

which gives 2 .||dx ρn ||

| ⎛ | ⎛ | ⎛ ⎞| ⎞| ⎞| | '' ||x|| | 1 | '' ||x|| | 1 | ' ||x|| | 1 2 | | | + 4. | | | ≤ |ρ + |ρ · ≤ |ρ | | 2 sn sn sn ||x|| sn | sn2 sn sn2

It follows by induction that there are constants .ai > 0 for .i ∈ N such that ||dxi ρn || ≤

.

ai sni

for x ∈ Rd .

Note that since .f¯n (0) = 0 and .d0 f¯n = 0, by (7.4) we have 1 ||f¯n (x)|| ≤ c¯n,2 ||x||2 2

.

Therefore,

and

||dx f¯n || ≤ c¯n,2 ||x||.

94

7 Stable and Unstable Foliations

||gn (x)|| ≤ ||f¯n (x)|| ≤

.

1 c¯n,2 (2sn )2 = 2c¯n,2 sn2 . 2

Moreover, dx gn u = ρn (x)dx f¯n u + (dx ρn u)f¯n (x),

.

which gives ||dx gn || ≤ sup ||dx f¯n || +

2 sn

≤ sup c¯n,2 ||x|| +

2 sn

||x|| 0, for q sufficiently small, we have sup ||R1,m || ≤ δ||ζ¯ − ζ ||.

.

m≤n

Similarly, we obtain ||R2,m || ≤

n−1 Σ

e−λ0 (m−n) Dl+1 eλ(m−l−1) Sl (ζ¯ , ζ )||xl (ζ¯ ) − xl (ζ )||

l=q .

≤

n−1 Σ

(7.18) e

−λ0 (m−n)

Dl+1 e

λ(m−l)

Sl (ζ¯ , ζ )2Dn ||ζ¯ − ζ ||e

(λ/2)(l−n)

l=q

both for .m < q and .m ≥ q. By (7.17) the functions .ζ |→ xl (ζ ) and so also .(ζ¯ , ζ ) |→ Sl (ζ¯ , ζ ) are continuous. Since the right-hand side of (7.18) is a finite sum, given .δ > 0, we have .

sup ||R2,m || ≤ δ||ζ¯ − ζ ||

m≤n

7.2 Invariant Manifolds

101

for .||ζ¯ − ζ || sufficiently small. Analogous arguments show that .

sup ||Ri,m || ≤ δ||ζ¯ − ζ || for i = 3, 4

m≤n

for .||ζ¯ −ζ || sufficiently small. Therefore, .||R||λ0 ≤ 4δ||ζ¯ −ζ || for .||ζ¯ −ζ || sufficiently small, and the desired result follows from the arbitrariness of .δ. ⨆ ⨅ Now let .T : Fn → S(λ0 ) be the linear operator defined by .T (ζ ) = (Cm,n ζ )m≤n . Since e−λ0 (m−n) ||Cm,n ζ || ≤ e−λ0 (m−n) Dn e−λ(n−m) ||ζ || ≤ Dn ||ζ ||

.

for .m ≤ n, we have .||T ||λ0 ≤ Dn . We denote by .L(c) the set of bounded linear operators from .Fn to .S(c). Lemma 7.4 For any sufficiently small a, the map .Fn ϶ ζ |→ x(ζ ) ∈ S(λ0 ) is differentiable, .d0 x = T and .||dζ x||L(λ0 ) ≤ 2Dn for .ζ ∈ Fn . Proof of the lemma Take .n ∈ Z and .ζ ∈ Fn . Moreover, let .Uζ : S(λ0 ) → S(λ0 ) be the linear operator defined by (Uζ v)m = −

n−1 Σ

Cm,l+1 dxl (ζ ) fl vl +

.

l=m

m−1 Σ

Bm,l+1 dxl (ζ ) fl vl

(7.19)

l=−∞

for .m ≤ n, where .v = (vm )m≤n . Proceeding as in (7.15), one can show that .Uζ is a bounded linear operator from .S(λ0 ) to itself. Moreover, .||Uζ || < 12 for a sufficiently small and so .Id − Uζ has a bounded inverse on .S(λ0 ). By Lemma 7.3 we have .

x(ζ¯ ) − x(ζ ) − Uζ (x(ζ¯ ) − x(ζ )) = T (ζ¯ − ζ ) + R,

which gives x(ζ¯ ) − x(ζ ) − (Id − Uζ )−1 T (ζ¯ − ζ ) → 0

.

when .ζ¯ → ζ . This shows that the map .x : Fn → S(λ0 ) is differentiable with dζ x = (Id − Uζ )−1 T ∈ L(λ0 )

.

and ||dζ x||L(λ0 ) ≤

.

Since .U0 = 0, we also get .d0 x = T .

||T || ≤ 2Dn . 1 − ||Uζ ||

(7.20) ⨆ ⨅

102

7 Stable and Unstable Foliations

Step 3. Higher derivatives Since .S(λ0 ) ⊂ S(λ1 ), the map .dζ x can also be seen as an element of .L(λ1 ). By (7.20) we have ||dζ x||L(λ1 ) ≤ ||dζ x||L(λ0 ) ≤ 2Dn .

(7.21)

.

Lemma 7.5 For any sufficiently small a, the map .Fn ϶ ζ |→ dζ x ∈ L(λ1 ) is continuous. Proof of the lemma Write ) ( hl (ζ¯ , ζ ) = dxl (ζ¯ ) fl − dxl (ζ ) fl dζ xl

.

and let n−1 Σ

' Rm =−

.

m−1 Σ

Cm,l+1 hl (ζ¯ , ζ ) +

l=m

Bm,l+1 hl (ζ¯ , ζ )

l=−∞

for .m ≤ n. Take .q ∈ Z with .q ≤ n − 1 and let ⌠ ' .R1,m

=

=

= ⌠

' .R4,m

' = Note that .Rm

=

¯

l=m Cm,l+1 hl (ζ , ζ )

e−λ1 (m−n)

e−λ1 (m−n)

Σ n−1

Cm,l+1 hl (ζ¯ , ζ ) ¯ l=m Cm,l+1 hl (ζ , ζ ) l=q

Σ n−1

Σ m−1 l=q

Bm,l+1 hl (ζ¯ , ζ )

e−λ1 (m−n) e−λ1 (m−n)

' i=1 Rm,i .

Σ q−1

Bm,l+1 hl (ζ¯ , ζ ) ¯ l=−∞ Bm,l+1 hl (ζ , ζ )

Σ l=−∞ m−1

It follows from (7.21) that

||dζ xl || ≤ 2Dn eλ0 (l−n)

.

We obtain

if m < q, if m ≥ q, if m > q, if m ≤ q,

0

Σ 4

if m < q, if m ≥ q,

e−λ1 (m−n) ⌠

' .R3,m

Σ q−1

0 ⌠

' .R2,m

e−λ1 (m−n)

for l ≤ n.

if m > q, if m ≤ q.

7.2 Invariant Manifolds

103

' ||R1,m || ≤ 2e−λ1 (m−n)

q−1 Σ

Dl+1 cl,1 2Dn eλ(m−l−1) eλ0 (l−n)

l=m

≤ 4aDn .

q−1 Σ

e(3λ/4)(m−l) e(λ/12)(l−n)

l=m

= 4aDn

e(λ/12)(m−n) (1 − e(2λ/3)(m−q) ) 1 − e−2λ/3

< 4aDn

e(λ/12)(q−n) . 1 − e−2λ/3

' || ≤ δ. Similarly, Given .δ > 0, for q sufficiently small, we have .supm≤n ||R1,m

' ||R2,m || ≤ e−λ1 (m−n)

n−1 Σ

.

Dl+1 eλ(m−l−1) ||dxl (ζ¯ ) fl − dxl (ζ ) fl ||2Dn eλ0 (l−n) .

l=q ' || ≤ δ Since the map .ζ |→ dxl (ζ ) fl is continuous, given .δ > 0, we have .supm≤n ||R2,m for .||ζ¯ − ζ || sufficiently small. Analogous arguments show that .

' sup ||Ri,m || ≤ δ

m≤n

for i = 3, 4

for .||ζ¯ − ζ || sufficiently small and so .||R ' ||L(λ1 ) → 0 when .ζ¯ → ζ . On the other hand, dζ¯ xm − dζ xm = −

n−1 Σ

.

( ) Cm,l+1 dxl (ζ¯ ) fl dζ¯ xl − dxl (ζ ) fl dζ xl

l=m

+

m−1 Σ

( ) Bm,l+1 dxl (ζ¯ ) fl dζ¯ xl − dxl (ζ ) Fl dζ xl

l=−∞ ' = Rm −

n−1 Σ

Cm,l+1 dxl (ζ ) fl (dζ¯ xl − dζ xl )

l=m

+

m−1 Σ

Bm,l+1 dxl (ζ ) fl (dζ¯ xl − dζ xl ).

l=−∞

Hence, ||dζ¯ xm − dζ xm ||e−λ1 (m−n)

.

104

7 Stable and Unstable Foliations

≤ ||R ' ||L(λ1 ) + e−λ1 (m−n)

n−1 Σ

Dl+1 cl,1 eλ(m−l−1) ||dζ¯ xl − dζ xl ||

l=m

+ e−λ1 (m−n)

m−1 Σ

Dl+1 cl,1 e−λ(m−l−1) ||dζ¯ xl − dζ xl ||

l=−∞ '

≤ ||R ||L(λ1 ) + a

n−1 Σ

e(3λ/4)(m−l) ||dζ¯ xl − dζ xl ||e−λ1 (l−n)

l=m m−1 Σ

+ ae3λ/2

e−(5λ/4)(m−l) ||dζ¯ xl − dζ xl ||e−λ1 (l−n) .

l=−∞

In a similar manner to that in the proof of Lemma 7.2, one can show that .

1 ||dζ¯ x − dζ x||L(λ1 ) ≤ ||R ' ||L(λ1 ) + ||dζ¯ x − dζ x||L(λ1 ) 2

for a sufficiently small. Therefore, ||dζ¯ x − dζ x||L(λ1 ) ≤ 2||R ' ||L(λ1 ) → 0

.

when .ζ¯ → ζ , which yields the desired statement.

⨆ ⨅

Now we consider the higher derivatives of .x(ζ ). Lemma 7.6 For any sufficiently small a, given .1 ≤ i ≤ p, the map .Fn ϶ ζ |→ x(ζ ) ∈ S(λi ) is of class .C i and .||dζi x||L(λi ) ≤ kn,i for .ζ ∈ Fn , where each sequence .(kn,i )n∈Z is upper tempered. Proof of the lemma We proceed by induction. By (7.21) the property holds for .i = 1. Let .Lj (c) be the set of bounded j -linear operators from .Fn to .S(c). Now take j .2 ≤ i ≤ p. By the induction hypothesis, the map .d x : Fn → Lj (λj ) is continuous and j

||dζ x||Lj (λj ) ≤ kn,j

.

(7.22)

for .ζ ∈ Fn and .1 ≤ j ≤ i − 1, where each sequence .(kn,j )n∈Z is upper tempered. We have dζi−1 xm

=−

n−1 Σ

Cm,l+1 dxl (ζ ) fl dζi−1 xl

+

l=m .

−

n−1 Σ l=m

m−1 Σ

Bm,l+1 dxl (ζ ) fl dζi−1 xl

l=−∞

Cm,l+1 Σ i,l (ζ ) +

m−1 Σ l=−∞

Bm,l+1 Σ i,l (ζ )

7.2 Invariant Manifolds

105

for .m ≤ n, where ⎞ i−3 ⎛ Σ i − 2 i−2−j j +1 dζ (dxl (ζ ) fl )dζ xl . j

Σ i,l (ζ ) =

.

j =0

j

Note that .Σ i,l (ζ ) is a linear combination of derivatives .dxl (ζ ) fl with .2 ≤ j ≤ i − 1 j

applied to two or more derivatives .dζ xl with .1 ≤ j ≤ i − 2. Since each map .fl is j

of class .C p and .ζ |→ dζ xl ∈ Lj (λj ) is continuous for .1 ≤ j ≤ i − 2, the map 1 .Σ i,l is of class .C . On the other hand, by (7.22) and the structure of .Σ i,l (ζ ), letting .cl = max2≤i≤p cl,i , we have ||dζ Σ i,l || ≤ cl Nn,i e2λi−1 (l−n)

.

for .l ≤ n and some upper tempered sequence .(Nn,i )n∈Z . We illustrate the estimate in the simplest case of .i = 3. We have Σ 3,l (ζ ) = dζ (dxl (ζ ) fl )dζ xl

.

and so Σ 3,l (ζ )(u, v) = dx2l (ζ ) fl (dζ xl u, dζ xl v).

.

Similarly, one can obtain a formula for .dζ Σ 3,l (u, v, w) that then gives ||dζ Σ 3,l || ≤ ||dx3l (ζ ) fl || · ||dζ xl ||3 + 2||dx2l (ζ ) fl || · ||dζ xl || · ||dζ2 xl || .

≤ cl,3 (kn,1 eλ1 (l−n) )3 + 2cl,2 kn,1 eλ1 (l−n) kn,2 eλ2 (l−n) ≤ cl Nn,3 e2λ2 (l−n) ,

3 + 2k k . taking the upper tempered sequence .Nn,3 = kn,1 n,1 n,2 Now let .V (ζ ) = (Vm (ζ ))m≤n , where

.

Vm (ζ ) = −

n−1 Σ l=m

Cm,l+1 Σ i,l (ζ ) +

m−1 Σ

Bm,l+1 Σ i,l (ζ )

(7.23)

l=−∞

for .m ≤ n. Note that the sequence .(Dn+1 cn )n∈Z is upper tempered, and so by Proposition 7.1, there is an upper tempered sequence .(c¯n )n∈Z (possibly depending on i) such that Dl+1 cl ≤ c¯l ≤ c¯n eλi−1 (n−l)

.

for l < n.

(7.24)

106

7 Stable and Unstable Foliations

Since ||Cm,l+1 dζ Σ i,l || ≤ Dl+1 eλ(m−l−1) cl Nn,i e2λi−1 (l−n)

.

for .m ≤ l ≤ n − 1 and ||Bm,l+1 dζ Σ i,l || ≤ Dl+1 e−λ(m−l−1) cl Nn,i e2λi−1 (l−n)

.

for .l ≤ m − 1, it follows from (7.24) that e−λi−1/2 (m−n)

n−1 Σ

.

||Cm,l+1 dζ Σ i,l (ζ )||

l=m

≤ e−λi−1/2 (m−n)

n−1 Σ

Dl+1 eλ(m−l−1) cl Nn,i e2λi−1

l=m

=

n−1 Σ

Dl+1 cl Nn,i e−λ eλi−1 (l−n) e((1+i)λ/(2+i))(m−l) e(λi−1 −λi−1/2 )(m−n)

l=m n−1 Σ

≤

Dl+1 cl Nn,i eλi−1 (l−n) e((1+i)λ/(2+i))(m−l)

l=m n−1 Σ

≤

c¯n Nn,i e((1+i)λ/(2+i))(m−l) .

l=m

Similar estimates hold for the second term in (7.23), which readily implies that the map .Fn ϶ ζ |→ V (ζ ) ∈ L(λi−1/2 ) is of class .C 1 . Now we consider the linear operator .U¯ ζ from .Li−1 (λi−1/2 ) to itself given by (7.19) for .v ∈ Li−1 (λi−1/2 ). We have i−1 i−1 i−1 i ¯ ¯ (dζi−1 ¯ x − dζ x) − Uζ (dζ¯ x − dζ x) = V (ζ ) − V (ζ ) + Σ ,

.

i ) where .Σ i = (Σ m m≤n with

i Σ m =−

n−1 Σ

( ) Cm,l+1 dxl (ζ¯ ) fl − dxl (ζ ) fl dζi−1 ¯ xl

l=m .

+

m−1 Σ l=−∞

for .m ≤ n. One can verify that

( ) Bm,l+1 dxl (ζ¯ ) fl − dxl (ζ ) fl dζi−1 ¯ xl

7.2 Invariant Manifolds

107

||Σ i ||Li−1 (λi−1/2 ) → 0

.

when ζ¯ → ζ.

Moreover, in a similar manner that in the proof of Lemma 7.4, the operator .Id − Uζ has a bounded inverse in .Li−1 (λi−1/2 ) for a sufficiently small and so .

( ) i−1 ¯ −1 V (ζ¯ ) − V (ζ ) + Σ i . dζi−1 ¯ x − dζ x = (Id − Uζ )

Therefore, .dζi x = (Id − U¯ ζ )−1 dζ V and .dζi x ∈ Li (λi−1/2 ). One can also show that ⨆ ⨅ the map .d i x : Fn → Li (λi ) is continuous and that (7.22) holds for .j = i. Step 4. Unstable manifold as a graph We define a function .ϕn : Fn → En by ϕn (ζ ) := Pn xn (ζ ) =

n−1 Σ

.

Bn,l+1 fl (xl (ζ )),

(7.25)

l=−∞

where .x(ζ ) is the unique solution given by Lemma 7.2. It follows from Lemma 7.6 that the map .ϕn is of class .C p . It is also clear that .0 ∈ Wn and .T0 Wn = Fn . It follows from Lemmas 7.1 and 7.2 that .v ∈ Wn if and only if there is a unique sequence .(xm )m≤n ∈ S(c) such that .xn = v and xm+1 = Am xm + fm (xm )

for m < n.

.

Now let .xm be this sequence and define ym+1 = Am xm + fm (xm )

.

for m < n + 1.

We have .yn+1 = An v + fn (v) and ym+1 = Am ym + fm (ym )

.

for m < n + 1.

Note that .

( ) { } sup ||ym ||e−c(m−n−1) = max ec ||(xm )m≤n ||, ||yn+1 || < +∞, m≤n+1

because .(xm )m≤n ∈ S(c). Therefore, .(ym )m≤n+1 ∈ S'c with .S'c defined as .S(c) replacing n by .n + 1. Therefore, .(An + fn )(Wn ) ⊂ Wn+1 . The inclusion .Wn+1 ⊂ (An + fn )(Wn ) can be obtained similarly. For the last statement in the theorem, note that .fl (xl (ζ )) = 0 for .||xl (ζ )|| ≥ 2rl /3 and ||fl (xl (ζ ))|| ≤ cl,1 ||xl (ζ )|| ≤

.

2 cl,1 rl < cl,1 3

108

7 Stable and Unstable Foliations

for .||xl (ζ )|| ≤ 2rl /3. By (7.25) we obtain ||ϕn (ζ )|| ≤

n−1 Σ

eλ(l+1−n) Dl+1 cl,1 ≤

.

l=−∞

aeλ/2 , 1 − e−λ

which gives the bound for .i = 0. On the other hand, since Dn ≤ D¯ n ≤ D¯ l e(λ/2)(n−l)

.

for l < n,

by (7.17) with .c = λ/2, we obtain ||ϕn (ζ¯ ) − ϕn (ζ )|| ≤

n−1 Σ

eλ(l+1−n) Dl+1 cl,1 2Dn ||ζ¯ − ζ ||ec(l−n)

l=−∞

≤

.

n−1 Σ

2eλ(l+1−n) Dl+1 cl,1 D¯ l ||ζ¯ − ζ ||

l=−∞

≤ 2aeλ/2

n−1 Σ

eλ(l+1−n) ||ζ¯ − ζ || =

l=−∞

2aeλ/2 ||ζ¯ − ζ ||, 1 − e−λ (7.26)

using (7.2). This gives the bound for .i = 1, also with .supn∈Z kn,1 < 1 for a sufficiently small. Finally, for .i > 1, it follows from Lemma 7.6 that ||dζi ϕn || = ||dζi xn || ≤ ||dζi x||Li (λi ) ≤ kn,i .

.

This completes the proof of the theorem.

⨆ ⨅

One can also construct stable invariant manifolds for the dynamics in (7.6), simply by reversing time. The proof is analogous and thus is omitted. Theorem 7.2 For the dynamics in (7.6), if the constant a in (7.2) is sufficiently small, then there are maps .ψn : En → Fn of class .C p , for .n ∈ Z, such that each set } { Vn = ξ + ψn (ξ ) : ξ ∈ En

.

is a .C p manifold satisfying the properties: 1. .0 ∈ Vn , .T0 Vn = En , and .(An + fn )(Vn ) = Vn+1 . 2. .||dξi ψn || ≤ k¯n,i for .i = 0, . . . , p and .ξ ∈ En , where each .(k¯n,i )n∈Z is upper tempered and .supn∈Z k¯n,1 < 1.

7.3 Invariant Foliations

109

7.3 Invariant Foliations In this section we construct an unstable foliation for the dynamics in (7.6) in a neighborhood of the origin. We continue to make the same assumptions as in Section 7.2. Given a vector .v ∈ Rd , let .(xm (v))m≤n be the sequence such that .xn = v and xm+1 = Am xm + fm (xm )

.

for m < n.

For each .n ∈ Z and .c ∈ (0, λ/2], we consider the set { } Wn (v) = w ∈ Rd : (xm (w) − xm (v))m≤n ∈ S(c) .

.

Theorem 7.3 For the dynamics in (7.6), if the constant a in (7.2) is sufficiently small, then there are maps .ϕ¯n : Fn × Rd → En for .n ∈ Z such that: 1. For .ζ1 , ζ2 ∈ En and .v ∈ Rd , we have .ϕ¯n (0, v) = 0 and ||ϕ¯n (ζ1 , v) − ϕ¯n (ζ2 , v)|| ≤

.

2aeλ/2 ||ζ1 − ζ2 ||. 1 − e−λ

(7.27)

2. .Wn (0) = Wn and } { Wn (v) = v + ζ + ϕ¯n (ζ, v) : ζ ∈ Fn .

.

3. .Vn ∩ Wn (v) consists of a single point w and ||xm (v) − xm (w)|| ≤ 2Dn ec(m−n) ||v − w|| for m ≤ n.

.

4. .Rd =

U v∈Vn

(7.28)

Wn (v).

Proof Given .v, w ∈ Rd and proceeding in a similar manner to that in Lemma 7.1, one can verify that if the sequence .ym = xm (w) − xm (v) for .m ≤ n is in .S(c), then ym = Cm,n ζ +

n−1 Σ

┌ ┐ Cm,l+1 fl (xl (v) + yl ) − fl (xl (v))

l=m .

+

m−1 Σ

┌ ┐ Bm,l+1 fl (xl (v) + yl ) − fl (xl (v))

(7.29)

l=−∞

for .m ≤ n, where .ζ = Qn (w − v). For each .y ∈ S(c), let .G(y) = (y¯m )m≤n , where .y¯m is given by the right-hand side of (7.29). Proceeding as in the proof of Lemma 7.2, one can show that .G(S(c)) ⊂ S(c) for a sufficiently small. Moreover, G is a contraction again for a sufficiently small and so it has a unique fixed point

110

7 Stable and Unstable Foliations

y(ζ, v) ∈ S(c). Now let

.

ϕ¯n (ζ, v) := Pn yn (ζ, v) =

n−1 Σ

.

┌ ┐ Bn,l+1 fl (xl (v) + yl (ζ, v)) − fl (xl (v)) .

l=−∞

Analogous estimates to those in (7.26) yield the first property in the theorem. Moreover, by construction we have } { } { Wn (v) = v + yn (ζ, v) : ζ ∈ Fn = v + ζ + ϕ¯n (ζ, v) : ζ ∈ Fn .

.

Note that for .v = 0 we have .G(y) = F (y) with F as in (7.14). Hence, .Wn (0) = Wn . For the third property in the theorem, it suffices to show that for each .v ∈ Rd there are unique .ζ ∈ Fn and .ξ ∈ En such that v + ζ + ϕ¯n (ζ, v) = ξ + ψn (ξ ).

(7.30)

.

Note that .Qn v +ζ = ψn (ξ ) and so .ζ is specified by .ξ . Therefore, it suffices to prove the existence and uniqueness of .ξ . Setting .ζ = ψn (ξ ) − Qn v in (7.30), we obtain ( ) Pn v + ϕ¯n ψn (ξ ) − Qn v, v = ξ.

(7.31)

.

Now we consider the map .H : En → En defined by ( ) H (ξ ) = Pn v + ϕ¯ n ψn (ξ ) − Qn v, v .

.

By (7.27) and the analogous inequality in Theorem 7.2, we have .

||H (ξ¯ ) − H (ξ )|| ≤

2aeλ/2 ||ψn (ξ¯ ) − ψn (ξ )|| ≤ 1 − e−λ

⎛

2aeλ/2 1 − e−λ

⎞2

||ξ¯ − ξ ||.

Therefore, H is a contraction for a sufficiently small, and there is a unique .ξ ∈ En satisfying (7.31). Moreover, in a similar manner to that in the proof of Lemma 7.2, one can verify that ||y(ζ, v)||c ≤ 2Dn ||w − v||

.

for a sufficiently small, which yields property (7.28). This completes the proof.

⨆ ⨅

Notes Proposition 7.1 presents a general property of upper tempered sequences whose proof is reminiscent of arguments in [10] relating the existence of a negative Lyapunov exponent for a cocycle and its nonuniform partial hyperbolicity. The example considered in Proposition 7.2 is essentially taken from [15]. Its main aim is to describe a way to guarantee the properties required for the maps .fn in Section 7.1.

7.3 Invariant Foliations

111

Theorem 7.1 is based on [15] and constructs unstable invariant manifolds for any sufficiently small perturbations .fn of a sequence of invertible matrices with a tempered exponential dichotomy. The argument is long and somewhat technical, which led us to rewrite it substantially to give as possible a streamlined simplified proof. The main elements are careful estimates of the dynamics and its derivatives, together with fixed point problems. To the possible extent, some of the arguments are inspired by related work of Li and Lu in [57] for random dynamical systems. The stable manifold theorem in the context of nonuniform hyperbolicity was established by Pesin in [77] with an elaborate proof based on the approach of Perron. Katok and Strelcyn [52] extended the result to maps with singularities. Ruelle [86] gave a proof based on perturbations of products of matrices related to the multiplicative ergodic theorem in [70]. Other proofs are due to Pugh and Shub [85] and Fathi, Herman and Yoccoz [34]. For versions of the stable manifold theorem in infinite-dimensional spaces, see the works of Ruelle [87] for maps in Hilbert spaces, and Mañé [62] for maps in Banach spaces under some compactness assumptions. Theorem 7.2 is a version of Theorem 7.1 for stable invariant manifolds. Its proof could be obtained following closely the proof of Theorem 7.1 and thus is omitted. Finally, Theorem 7.3 gives a construction of an unstable foliation for the same perturbed dynamics. Its proof is based on [15]. One could construct similarly a stable foliation. Each of them can be used to give a set of coordinates for which the stable and unstable invariant manifolds become the stable and unstable spaces. This idea is used in Chapter 8 in the construction of smooth conjugacies.

Chapter 8

Construction of Smooth Conjugacies

This chapter uses the machinery developed in the former chapters, together with a few other elements, to linearize via a smooth map any sufficiently small tempered perturbation of a tempered exponential dichotomy in the absence of resonances. More precisely, we show that if there are no resonances up to a given order, then the nonlinear dynamics can be linearized by local diffeomorphisms with any prescribed regularity.

8.1 Preparations Let .(An )n∈Z be a sequence of invertible .d × d matrices with a tempered exponential dichotomy (see Definition 1.1) and with tempered growth (see Definition 2.1). The latter is equivalent to require that there are a constant .γ ≥ 0 and an upper tempered sequence .(Kn )n∈Z in .[1, +∞) such that ||Am,n || ≤ Kn eγ |m−n|

.

for m, n ∈ Z.

(8.1)

Let also .(Bn )n∈Z be the cohomologous sequence constructed in Theorem 3.1, which thus also has a tempered exponential dichotomy. Moreover, we consider maps d → Rd for .n ∈ Z of class .C p , for some .p ≥ 2, with .f (0) = 0 and .fn : R n .d0 fn = 0 for .n ∈ Z such that .

lim sup n→±∞

1 log||∂0r fn || ≤ 0 for 2 ≤ |r| ≤ p. n

(8.2)

Again as in Section 7.1, we assume that there are upper tempered sequences (cn,i )n∈Z for .0 ≤ i ≤ p, and a lower tempered sequence .(rn )n∈Z in .(0, 1] satisfying condition (7.2) for some .a > 0 sufficiently small as in Theorems 7.1 and 7.2 such that for each .n ∈ Z:

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 L. Barreira, C. Valls, Spectra and Normal Forms, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-51897-3_8

113

114

8 Construction of Smooth Conjugacies

1. .fn (x) = 0 whenever .||x|| ≥ 2rn /3. 2. .||dxi fn || ≤ cn,i for .x ∈ Rd and .0 ≤ i ≤ p. We consider the dynamics xn+1 = Bn xn + fn (xn )

.

for n ∈ Z.

(8.3)

Now we use the normal form given by Theorem 3.2. Namely, assume that .(An )n∈Z has no resonances up to order p. By Proposition 3.1, two cohomologous sequences have the same tempered spectrum and so .(Bn )n∈Z also has no resonances up to order p. Applying Theorem 3.2 we find that there are polynomial coordinate changes that take the dynamics in (8.3) to yn+1 = Bn yn + gn (yn )

.

for n ∈ Z,

(8.4)

where .gn : Rd → Rd are maps of class .C p with d0i gn = 0

.

for n ∈ Z and i = 0, . . . , p.

In view of property (3.17), we also have .

lim sup n→±∞

1 log||∂0r gn || ≤ 0 for 2 ≤ |r| ≤ p. n

Now we make a further coordinate change so that the stable and unstable manifolds .Vn and .Wn coincide, respectively, with the stable and unstable spaces .En and .Fn . Consider the coordinates .(ξ, ζ ) ∈ En × Fn for .n ∈ Z (on purpose we avoid indicating explicitly their dependence on n). We introduce new coordinates d .(y, z) on .R , given by y = ξ − ϕn (ζ ),

.

z = ζ − ψn (ξ )

(8.5)

with the maps .ϕn and .ψn as in Theorems 7.1 and 7.2 (again without indicating explicitly their dependence on n). To verify that this is indeed a coordinate change, we first note that since the perturbations .fn vanish outside the balls .B(0, rn ), the maps .ϕn and .ψn also vanish outside some balls where thus .(y, z) = (ξ, ζ ). This guarantees that the map defined by (8.5) is onto. Moreover, one can use a fixed point problem to show that it is also one to one. Namely, substituting .ξ = y + ϕn (ζ ) in the second equation in (8.5) gives ζ = z + ψn (y + ϕn (ζ )),

.

and in view of Theorems 7.1 and 7.2, the map ζ |→ z + ψn (y + ϕn (ζ ))

.

8.1 Preparations

115

is a contraction (because .ψn and .ϕn have Lipschitz constants less than 1). It also follows from these theorems that the coordinate change is of class .C p . In the new coordinates in (8.5), the dynamics in (8.4) becomes x¯n+1 = Bn x¯n + f¯n (x¯n )

.

for n ∈ Z,

for some maps .f¯n : Rd → Rd of class .C p such that: 1. d0i f¯n = 0

.

for n ∈ Z and i = 0, . . . , p.

(8.6)

2. Pn f¯n (0, z) = 0

.

and

Qn f¯n (y, 0) = 0

for n ∈ Z and (y, z) ∈ Rd .

(8.7)

Here, .Pn and .Qn are the projections associated to the splitting .Rd = En ⊕ Fn , respectively, onto the stable and unstable spaces. Property (8.7) follows readily from the fact that now .y = 0 is the unstable manifold .Wn and .z = 0 is the stable manifold .Vn . Moreover, the maps .f¯n also have controlled derivatives, in the following sense. Proposition 8.1 If the sequence .(An )n∈Z has tempered growth and the maps .fn satisfy property (8.2), then the sequence bn,i =

.

sup ||dxi f¯n ||

(8.8)

x∈B(0,rn )

is upper tempered for .0 ≤ i ≤ p. Proof Define a function .qn : Rd → Rd by qn (x) = (ξ − ϕn (ζ ), ζ − ψn (ξ )) for x = (ξ, ζ ) ∈ Rd .

.

Then

.

x¯n+1 = qn+1 (xn+1 ) = qn+1 (Bn xn + gn (xn )) ( ) = qn+1 Bn qn−1 (x¯n ) + gn (qn−1 (x¯n )) = Bn x¯n + f¯n (x¯n ),

where f¯n = (qn+1 − Id) ◦ (Bn + gn ) ◦ qn−1 + gn ◦ qn−1 + Bn ◦ (qn−1 − Id).

.

(8.9)

By hypothesis, the sequence .(An )n∈Z and so also the sequence .(Bn )n∈Z have tempered growth, and the derivatives of the maps .fn satisfy property (8.2). Moreover, in view of Theorems 7.1 and 7.2, the maps .ϕn and .ψn have derivatives, up to order p,

116

8 Construction of Smooth Conjugacies

bounded by upper tempered sequences, and so the same happens to the derivatives of the maps .gn . Together with (8.9), this readily implies that the sequence .(bn,0 )n∈Z is upper tempered. Taking derivatives in (8.9), we obtain d f¯n = (dqn+1 − Id)(Bn + dgn )d(qn−1 ) + dgn d(qn−1 ) + Bn (d(qn−1 ) − Id).

.

(8.10)

We have bounds by upper tempered sequences for all maps in (8.10) except possibly for the derivatives .d(qn−1 ). By (8.5), letting ξ = y + un (y, z)

and

ζ = z + vn (y, z),

un (y, z) = ϕn (z + vn (y, z))

and

vn (y, z) = ψn (y + un (y, z)).

.

we obtain .

Hence, ψn (y + ϕn (z + vn (y, z))) = vn (y, z),

.

which gives ∂y vn = dψn (Id + dϕn ∂y vn ) = dψn + dψn dϕn ∂y vn .

.

2 < 1 and so On the other hand, by Theorems 7.1 and 7.2, we have .||dψn dϕn || ≤ kn,1

∂y vn = (Id − dψn dϕn )−1 dψn

.

and ||(Id − dψn dϕn )−1 || ≤

.

1 1 ≤ . 2 1 − ||dψn dϕn || 1 − kn,1

Similar computations give ||d(qn−1 )|| = ||d(un , vn )|| ≤

.

2 kn,1 + kn,1 2 1 − kn,1

≤

kn,1 . 1 − supn∈Z kn,1

This ensures that the sequence .(bn,1 )n∈Z is upper tempered. By (8.9), all higher-order derivatives of .f¯n are linear combinations of finite products involving the terms d m (dqn+1 − Id), d m (Bn + dgn ), d m (qn−1 ), d m gn , Bn or d m (d(qn−1 ) − Id),

.

8.2 Bounds for the Perturbations

117

for some .m ≥ 0. All of them are bounded by upper tempered sequences, which implies that .(bn,i )n∈Z is an upper tempered sequence for each .i > 1. ⨆ ⨅

8.2 Bounds for the Perturbations Now we describe a useful decomposition of the maps .f¯n , and we establish some upper bounds for these maps and for their derivatives. We denote by .[x] the least integer .m ≥ x. Proposition 8.2 For each .n ∈ Z there are functions .f¯1n and .f¯2n of class .C [p/2] such that for .j = 1, 2 and .i = 0, . . . , [p/2] we have f¯n = f¯1n + f¯2n ,

.

Pn f¯j n (0, z) = Qn f¯j n (y, 0) = 0

.

d0i f¯j n = 0, for (y, z) ∈ Rd ,

(8.11) (8.12)

and ||dxi f¯1n || ≤ an,i ||y||[p/2]−i ,

.

||dxi f¯2n || ≤ an,i ||z||[p/2]−i

(8.13)

for all .x = (y, z) ∈ B(0, rn ) and some upper tempered sequences .(an,i )n∈Z . Proof Let Σ

f¯n (x) =

.

r∈Nk0 ,0≤|r|≤[p/2]

1 r,0 ¯ r ∂ fn y + R[p/2] (x) r! (0,z)

be the Taylor expansion of .f¯n up to order .[p/2] with respect to y, where r,0 ¯ r ∂(0,z) fn y = ∂yr f¯n |x=(0,z) y r .

.

This last function is of class .C p−[p/2] ⊂ C [p/2] (since .f¯n is of class .C p ). Now take k .l, m ∈ N with .|l| + |m| = i ≤ [p/2]. Note that 0 ∂yl ∂zm (f¯n − R[p/2] ) =

Σ r∈Nk0 ,0≤|r|≤[p/2]

.

=

Σ r∈Nk0 ,0≤|r|≤[p/2]

1 m r,0 ¯ l r fn )∂y y ∂ (∂ r! z (0,z) 1 ∂ m ∂ r,0 f¯n y r−l . (r − l)! z (0,z)

118

8 Construction of Smooth Conjugacies

Therefore, .∂yl ∂zm R[p/2] is the error term of the Taylor expansion of the function l m .∂y ∂z f¯n of order .[p/2] − |l| − |m|. Hence, it follows from (8.8) that ||dxi R[p/2] || ≤ an,i ||y||[p/2]−i

.

for x ∈ B(0, rn )

(8.14)

and some upper tempered sequence .(an,i )n∈Z . On the other hand, r,0 ¯ r ||∂yl ∂zm (∂(0,z) fn y )|| =

.

r! r! ||∂ m ∂ r,0 f¯n || ||∂ m (∂ r,0 f¯n )y r−l || ≤ (r − l)! z (0,z) (r − l)! z (0,z)

for .(y, z) ∈ B(0, rn ), taking .rn < 1. By the mean value theorem, we have r,0 ¯ r,0 ¯ r,0 ¯ r,1 ||∂(0,z) fn || = ||∂(0,z) fn − ∂(0,0) fn || ≤ ||∂(0,w) f¯n || · ||z|| ≤ bn,|r|+1 ||z||

.

for some point .(0, w) ∈ B(0, rn ). In fact, one can proceed inductively to obtain r,0 ¯ ||∂(0,z) fn || ≤ bn,|r|+[p/2] ||z||[p/2]

.

(because .d0i f¯n = 0 for .0 ≤ i ≤ p). Similarly, we have r,0 ¯ r,m ¯ ||∂zm ∂(0,z) fn || = ||∂(0,z) fn || ≤ bn,|r|+|m|+[p/2] ||z||[p/2]−|m|

.

and so also r,0 ¯ r ||∂yl ∂zm (∂(0,z) fn y )|| ≤

.

r! bn,|r|+|m|+[p/2] ||z||[p/2]−|m| . (r − l)!

This readily implies that || ⎛ ⎞|| || || i 1 r,0 r || || ¯ ∂ . dx fn y || ≤ dn,i ||z||[p/2]−i || r! (0,z)

for x ∈ B(0, rn )

(8.15)

and some upper tempered sequence .(dn,i )n∈Z . Finally, by (8.14) and (8.15), letting f¯1n = R[p/2]

.

and

f¯2n = f¯n − f¯1n

gives property (8.13), which also yields the second part of (8.11). Property (8.12) follows now readily from (8.6) and (8.7). ⨆ ⨅ Before proceeding, we make an appropriate cutoff of the perturbations .f¯n since a priori the constructions made to give the perturbations .gn and .f¯n may not assure that each .f¯n vanishes outside a small neighborhood of the origin. Let .ρ : R → [0, 1] be a .C ∞ function satisfying (7.5). We define .g¯ n , g¯ j n : Rd → Rd by

8.3 Smooth Linearization

119

g¯ n (x) = ρn (x)f¯n (x),

.

g¯ j n (x) = ρn (x)f¯j n (x),

where ρn (x) = ρ(||x||/sn )

.

with sn = rn /3.

Note that .g¯ n is of class .C p and that .g¯ 1n and .g¯ 2n are of class .C [p/2] , in view of Proposition 8.2. Proceeding as in the proof of Proposition 7.2 yields the following result. Proposition 8.3 For each .n ∈ Z, .j = 1, 2, and .0 ≤ i ≤ [p/2], we have: 1. .g¯ j n (x) = f¯j n (x) for .x ∈ B(0, sn ). 2. .||g¯ j n (x)|| ≤ 29 bn,2 rn2 and .||dx g¯ j n || ≤ 2bn,2 rn for .x ∈ Rd . 3. .||dxi g¯ j n || ≤ c¯n,i for .x ∈ Rd and some upper tempered sequence .(c¯n,i )n∈Z . 4. ||dxi g¯ 1n || ≤ a¯ n,i ||y||[p/2]−i ρn (x),

.

||dxi g¯ 2n || ≤ a¯ n,i ||z||[p/2]−i ρn (x)

(8.16)

for .x = (y, z) ∈ Rd and some upper tempered sequence .(a¯ n,i )n∈Z .

8.3 Smooth Linearization We finally construct conjugacies given by .C l local diffeomorphisms between the dynamics defined by the maps .Bn and .Bn + f¯n in a neighborhood of the origin. Theorem 8.1 Let .(An )n∈Z be a sequence of invertible .d × d matrices with a tempered exponential dichotomy and with tempered growth, and let .(Bn )n∈Z be the cohomologous sequence constructed in Theorem 3.1. Moreover, let .fn : Rd → Rd for .n ∈ Z be maps of class .C p , for some .p ≥ 2, with .fn (0) = 0 and .d0 fn = 0 for .n ∈ Z satisfying (8.2) and 2Kn+1 bn,2 rn ≤ 1

.

for n ∈ Z.

(8.17)

Given .l ∈ N, there is .p = p(l) ∈ N such that if there are no resonances up to order p, then there are .C l local diffeomorphisms .hn : B(0, r¯n ) → Rd with .hn (0) = 0 and .d0 hn = Id, for some numbers .r¯n ∈ (0, rn ), such that hn+1 ◦ Bn = (Bn + f¯n ) ◦ hn

.

on B(0, r¯n ) for n ∈ Z.

Proof Step 1. Some upper bounds For each .n ∈ Z and .τ ∈ [0, 1], we define new maps .Fnτ , Gτn : Rd → Rd by Fnτ = Bn + τ g¯ 1n

.

and

Gτn = Bn + g¯ 1n + τ g¯ 2n .

(8.18)

120

8 Construction of Smooth Conjugacies

Moreover, let

=

τ .Fm,n

⎧ τ τ ⎪ ⎪ ⎨Fm−1 ◦ · · · ◦ Fn

if m > n,

Id ⎪ ⎪ ⎩(F τ )−1 ◦ · · · ◦ (F τ )−1 m n−1

if m = n, if m < n

and ⎧ ⎪ ⎪Gτm−1 ◦ · · · ◦ Gτn ⎨

=

τ .Gm,n

if m > n,

Id ⎪ ⎪ ⎩(Gτ )−1 ◦ · · · ◦ (Gτ )−1 m n−1

if m = n, if m < n.

Recall that the sequence .(Bn )n∈Z also has tempered growth, that is, there are a constant .γ ≥ 0 and an upper tempered sequence .(Kn )n∈Z in .[1, +∞) such that ||Bm,n || ≤ Kn eγ |m−n|

.

for m, n ∈ Z.

Lemma 8.1 For each .i = 1, . . . , [p/2], there are a constant .μi > 0 and an upper tempered sequence .(κn,i )n∈Z such that .

|m|

r τ ±1 max sup ||∂(x,τ ) ((Fn+m,n ) )|| ≤ κn,i μi

1≤|r|≤i x∈Rd

(8.19)

and .

|m|

r τ ±1 max sup ||∂(x,τ ) ((Gn+m,n ) )|| ≤ κn,i μi

1≤|r|≤i x∈Rd

(8.20)

for .m, n ∈ Z and .τ ∈ [0, 1]. τ Proof of the lemma We prove (8.19) for .m ≥ 0 taking the plus sign in .(Fn+m,n )±1 . The remaining cases, as well as all cases for property (8.20), can be treated similarly. Let τ xn+m = Fm+n,n (x)

.

for m ≥ 0.

Then xn+m = An+m,n x + τ

n+m−1 Σ

.

Bn+m,l+1 g¯ 1l (xl ),

l=n

and since .(An )n∈Z has tempered growth (see (8.1)), we obtain

(8.21)

8.3 Smooth Linearization

121

||∂x xn+m || ≤ eγ m Kn +

n+m−1 Σ

.

eγ (n+m−l−1) Kl+1 ||dxl g¯ 1l || · ||∂x xl ||.

l=n

It follows from (8.17) and Proposition 8.3 that e

.

−γ m

||∂x xn+m || ≤ Kn + e

−γ

n+m−1 Σ

e−γ (l−n) ||∂x xl ||.

l=n

Writing .al = e−γ (l−n) ||∂x xl ||, this is the same as an+m ≤ Kn + e−γ

n+m−1 Σ

.

al .

l=n

It follows from an appropriate discrete-time version of Gronwall’s lemma (see, for example, [1]) that an+m ≤ Kn (1 + e−γ )m

.

⇔

||∂x xn+m || ≤ Kn (1 + eγ )m .

(8.22)

Analogously, we have ∂τ xn+m =

n+m−1 Σ

.

Bn+m,l+1 g¯ 1 + τ

l=n

n+m−1 Σ

Bn+m,l+1 dxl g¯ 1l ∂τ xl ,

l=n

and it follows again from (8.17) and Proposition 8.3 that e−γ m ||∂τ xn+m || ≤

n+m−1 Σ

eγ (n−l−1) + e−γ

l=n .

≤

n+m−1 Σ

e−γ (l−n) ||∂τ xl ||

l=n

n+m−1 Σ 1 −γ + e e−γ (l−n) ||∂τ xl ||. γ e −1 l=n

Therefore, using once more the discrete-time version of Gronwall’s lemma, we obtain ||∂τ xn+m || ≤

.

eγ

1 (1 + eγ )m . −1

Together with (8.22), this establishes (8.19) for .i = 1 and .m ≥ 0 taking the plus sign. We proceed by induction on i. Assume that (8.19) holds up to order i for .m ≥ 0 taking the plus sign. Write .d = (∂x , ∂τ ). By (8.21) we have

122

8 Construction of Smooth Conjugacies

.

d i+1 xn+m = τ

n+m−1 Σ

Bn+m,l+1 dxl g¯ 1l d i+1 xl + Li ,

(8.23)

l=n

where .Li is a linear combination of finite products involving derivatives .d j xl of order .j ≤ i. Using (8.16) and the induction hypothesis, one can show that .||Li || ≤ κ¯ n,i νim for some constant .νi > 0 and some upper tempered sequence .(κ¯ n,i )n∈Z . On the other hand, by (8.17) and Proposition 8.3, we have ||dxl g¯ 1l || ≤ 2bn,2 rn ≤ 1 since Kn+1 ≥ 1.

.

It follows from (8.23) that e−γ m ||d i+1 xn+m || ≤ e−γ

n+m−1 Σ

.

e−γ (l−n) ||d i+1 xl || + κ¯ n,i (e−γ νi )m .

l=n

Again by the discrete-time version of Gronwall’s lemma, we obtain e−γ m ||d i+1 xn+m || ≤ κ¯ n,i (e−γ νi )m (1 + e−γ )m

.

and so ||d i+1 xn+m || ≤ κ¯ n,i μm i+1

.

with μi+1 = νi (1 + e−γ ).

Therefore, (8.19) holds with i replaced by .i + 1 for .m ≥ 0 taking the plus sign.

⨆ ⨅

Step 2. Conjugacies via the homotopy method We continue to consider the maps .Fnτ in (8.18), and we describe a criterion for the existence of local conjugacies between the maps .Fn0 = Bn and .Fn1 = Bn + g¯ 1n . Lemma 8.2 Assume that there are .C l maps .Xn : Rd × [0, 1] → Rd and constants .cn > 0 for .n ∈ Z such that ||Xn (x, τ )|| ≤ cn ||x||2

for ||x|| < 1, τ ∈ [0, 1]

(8.24)

dx Fnτ Xn (x, τ ) − Xn+1 (Fnτ (x), τ ) = −g¯ 1n (x).

(8.25)

.

and .

Then there are .C l local diffeomorphisms .hn : B(0, sn ) → Rd , for .sn > 0 sufficiently small, such that hn+1 ◦ Bn = (Bn + g¯ 1n ) ◦ hn

.

on B(0, sn ) for n ∈ Z.

(8.26)

8.3 Smooth Linearization

123

Proof of the lemma For each .n ∈ Z, define functions .Un , Vn : Rd × [0, 1] → Rd × [0, 1] by Un (x, τ ) = (Fnτ (x), τ )

Vn (x, τ ) = (Xn (x, τ ), 1).

and

.

Letting .d = (∂x , ∂τ ), it follows from (8.25) that dUn Vn = Vn+1 ◦ Un .

.

(8.27)

Now let .φnt be the flow of class .C l generated by the equation .(x, τ )' = Vn (x, τ ). It follows from (8.27) that t Un ◦ φnt = φn+1 ◦ Un .

.

Observe also that ⎧

t

x(t) = x(0) +

.

( ) Xn x(s), τ (0) + s ds,

0

which gives ⎧

t

||x(t)|| ≤ ||x(0)|| +

.

cn ||x(s)||2 ds.

0

Now we consider the equation ⎧

t

y(t) = ||x(0)|| +

.

cn y(s)2 ds.

0

It follows from usual comparison criteria for the solutions of ordinary differential equations that if both .x(t) and .y(t) are defined in an interval .[0, t0 ), then ||x(t)|| ≤ y(t)

.

for t ∈ [0, t0 ).

On the other hand, given a sufficiently small .sn > 0, for .||x(0)|| < sn , the solution y(t) is defined at least for .t ∈ [0, 1]. This readily implies that .x(t) is defined for .t ∈ [0, 1] whenever .x(0) ∈ B(0, sn ). Finally, writing .

φn1 (x, 0) = (hn (x), 1),

.

we obtain a .C l map .hn : B(0, sn ) → Rd . Then (Un ◦ φn1 )(x, 0) = Un (hn (x), 1) = (Fn1 (hn (x)), 1) = ((Bn + g¯ 1n )(hn (x)), 1)

.

124

8 Construction of Smooth Conjugacies

and 1 1 1 (φn+1 ◦ Un )(x, 0) = φn+1 (Fn0 (x), 0) = φn+1 (Bn (x), 0) = (hn+1 (Bn (x)), 1).

.

This establishes property (8.26). Moreover, one can verify that writing φn−1 (x, 1) = (h¯ n (x), 0),

.

l we have .h¯ n = h−1 n and so .hn is a .C local diffeomorphism.

⨆ ⨅

Step 3. Conjugacies between .Bn and .Bn + g¯ n In this step we verify the conditions of Lemma 8.2 for certain maps, which thus leads to the construction of topological conjugacies (see (8.26)). Let In,τ = τ g¯ 1n ◦ (Fnτ )−1

.

and

Jn,τ = (g¯ 1n + τ g¯ 2n ) ◦ (Gτn )−1 .

Given maps .f, g : Rd → Rd such that f is a diffeomorphism, we shall write (f |g)(x) = df −1 (x) f g(f −1 (x)).

.

Lemma 8.3 Given .l ∈ N, there is .p ∈ N such that Xn (x, τ ) = −

+∞ Σ

τ (Fn,n+m |Im+n−1,τ )(x)

(8.28)

(Gτn,n−m |J−m+n−1,τ )(x)

(8.29)

.

m=1

and Yn (x, τ ) =

+∞ Σ

.

m=0

are of class .C l and satisfy (8.24) and (8.25) for some upper tempered sequence .(cn )n∈Z . Proof of the lemma Proceeding formally, we obtain (Fnτ |Xn )(x, τ ) = −

+∞ Σ ( τ ) τ Fn ◦ Fn,n+m |Im+n−1,τ (x) m=1

= −In,τ (x) − .

+∞ Σ

τ (Fn+1,n+m |Im+n−1,τ )(x)

m=2

= −In,τ (x) −

+∞ Σ

τ (Fn+1,n+m+1 |Im+n,τ )(x)

m=1

= −In,τ (x) + Xn+1 (x, τ )

8.3 Smooth Linearization

125

and, similarly, (Gτn |Yn )(x, τ ) =

.

+∞ Σ ( τ ) Gn ◦ Gτn,n−m |J−m+n−1,τ (x) m=0

= −Jn,τ (x) +

+∞ Σ

(Gτn+1,n−m |J−m+n−1,τ )(x)

m=−1

= −Jn,τ (x) +

+∞ Σ

(Gτn+1,n+1−m |J−m+n,τ )(x)

m=0

= −Jn,τ (x) + Yn+1 (x, τ ). Therefore, once we show that the series in (8.28) and (8.29) are .C l functions, they shall become solutions of (8.25). We show that .Xn is well defined and is of class .C l (the argument for .Yn is analogous). Observe that by (8.13) with .i = 0 and Lemma 8.1 we have +∞ Σ

X:=

τ ||(Fn,n+m |In+m−1,τ )(x)||

m=1

.

=τ

+∞ Σ

τ τ τ ||dFn+m,n (x) Fn,n+m g¯ 1n (Fn+m−1,n (x))||

(8.30)

m=1 +∞ Σ

≤

[p/2] κn,1 μm , 1 an,0 ||yn+m−1 ||

m=1 τ (x) in coordinates .(y, z). Note that where .y1 denotes the first component of .Fl,n in view of (8.12) the stable and unstable manifolds for the dynamics defined by .Bn + τ g ¯ 1n are also the stable and unstable spaces, and so the coordinates .(y, z) associated to these maps coincide for each .τ with those associated to .Bn + g¯ n . By the stable version of Theorem 7.3, for .v = yn , there is a point w in the unstable manifold (which now is simply the unstable space) such that τ τ ||Fm,n (v) − Fm,n (w)|| ≤ 2Dn e−(λ/2)(m−n) ||v − w||

.

(8.31)

for .m ≥ n. The points v and w are in the same stable leaf, which by Theorem 7.2 is a graph of a Lipschitz function with Lipschitz constant .kn,1 such that .supn∈Z kn,1 < 1. τ (v) in Therefore, there is a constant .C > 0 such that the first component .ym of .Fm,n coordinates .(y, z) satisfies τ τ ||ym || ≤ C||Fm,n (v) − Fm,n (w)|| for m ≥ n

.

126

8 Construction of Smooth Conjugacies

τ (w) vanishes). By (8.31) we obtain (note that the first component of .Fm,n

||yn+m−1 || ≤ 2CDn e−(λ/2)(m−1) ||y||

.

for m > 0.

Now take p sufficiently large so that .μ1 e−(λ/2)[p/2] < 1. By (8.30) we have X≤

+∞ Σ

( )[p/2] −(λ/2)(m−1) κn,1 μm ||y|| 1 an,0 2CDn e

m=1

.

≤

+∞ Σ

μ1 κn,1 an,0 (2CDn )[p/2] (μ1 e−(λ/2)[p/2] )m−1 ||x||[p/2]

m=1

≤ μ1 κn,1 an,0 (2CDn )[p/2]

1 1 − μ1

e−(λ/2)[p/2]

||x||[p/2]

≤ d¯n,1 ||x||[p/2] < +∞, for some upper tempered sequence .(d¯n,1 )n∈Z . This shows that the function .Xn (x, τ ) is well defined. Similarly, using Proposition 8.3 and Lemma 8.1, one can show that given .l ∈ N, there is .p ∈ N such that for each .j ≤ l we have

.

+∞ Σ j τ ||dx (Fn,n+m |In+m−1,τ )|| ≤ d¯n,j ||x||[p/2]−j

(8.32)

n=1

for some upper tempered sequence .(d¯n,j )n∈Z . Indeed, taking formally the derivative j .d (x,τ ) Xn , we obtain a series of linear combinations of terms involving derivatives of τ τ τ the maps .Fn,n+m , .Fn+m,n , .Fn+m−1,n , and .g¯ 1n , that is, τ

j +∞ Σ Σ

.

Am,i (x, τ )dFi τ

m=1 i=0

n+m−1,n (x)

g¯ 1n Bm,i (x, τ ),

where the expressions .Am,i (x, τ ) and .Bm,i (x, τ ) involve at most .j + 1 derivatives of .F τ up to order .j + 1 but not .g¯ 1n . By Lemma 8.1 there are a constant .μ¯ j +1 > 0 and upper tempered sequences .(dn,i )n∈Z such that ||Bm,i (x, τ )|| ≤ dn,i μ¯ m j +1 ,

.

Therefore, by (8.16), we have

||Am,i (x, τ )|| ≤ dn,i μ¯ m j +1 .

8.3 Smooth Linearization

127

j +∞ Σ Σ ||Am,i (x, τ )dFi τ

n+m−1,n (x)

m=1 i=0 .

≤

j +∞ Σ Σ

g¯ 1n Bm,i (x, τ )|| (8.33)

( )[p/2]−i 2 dn,i μ¯ 2m ¯ n,i 2CDn e−(λ/2)(m−1) ||y|| . j +1 a

m=1 i=0

Taking p sufficiently large so that μ¯ 2j +1 e−(λ/2)([p/2]−l) < 1

.

for all j ≤ l,

each series on the right-hand side of (8.33) converges and property (8.32) holds for j ≤ l. In particular, the function .Xn is of class .C l . ⨆ ⨅

.

We proceed with the proof of Theorem 8.1. By Lemma 8.3, given .l ∈ N, there is .p = p(l) ∈ N for which there are functions .Xn (x, τ ) and .Yn (x, τ ) of class l .C satisfying (8.24) and (8.25). Hence, it follows from Lemma 8.2 that there are conjugacies given by .C l local diffeomorphisms between .Bn and .Bn + g¯ 1n , and between .Bn + g¯ 1n and Bn + g¯ 1n + g¯ 2n = Bn + g¯ n = Bn + f¯n

.

on balls .B(0, sn ) of sufficiently small radius .sn for each .n ∈ Z. This completes the proof of the theorem. ⨆ ⨅ Notes The construction of smooth conjugacies between a linear dynamics with a tempered exponential dichotomy and its sufficiently small tempered perturbations is done in this chapter taking advantage of the material discussed in the former chapters. We first consider a sequence of invertible matrices .(An )n∈Z and the cohomologous sequence .(Bn )n∈Z constructed in Theorem 3.1 (without loss of generality, one could in fact consider only the latter since one is brought to the other by a tempered sequence of matrices). Then we consider a coordinate change such that the stable and unstable invariant manifolds of the origin become the stable and unstable spaces. Proposition 8.1 considers the nonlinear perturbations .f¯n that are obtained after these coordinate changes and shows that their derivatives are again bounded by upper tempered sequences. Proposition 8.2 shows that these perturbations can be decomposed into two components that vanish either along the stable space or the unstable space, also with upper tempered bounds (see (8.12) and (8.13)). These propositions and their proofs are based on [15], although we added substantial details particularly to the proof of Proposition 8.2. Given .l ∈ N, Theorem 8.1 constructs .C l local conjugacies between the perturbed dynamics and its linear part provided that there are no resonances up to some order .p = p(l). The main element of the proof is the use of the homotopy method to

128

8 Construction of Smooth Conjugacies

give a criterion for the existence of conjugacies. This criterion is then verified for certain maps obtained from the perturbations .f¯n and their decompositions given by Proposition 8.2. Most of the argument is based on [15] that to the possible extent is inspired by earlier work of Li and Lu in [57] for random dynamical systems. The problem of bringing a dynamics to one in a simpler form goes back to Poincaré in [83] and continues being central in dynamics. He proved in particular that if the eigenvalues of an analytic diffeomorphism have all absolute values less than 1 or have all absolute values bigger than 1, and there are no resonances, then the diffeomorphism is analytically conjugate to the linear part. Siegel [95] proved the same result without restrictions on the eigenvalues under a small divisor condition. For further developments, see the works of Brjuno [19], Arnold [4], Moser [66, 67], and Zehnder [108]. For related results for diffeomorphisms of the circle, see the works of Arnold [3], Herman [47], Yoccoz [107], Katznelson and Ornstein [53], and Sinai and Khanin [101]. See also the related works of Li and Lu [56, 58] for the study of random dynamical systems. On the other hand, the work of Sternberg [102, 103] showed that for the perturbations of an autonomous linear dynamics the existence of resonances prevents the existence of conjugacies with a prescribed smoothness. The same happens in the results of this chapter once the notion of resonance is replaced by an appropriate notion in the nonautonomous setting. We avoided discussing the construction of topological conjugacies as in the classical Grobman–Hartman theorem, due to Grobman [36, 37] and Hartman [43, 44]. Using work of Moser in [68] on the structural stability of Anosov diffeomorphisms, the theorem was extended to Banach spaces by Palis [71] and Pugh [84]. We emphasize that in general these conjugacies are at most locally Hölder continuous.

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