Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs 3031120302, 9783031120305

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Frontiers in Mathematics

Jihoon Lee Carlos Morales

Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs

Frontiers in Mathematics Editorial Board Members William Y. C. Chen, Nankai University, Tianjin, China Laurent Saloff-Coste, Cornell University, Ithaca, NY, USA Igor Shparlinski, The University of New South Wales, Sydney, NSW, Australia Wolfgang Sprößig, TU Bergakademie Freiberg, Freiberg, Germany

Jihoon Lee • Carlos Morales

Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs

Jihoon Lee Department of Mathematics Chonnam National University Gwangju, Korea (Republic of)

Carlos Morales Mathematics Institute Federal University of Rio de Janeiro Rio de Janeiro, Brazil

ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-3-031-12030-5 ISBN 978-3-031-12031-2 (eBook) https://doi.org/10.1007/978-3-031-12031-2 Mathematics Subject Classification: 37B05, 35K57, 53C23 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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 book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

A common definition of a dynamical system is any phenomenon of nature evolving in “time”. One of the most important classes of dynamical systems is described by differential equations or difference equations. We can classify dynamical systems according to discrete or continuous time. Specific questions in the qualitative theory of dynamical systems are the long-term behavior of solutions and the stability problem. A main goal is to study the preservation of geometric structure of solutions under perturbations. For this problem, in early 1960s, Smale used the theory of differential topology to understand the structure of solutions on compact smooth manifolds. One of the most successful notions is hyperbolicity, which has been well developed due to Anosov, Smale, Palis, Mañé, etc. In the light of the rich consequence of differentiable dynamical systems, there are many works that have studied the dynamics properties from a topological viewpoint. For example, instead of hyperbolic systems, it was also shown that any expansive system with shadowing property is topologically stable and admits the spectral decomposition. These results are known as Walters’ stability theorem and spectral decomposition theorem, respectively. One of the important concepts in geometry is the Gromov-Haudorff distance. Roughly, the Gromov-Haudorff distance measures how far two compact metric spaces are from being isometric. Motivated from this, Arbieto and Morales [5] introduced the GromovHausdorff distance between two maps on two compact metric spaces and applied it to study the stability of dynamical systems under perturbations of both maps and phase spaces. We observe that the Gromov-Hausdorff distance is a strong tool to study the stability of dynamical systems. In the first part of the book, we introduce the notion of Gromov-Hausdorff distance DGH between two dynamical systems and study the stability of dynamical systems under Gromov-Hausdorff perturbations. Note that distance DGH induces a topology on the collection DS of all dynamical systems up to isometries. In the second part of the book, we study the stability of dynamical systems induced by dissipative partial differential equations under perturbations of the domain and equation. One of the difficulties in this direction is that the phase space of the induced dynamical system can be changed as we perturb the domain. To overcome this difficulty, we use the Gromov-Hausdorff distance between two dynamical systems. v

vi

Preface

Part I consists of four chapters to study the abstract Gromov-Hausdorff perturbation theory of dynamical systems. More precisely, we introduce some background concerning the Gromov-Hausdorff space and the Gromov-Hausdorff distance between two dynamical systems in Chap. 1. In Chap. 2, we study a notion of stability, called the topological Gromov-Hausdorff stability, for maps and group actions on compact metric spaces. In Chap. 3, we introduce the notion of orbit shift Gromov-Hausdorff topological stability and study the relationship with the topologically Gromov-Hausdorff stable maps or Anosov maps. In Chap. 4, we analyze the shadowing property of dynamical systems under Gromov-Hausdorff perturbations. Part II consists of three chapters in which we apply the Gromov-Hausdorff perturbation theory in Part I to study the stability of dynamical systems induced by the dissipative partial differential equations like reaction diffusion equations or Chafee-Infante equations. Gwangju, Korea (Republic of) Rio de Janeiro, Brazil October 2022

Jihoon Lee Carlos Morales

Contents

Part I Abstract Theory 1

Gromov-Hausdorff Distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Preliminary Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Gromov-Hausdorff Distance for Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Precompactness, Completeness, and a Variational Principle. . . . . . . . . . . . . . . . . 1.5 A Variational Principle for the Gromov-Hausdorff Distance . . . . . . . . . . . . . . . . 1.6 Distances Characterizing Homeomorphic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 C 0 -Gromov-Hausdorff Distance for Dynamical Systems. . . . . . . . . . . . . . . . . . . . 1.8 Gromov-Hausdorff Space of Continuous Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 6 9 12 15 17 33

2

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definitions and Statement of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Isometric Stability: Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Gromov-Hausdorff Stability for Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Gromov-Hausdorff Stability of Global Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 45 46 48 52 54 58

3

Continuity of the Shift Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminary Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Application to Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 67 69 74

vii

viii

4

Contents

Shadowing from the Gromov-Hausdorff Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Definitions, Statement of Main Results, and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 79

Part II Applications to PDEs 5

GH-Stability of Reaction-Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6

Stability of Inertial Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Proof of Theorem 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 116 137

7

Stability of Chafee-Infante Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 L-Morse-Smale and Equivalence of Global Attractors . . . . . . . . . . . . . . . . . . . . . . 7.3 Continuity of Global Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Geometric Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 143 149 152

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Part I Abstract Theory

1

Gromov-Hausdorff Distances

1.1

Introduction

Metric spaces may be considered as elements in a large space. This was the basis of J. A. Wheeler’s suggestion that the dynamic object in Einstein’s general relativity is not spacetime, but space. He then defined the superspace, namely, the set of isometry classes of Riemannian metrics. He considered this space as the arena where geometrodynamics takes place [77]. In 1975, Edwards [27] defined a distance on Wheeler’s superspace which a fortiori extends to the space of isometry classes of compact metric spaces. This distance was rediscovered by Gromov, who used it in his celebrated Polynomial Growth Theorem [31]. This distance is now called Gromov-Hausdorff distance. We start this chapter with a review of the Gromov-Hausdorff distance. It includes not only its precompactness and completeness, but also a variational principle which seems to be new. We also discuss the possible existence of metric space distances characterizing homeomorphic rather than isometric spaces. Afterwards, we extend the Gromov-Hausdorff distance to continuous maps between metric spaces. This leads to the Gromov-Hausdorff space of continuous maps. Finally, we extend this distance to dynamical systems such as flows or finitely generated group actions on metric spaces.

1.2

Preliminary Facts

When two compact metric spaces X and Y are regarded as elements of a large space, it may be of interest to consider possible relations between them. A natural one is as follows. An isometry between X and Y is a surjective map i : X → Y such that d(i(x), i(x  )) = d(x, x  ) for every x, x  ∈ X. We say that X and Y are isometric if there is an isometry © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Lee, C. Morales, Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-12031-2_1

3

4

1 Gromov-Hausdorff Distances

between X and Y . The composition of (two or more) isometries is again an isometry. Therefore, existence of an isometry is an equivalence relation in the space of all compact metric spaces. Let us introduce the notion of approximate isometry. Fix δ > 0. A δ-isometry is a (not-necessarily continuous) map i : X → Y such that sup |d(i(x), i(x  )) − d(x, x  )| ≤ δ

x,x  ∈X

and

dH (i(X), Y ) ≤ δ.

Here dH (A, C) denotes the Hausdorff distance, defined by dH (A, C) = inf{ε > 0 : A ⊂ B(C, ε) and C ⊂ B(A, ε)},  where B(A, ε) = a∈A B(a, ε) is the ε-ball centered at A. The diameter of a nonempty subset A ⊂ X is defined as diam(A) = sup d(a, a  ). a,a  ∈A

The composition of an isometry and a δ-isometry is considered in the next lemma. Lemma 1.1 If i : X → Y is a δ-isometry (resp., an isometry) and j : Y → Z is an isometry (resp., a δ-isometry), then j ◦ i : X → Z is a δ-isometry. Proof Let us consider first the case when i and j are a δ-isometry and an isometry, respectively. For x, x  ∈ X, we have |d(j ◦ i(x), j ◦ i(x  )) − d(x, x  )| = |d(i(x), i(x  )) − d(x, x  )| ≤ δ, and so sup |d(j ◦ i(x), j ◦ i(x  )) − d(x, x  )| ≤ δ.

x,x  ∈X

On the other hand, if z ∈ Z, then j (y) = z for some y ∈ Y . Moreover, there is x ∈ X such that d(i(x), y) ≤ δ. Then d(j ◦ i(x), z) = d(j (i(x)), j (y)) = d(i(x), y) ≤ δ, proving that dH (j ◦ i(X), Z) ≤ δ. We conclude that j ◦ i is a δ-isometry. Now suppose that i and j are an isometry and a δ-isometry, respectively. Then |d(j ◦ i(x), j ◦ i(x  )) − d(x, x  )| = |d(j (i(x)), j (i(x  ))) − d(i(x), i(x  ))| ≤ δ for x, x  ∈ X, and so sup |d(j ◦ i(x), j ◦ i(x  )) − d(x, x  )| ≤ δ.

x,x  ∈X

1.2 Preliminary Facts

5

If z ∈ Z, then d(j (y), z) ≤ δ for some y ∈ Y and y = i(x) for some x ∈ X. So, d(j ◦ i(x), z) = d(j (y), z) ≤ δ. Therefore, dH (j ◦ i(X), Z) ≤ δ, completing the proof.  We now consider the composition of δ-isometries. Proposition 1.1 If i : X → Y and j : Y → Z are a δ1 -isometry and a δ2 -isometry respectively, of compact metric spaces, then j ◦ i : X → Z is a 2(δ1 + δ2 )-isometry. Proof For all x, x  ∈ X, we have d(j ◦ i(x), j ◦ i(x  )) ≤ δ2 + d(i(x), i(x  )) ≤ (δ1 + δ2 ) + d(x, x  ) and d(x, x  ) ≤ δ1 + d(i(x), i(x  )) ≤ (δ1 + δ2 ) + d(j ◦ i(x), j ◦ i(x  )). Consequently, sup |d(j ◦ i(x), j ◦ i(x  )) − d(x, x  )| ≤ δ1 + δ2 .

x,x  ∈X

Also, if z ∈ Z, then there is an y ∈ Y such that d(j (y), z) ≤ δ2 ; and for this y, there is an x ∈ X such that d(i(x), y) ≤ δ1 . Then, d(j ◦ i(x), z) ≤ d(j (i(x)), j (y)) + d(j (y), z) ≤ 2δ2 + d(i(x), y) ≤ δ1 + 2δ2 ≤ 2(δ1 + δ2 ). Therefore, dH (j ◦i(X), Z) ≤ 2(δ1 +δ2 ), and we conclude that j ◦i is a 2(δ1 +δ2 )-isometry from X to Z.  The following theorem is the basis of the definition of the Gromov-Hausdorff distance. It just characterizes isometric spaces in terms of δ-isometries. Theorem 1.1 Two compact metric spaces X and Y are isometric if and only if inf{Δ > 0 : there is a Δ-isometry i : X → Y } = 0.

6

1 Gromov-Hausdorff Distances

Proof The sufficiency follows from the obvious fact that every isometry is a Δ-isometry for every Δ > 0. For the necessity, suppose that the infimum in the statement is zero. Then there is a sequence of Δn -isometries {in : X → Y }n∈N with Δn → 0 as n → ∞. Since X is compact, there exists a countable dense subset A of X. Since Y is compact, the sequence {in (a)}n∈N has a convergent subsequence for every a ∈ A. Since A is countable, by a diagonal argument, there is a subsequence {ink }k∈N such that (ink (a))k∈N is convergent for every a ∈ A. Hence, there exists a map i : A → Y such that the sequence ink : A → Y converges pointwise to i. Since d(i(a), i(a  )) = lim d(ink (a), ink (a  )) ≤ lim (Δnk + d(a, a  )) = d(a, a  ) k→∞

k→∞

for every a, a  ∈ A, we conclude that i is uniformly continuous. Hence, since A is dense, i : A → Y admits a continuous extension, still denoted by i : X → Y , such that d(i(x), i(x  )) ≤ d(x, x  ) for every x, x  ∈ X. Now take y ∈ X. Then there is xnk ∈ X such that d(ink (xnk ), y) ≤ δnk for every k ∈ N. Since X is compact, we can assume that xnk → x for some x ∈ X. Fix Δ > 0 and take a ∈ A such that d(x, a) < Δ2 . We can choose k such that d(xnk , a) < Δ2 . Then d(i(x), y) ≤ d(i(x), i(a)) + d(i(a), ink (a)) + d(ink (a), ink (xnk )) + d(ink (xnk ), y), and so d(i(x), y)
0 for which there is a Δ-isometry i : X → Y is zero. Therefore, thus infimum represents a measure of how far X and Y are from being isometric. Definition 1.1 The Gromov-Hausdorff distance between two compact metric spaces X and Y is defined by dGH (X, Y ) = inf{Δ > 0 : there are Δ-isometries i : X → Y and j : Y → X}. The following theorem summarizes the main properties of dGH . Theorem 1.2 The following properties hold for every triple of compact metric spaces X, Y , and Z: (1) 0 ≤ dGH (X, Y ) < ∞ and dGH (X, Y ) = 0 if and only if X and Y are isometric; (2) dGH (X, Y ) = dGH (Y, X); and (3) dGH (X, Z) ≤ 2(dGH (X, Y ) + dGH (Y, Z)). Proof First, we prove Item (1). Clearly, dGH (X, Y ) ≥ 0. Fix y0 ∈ Y and define i : X → Y by i(x) = y0 for every x ∈ X. Since |d(i(x), i(x  )) − d(x, x  )| = d(x, x  ) ≤ diam(X) for x, x  ∈ X, we have sup |d(i(x), i(x  )) − d(x, x  )| ≤ diam(X).

x,x  ∈X

Also, d(i(x), y) = d(y0 , y) ≤ diam(Y ) for all y ∈ Y . So, dH (i(X), Y ) ≤ diam(Y ). All together, these facts imply that i is a max{diam(X), diam(Y )}-isometry from X to Y . Likewise, fixing x0 ∈ X and defining j : Y → X by j (y) = x0 for every y ∈ Y , we obtain a max{diam(X), diam(Y )}-isometry from Y to X. Thus shows that dGH (X, Y ) ≤ max{diam(X), diam(Y )} < ∞. The fact that dGH (X, Y ) = 0 if and only if X and Y are isometric is a direct consequence of Theorem 1.1. This proves Item (1). Item (2) is trivial (which is why the Δ-isometry j : Y → X in the definition of dGH was added). Finally, we prove Item (3). Fix δ > 0. It follows from the definition of dGH that there are (dGH (X, Y ) + δ)-isometries i : X → Y and j : Y → X; and (dGH (Y, Z) + δ)-isometries k : Y → Z and l : Z → Y . By Proposition 1.1, k ◦ i : X → Z and j ◦ l : Z → X are 2(dGH (X, Y ) + dGH (Y, Z) + 2δ)-isometries. Therefore, dGH (X, Z) ≤ 2(dGH (X, Y ) +  dGH (Y, Z)) + 4δ. Since δ is arbitrary, we get Item (3).

8

1 Gromov-Hausdorff Distances

Hereafter we will use the following definition M = {X : X is a compact metric space}. We write X ≈ Y for X, Y ∈ M if X and Y are isometric. Lemma 1.2 If X ≈ X and Y ≈ Y  for X, X , Y, Y  ∈ M, then dGH (X, Y ) = dGH (X , Y  ). Proof By hypothesis, there are isometries A : X → X and B : Y → Y  . Fix δ > 0. By definition, there are (dGH (X, Y ) + δ)-isometries i : X → Y and j : Y → X. By Lemma 1.1, B ◦i : X → Y  and j ◦B −1 : Y  → X are (dGH (X, Y )+δ)-isometries, whence dGH (X, Y  ) ≤ dGH (X, Y ) + δ. Since δ is arbitrary, dGH (X, Y  ) ≤ dGH (X, Y ). Reversing the roles of Y and Y  , we obtain dGH (X, Y ) ≤ dGH (X, Y  ). Therefore, dGH (X, Y ) =  dGH (X, Y  ). It follows that dGH (X, Y ) = dGH (X, Y  ) = dGH (X , Y  ). As we already mentioned, ≈ is an equivalence relation of M. We denote M = M/ ≈ the set of equivalence classes of ≈ and denote by [X] the equivalence class of X ∈ M. Lemma 1.2 implies that the map dGH : M × M → [0, ∞[ given by dGH ([X], [Y ]) = dGH (X, Y ),

[X], [Y ] ∈ M,

is well-defined. Hereafter, in order to simplify notations, we will still denote by X the elements of M, understanding that X is an equivalence class rather than a single metric space. Theorem 1.2 implies the following properties: 1. 0 ≤ dGH (X, Y ) < ∞ and dGH (X, Y ) = 0 if and only if X = Y , 2. dGH (X, Y ) = dGH (Y, X); and 3. dGH (X, Z) ≤ 2(dGH (X, Y ) + dGH (Y, Z)) for every X, Y, Z ∈ M. This means that (M, dGH ) is a so-called quasi-metric space. Definition 1.2 The space M = (M, dGH ) is called the Gromov-Hausdorff space. This space will be object of study in the next sections.

1.4 Precompactness, Completeness, and a Variational Principle

1.4

9

Precompactness, Completeness, and a Variational Principle

Many authors have studied the topology of the Gromov-Hausdorff space M. It is not difficult to see that M is separable, since the set of finite metric spaces with rationalvalued distance functions is dense in M. We can also prove that M is complete. Indeed, we will obtain this as a corollary of the so-called Gromov precompactness theorem stated as follows. Given ε > 0, an ε-net of a metric space X is a subset S ⊂ X such that dH (S, X) < ε, i.e., for every x ∈ X, there is s ∈ S such that d(s, x) < ε. The cardinality of S is denoted by |S|. A subset F of a metric space X is precompact if the closure of F in X is compact and, equivalently, if every sequence in F has a convergent subsequence. Theorem 1.3 A subset X of M is precompact if and only if the following properties hold: (1) supX∈X diam(X) < ∞; (2) For every ε > 0, there is N(ε) ∈ N such that every X ∈ X has an ε-net with at most N (ε) elements. Proof We will prove the sufficiency and leave the necessity as an exercise (see Exercise 1.20). Let {Xn }n∈N be a sequence in X . We need to prove that this sequence has a convergent subsequence. This is done as follows. First, take D = supX∈X diam(X). Then D < ∞ by Item (1). Next, take an arbitrary countable set X = {x1 , x2 , . . .}. We will define a map d : X × X → [0, ∞[ through the following procedure. Take ε = k1 for k ∈ N in Item (2) to obtain the sequence {N ( k1 )}k∈N . It follows from Item (2) that given n ∈ N, there is a sequence E1 , E2 , . . . , Ek , . . . ⊂ Xn such that Ek is a k1 -net of Xn with at most N( k1 ) elements. Then  we can organize Sn = k≥1 Ek = (xin )i∈N so that the first Nk elements form a k1 -net of Xn , where (Nk )k∈N is defined by N1 = N(1) and Nk = Nk−1 + N( k1 ) for k ≥ 2. By using Item (1) in the statement of the theorem, we have that {d(xin , xjn ) : n, i, j ∈ N} is bounded (in fact, contained in [0, D]). Then a diagonal argument allows us to choose a subsequence {Xnk }k∈N of {Xn }n∈N such that (d(xink , xjnk ))k∈N converges for every i, j ∈ N. For simplicity, we assume that nk = n for every k ∈ N. Hence, the real number sequence {d(xin , xjn )}n∈N converges for every i, j ∈ N. Finally, we define d : X × X → [0, ∞[ by d(xi , xj ) = lim d(xin , xjn ) n→∞

for all i, j ∈ N. Thus d is non-negative, finite, symmetric, and satisfies the triangle inequality, but may be zero at different points xi , xj . By identifying such points, we obtain a metric space still denoted by (X, d). Denote by X∞ the completion of (X, d). To finish the proof, we shall prove the following properties. •

X∞ is compact.

10

1 Gromov-Hausdorff Distances

Since X∞ is complete, it suffices to prove that X∞ has a finite k1 -net for all k ∈ N (Exercise 1.16). First, we prove that S (k) = {x1 , x2 , . . . , xNk } is a k1 -net of X. Take any xi ∈ X. It (k) follows from the choices that Sn = {xin : 1 ≤ i ≤ Nk } is a k1 -net of Xn . So, there is a sequence {jn }n∈N with 1 ≤ jn ≤ Nk such that d(xin , xjnn ) ≤ k1 for every n. Since Nk is finite and does not depend on n, there is 1 ≤ j ≤ Nk such that jn = j for infinitely many n’s. It follows that d(xin , xjn ) ≤ k1 for infinitely many n’s. By taking n → ∞, we get d(xi , xj ) ≤ k1 , concluding that S (k) is a k1 -net of X. Finally, since X is dense in X∞ , S (k) is also a k1 -net of X∞ . Since S (k) is finite, we are done. •

Xn → X∞ . We have dGH (Xn , X∞ ) ≤ 2(dGH (Xn , Sn(k) ) + dGH (Sn(k) , X∞ )),

and so   1 1 (k) (k) . dGH (Xn , X∞ ) ≤ 2 + 2 dGH (Sn , S ) + k k 

(k)

Since dGH (Sn , S (k) ) → 0 as n → ∞, we get lim dGH (Xn , X∞ ) ≤

n→∞

6 k

for all k ∈ N, proving the result.



Recall that a metric space X is complete if every Cauchy sequence in X is convergent (a Cauchy sequence, is a sequence {xn }n∈N such that for every ε > 0, there is N ∈ N such that d(xn , xm ) ≤ ε for all n, m ≥ N). We will prove that M is complete. For this, we need two additional lemmas. Lemma 1.3 If S ⊂ X is an ε-net of a compact metric space X and i : X → Y is a δ-isometry of compact metric spaces, then i(S) is an (ε + 2δ)-net of Y . Proof If y ∈ Y , then there is an x ∈ X such that d(i(x), y) ≤ δ. For this x, we have an s ∈ S such that d(x, s) ≤ ε. Then d(i(s), y) ≤ d(i(s), i(x)) + d(i(x), y) ≤ δ + d(x, s) + d(i(x), y) ≤ 2δ + ε, proving the result.



1.4 Precompactness, Completeness, and a Variational Principle

11

Lemma 1.4 If {Xn }n∈N is a Cauchy sequence in M and dGH (Xn , Xn+1 ) < 2−(n+1) for n ∈ N, then sup diam(Xn ) < ∞. n∈N

Proof By hypothesis, there is a sequence of 2−(n+1) -isometries {in : Xn+1 → Xn }n∈N . It follows that   d(xn+1 , xn+1 ) < 2−(n+1) + d(in (xn+1 ), in (xn+1 )) ≤ 2−(n+1) + diam(Xn )  for every xn+1 , xn+1 ∈ Xn+1 , whence diam(Xn+1 ) ≤ 2−(n+1) + diam(Xn ) for every n ∈ N. Inductively, we obtain

diam(Xn ) ≤ diam(X1 ) +

∞ 

2−k

k=1

for every n ∈ N. Therefore, sup diam(Xn ) ≤ diam(X1 ) + n∈N

∞ 

2−k < ∞,

k=1



proving the result. Now we can prove the following theorem. Theorem 1.4 The Gromov-Hausdorff space M is complete.

Proof Let {Xn }n∈N be a Cauchy sequence of M. By passing to a subsequence if necessary, we can assume that dGH (Xn , Xn+1 ) < 2−(n+1) for every n ∈ N. We shall prove that the family X = {X1 , X2 , . . .} satisfies the conditions of the Gromov precompactness theorem. First, we observe that supn∈N diam(Xn ) < ∞ by Lemma 1.4, and so Item (1) of Theorem 1.3 holds. Next, we prove Item (2) of Theorem 1.3. Fix ε > 0 and take n ∈ N such that ∞  r=n

2−r
n. We shall prove that Sk is an ε-net with no more than N(ε) elements for every k ∈ N. This is obviously true for 1 ≤ k ≤ n, and so we can assume k > n.  For this, we claim that Sk is an ( 2ε + kr=n 2−r )-net of Xk for every k > n. Since Sn is an 2ε -net of Xn , Sn+1 = in (Sn ) is an ( 2ε + 2 · 2−(n+1) )-net (and so an ( 2ε + 2−n )-net) of Xn+1 by Lemma 1.2. Then the assertion holds for k = n + 1. Now suppose that the assertion is true for k > n. Then by Lemma 1.2, we have that Sk+1 = ik+1 (Sk ) is an   −r ( 2ε + kr=n 2−r + 2 · 2−(k+2) )-net (and so an ( 2ε + k+1 r=n 2 )-net) of Xk+1 and the claim follows by induction. Since |Sk | ≤ |Sn | ≤ N(ε), we conclude that Sk is an ε-net with no more than N(ε) elements for every k ∈ N. Then Item (2) holds and so X is precompact by Theorem 1.3. It follows that {Xn }n∈N has a convergent subsequence. Since {Xn }n∈N is Cauchy, {Xn }n∈N itself is convergent (see  Exercise 1.16) and the result follows.

1.5

A Variational Principle for the Gromov-Hausdorff Distance

In this section we express the Gromov-Hausdorff distance as the solution of an optimization problem in the space of probability measures. Let X, Y be compact metric spaces, let Δ > 0, and let ν be a Borel probability measure on Y . We say that i : X → Y is a (ν,Δ)-isometry if • •

|d(i(x), i(x  )) − d(x, x  )| < Δ for all x, x  ∈ X, and ν((Im(i))Δ ) = 1.

We give two examples. Example 1.3 (a) i : X → Y is a Δ-isometry if and only if i is a (ν,Δ)-isometry for every Borel probability measure ν on Y . Indeed, the direct implication is obvious, while the opposite one follows by taking ν = δy0 for y0 ∈ Y , where δy0 is the Dirac measure at y0 . Then y0 ∈ (Im(i))Δ for all y0 ∈ Y . Hence, we get Y = (Im(i))Δ . (b) i : X → Y is (ν, Δ)-isometry for every Δ > 0 (ν is fixed) if and only if i is an isometric immersion such that Im(i) ⊇ supp(ν). The second example above suggests the following definition.

1.5 A Variational Principle for the Gromov-Hausdorff Distance

13

Definition 1.4 Let X, Y be compact metric spaces and Y0 ⊆ Y . We say that i : X → Y is a (Δ, Y0 )-isometry if • •

|d(i(x), i(x  )) − d(x, x  )| < Δ for all x, x  ∈ X, and (Im(i))Δ ⊇ Y0 .

Definition 1.5 Given the pair (X, X0 ) and (Y, Y0 ) of compact metric spaces X, Y and subsets X0 ⊆ X and Y0 ⊆ Y , we define a distance d((X, X0 ), (Y, Y0 )) as the infimum of the numbers Δ > 0 such that there are a (Δ, Y0 )-isometry i : X → Y and a (Δ, X0 )isometry j : Y → X. A priori, there is no comparison between d((X, X0 ), (Y, Y0 )) and dGH (X, Y ). However, we have the following result. Theorem 1.5 For every pair (X, X0 ) and (Y, Y0 ), we have d((X, X0 ), (Y, Y0 )) = 0

if and only if dGH (X, Y ) = 0.

Proof Suppose dGH (X, Y ) = 0, which hold if and only if X and Y are isometric. Then for any Δ > 0, there exist Δ-isometries (actually, isometries) i : X → Y and j : Y → X. So, i(X) ⊇ Y0 and j (Y ) ⊇ X0 ; hence, d((X, X0 ), (Y, Y0 )) = 0. Conversely, if d((X, X0 ), (Y, Y0 )) = 0, then by the standard procedure, there are isometric immersions i : X → Y and j : Y → X. The composition j ◦ i : X → X is an isometric immersion with X compact, so it is onto. Then if y ∈ Y we have that f (y) ∈ X. Consequently, j (y) = j i(x) for some x ∈ X. Hence, y = i(x) (since j is injective), proving that i is  onto and so an isometry. Then X and Y are isometric, proving dGH (X, Y ) = 0. By a measure metric space (X, ν), we mean a compact metric space X equipped with a Borel measure ν. The following will be the distance between measure metric spaces we will deal with. Definition 1.6 Given two measure metric spaces (X, ν) and (Y, μ), we define d((X, ν), (Y, μ)) as the infimum of Δ > 0 such that there are a (ν, Δ)-isometry i : X → Y and a (μ, Δ)-isometry j : Y → X. We denote by P(X) the set of Borel probability measures on X. The following variational principle for the Gromov-Hausdorff distance is obtained. Theorem 1.6 If X and Y are compact metric spaces without isolated points, then dGH (X, Y ) =

sup

d((X, ν), (Y, μ)).

(ν,μ)∈P(X)×P(Y )

14

1 Gromov-Hausdorff Distances

Proof Fix ε > 0 and define Δ = dGH (X, Y ) + ε. Then there exist Δ-isometries i : X → Y and j : Y → X. By Item (2) of Example 1.3, we have that i is a (Δ, ν)-isometry and j is a (Δ, μ)-isometry (for any ν, μ). Then d((X, ν), (Y, μ)) < Δ. Thus, d((X, ν), (Y, μ)) ≤ Δ = dGH (x, y) + ε.

sup

(ν,μ)∈P(X)×P(Y )

Since ε > 0 is arbitrary, we get sup

d((X, ν), (Y, μ)) ≤ dGH (x, y).

(ν,μ)∈P(X)×P(Y )

For the converse inequality, we deduce from the assumption of nonexistence of isolated points that there exists (ν0 , μ0 ) ∈ P(X)×P(y) such that supp(ν0 ) = X and supp(μ0 ) = Y . By taking ε > 0, Δ = d((X, ν0 ), (Y, μ0 )) + ε, and Δ = d((X, ν0 ), (Y, μ0 )) + 2ε, we get a (Δ, μ0 )-isometry i : X → Y and a (Δ, ν0 )-isometry j : Y → X. But then   μ0 (Im(i)Δ ) = 1, so Y = supp(μ0 ) ⊆ (Im(i))Δ . Similarly, X ⊆ (Im(i))Δ . So, i and j are Δ-isometries, implying that dGH (X, Y ) ≤ Δ = d((X, ν0 ), (Y, μ0 )) + 2ε. Since ε is arbitrary, we get dGH (X, Y ) ≤ d((X, ν0 ), (Y, μ0 )). Thus, dGH (x, y) ≤

sup

d((X, ν), (Y, μ)).

(ν,μ)∈P(X)×P(y)

This completes the proof.



1.6 Distances Characterizing Homeomorphic Spaces

1.6

15

Distances Characterizing Homeomorphic Spaces

This section deals with the problem of whether the Gromov-Hausdorff distance can be extended to some distance that is able to detect when two metric spaces are homeomorphic rather than isomorphic. This problem is quite natural and certainly familiar to some authors (though as far as we know nobody wrote about this before). Although in general the answer is negative, the readers (and of course ourselves) are encouraged to overcome the obstacles we found here. To start, let us recall the definition M = {X : X is a compact metric space}. Two spaces X, Y ∈ M are homeomorphic if there is a homeomorphism from X to Y . Like isomorphism, homeomorphism is an equivalence relation in M. Motivated by the Gromov-Hausdorff distance in M, it is natural to study non-zero “distances” in M that vanish on pairs of homeomorphic rather than isomorphic spaces. More precisely, we study non-zero maps D : M × M → [0, ∞[ satisfying the following properties: (P1) If X and Y are homeomorphic, then D(X, Y ) = 0. (P2) D(X, Y ) = D(Y, X) for every X, Y ∈ M. (P3) There is K > 0 such that D(X, Z) ≤ K(D(X, Y )+D(Y, Z)) for every X, Y, Z ∈ M.

We will give some negative answers with the following definition. Hereafter, {∗} will denote the one-point set. Definition 1.7 We say that D : M × M → [0, ∞[ is normalized if there is L > 0 such that L−1 D(X, {∗}) ≤ diam(X) ≤ LD(X, {∗}),

∀X ∈ M.

We can give two examples. The first is precisely the Gromov-Hausdorff distance. Example 1.8 dGH (X, {∗}) = diam(X) for every X ∈ M. In particular, the GromovHausdorff distance dGH is normalized. Proof Take ε > 0 and Δ < dGH (X, {∗}) + ε such that there is a Δ-isometry i : X → {∗}. Then d(x, x  ) = |d(∗, ∗) − d(x, x  )| = |d(i(x), i(x  )) − d(x, x  )| ≤ Δ

16

1 Gromov-Hausdorff Distances

for every x, x  ∈ X, proving diam(X) ≤ Δ < dGH (X, {∗}) + ε. Since ε is arbitrary, diam(X) ≤ dGH (X, {∗}). Now suppose that dGH (X, {∗}) < diam(X). Then there exist a Δ < diam(X) and a Δ-isometry i : X → {∗}. Again, we have d(x, x  ) ≤ Δ for every x, x  ∈ X. Thus, diam(X) ≤ Δ < diam(X), a contradiction. This completes the proof.  The second is as follows. Define the C 0 distance between maps r, l : X → Y by d(r, l) = sup d(r(x), l(x)). x∈X

Given X, Y ∈ M, we define D(X, Y ) as the infimum of δ > 0 such that there are continuous maps i : X → Y and j : Y → X such that d(i ◦ j, I ) ≤ δ

and

d(j ◦ i, I ) ≤ δ.

Example 1.9 The map D defined as above is normalized. Proof If i : X → {∗} and j : {∗} → X, then d(i ◦ j (∗), ∗) = 0 < diam(X) and d(j ◦ i(x), x) = d(j (∗), x) ≤ diam(X) for every x ∈ X. Thus, D(X, Y ) ≤ diam(X). Next, we show that inf sup d(x, x  ) ≥

x  ∈X x∈X

1 diam(X) 3

(1.1)

for all X ∈ M. Indeed, suppose that the inequality in (1.1) fails. Then there exists x  ∈ X such that d(x, x  ) < 13 diam(X) for every x ∈ X. So, for every x, x  ∈ X, d(x, x  ) ≤ d(x, x  ) + d(x  , x  )
L2 K. Define X as X endowed with the metric d  (x, x  ) = λd(x, x  ) for x, x  ∈ X, where d is the metric of X. Then X is a compact metric space with diam(X ) = λ, diam(X). Furthermore, the identity i : X → X , that is, i(x) = x for x ∈ X, is a homeomorphism. Then X ≡ X . Therefore, from the normalization inequality, we obtain (1.2) λ, diam(X) = diam(X ) ≤ LD(X , {∗}) ≤ KLD(X, {∗}) ≤ KL2 diam(X).

Dividing by diam(X) > 0, we get λ ≤ L2 K, a contradiction. This completes the proof.  Theorem 1.7 implies. Example 1.10 The map D defined in Example 1.9 does not satisfy (P3). See also Exercise 1.27.

1.7

C 0 -Gromov-Hausdorff Distance for Dynamical Systems

In this section, we introduce the C 0 -Gromov-Hausdorff distance for certain dynamical systems. This includes maps, flows, and finitely generated group actions. When necessary, we will let d X denote the distance on a metric space X. Let f : X → X and g : Y → Y be continuous maps of compact metric spaces. We say that f and g are isometrically conjugate if there is an isometry h : Y → X such that f ◦ h = h ◦ g. In the following theorem, we characterize isometrically conjugate maps. Theorem 1.8 Two maps f : X → X and g : Y → Y of compact metric spaces are isometrically conjugate if and only if inf{Δ > 0 : ∃Δ-isometry i : X → Y such that d(g ◦ i, i ◦ f ) ≤ Δ} = 0.

18

1 Gromov-Hausdorff Distances

Proof Suppose that f and g are isometrically conjugate. Namely, there is an isometry h : Y → X such that f ◦ h = h ◦ g. Then for any Δ > 0, h : Y → X and h−1 : Y → X are Δ-isometries satisfying d(g ◦ h−1 , h−1 ◦ f ) = d(h ◦ g, f ◦ h) = 0 < Δ. Since Δ is arbitrary, the infimum in the statement is zero. Conversely, suppose that the infimum in the statement is zero. Then there are sequences of n1 -isometries {in : X → Y }n∈N and {jn : Y → X}n∈N satisfying d(g ◦ in , in ◦ f )
0. Then there are Δ < dGH (X, Y ) + ε and Δisometries i : X → Y and j : Y → X. Clearly, d(IdY ◦ i, i ◦ IdX ) = d(j ◦ IdY , IdX ◦ j ) = 0 < Δ, and so dGH0 (IdX , IdY ) ≤ Δ < dGH (X, Y ) + ε. Since ε is arbitrary, dGH0 (IdX , IdY ) ≤ dGH (X, Y ). Consequently, that dGH0 (IdX , IdY ) = dGH (X, Y ). Item (3) is a direct consequence of Theorem 1.8 and Item (4) is a direct consequence of the definition. Let us prove Item (5). Fix ε > 0. It follows from the definition that there are 0 < Δ1 < dGH0 (f, g) + ε, 0 < Δ2 < dGH0 (g, r) + ε, Δ1 -isometries i : X → Y , j : Y → X, and Δ2 -isometries k : Y → Z, l : Z → Y such that d(g ◦ i, i ◦ f ) < Δ1 ,

d(j ◦ g, f ◦ j ) < Δ1 ,

d(r ◦ k, k ◦ g) < Δ2 , and d(l ◦ r, g ◦ l) < Δ2 . By Proposition 1.1, k ◦ i : X → Z and l ◦ j : Z → X are 2(Δ1 + Δ2 )-isometries. Since d(r ◦ k ◦ i, k ◦ i ◦ f ) ≤ d(r ◦ k ◦ i, k ◦ g ◦ i) + d(k ◦ g ◦ i, k ◦ i ◦ f ) ≤ 2Δ2 + d(g ◦ i, i ◦ f ) ≤ 2(Δ1 + Δ2 ), we get d(r ◦ k ◦ i, k ◦ i ◦ f ) ≤ 2(Δ1 + Δ2 ). Likewise, d(l ◦ j ◦ r, f ◦ l ◦ j ) ≤ 2(Δ1 + Δ2 ), proving that dGH0 (f, r) ≤ 2(Δ1 + Δ2 ) < 2(dGH0 (f, g) + dGH0 (g, r) + 2ε). Letting ε → 0, we get Item (5).

1.7 C 0 -Gromov-Hausdorff Distance for Dynamical Systems

21

To prove Item (6), we first observe that dGH0 (f, g) ≥ 0 is obvious by definition. The finiteness of dGH0 (f, g) is a direct consequence of the inequality dGH0 (f, g) ≤ max{dGH (X, Y ), diam(X), diam(Y )}. (see Exercise 1.23). Let us prove Item (7). Since dGH0 (f, gn ) → 0, there are a sequence {δn }n∈N with δn → 0 and δn -isometries in : X → Yn and jn : Yn → X such that d(gn ◦ in , in ◦ f ) < δn

and

d(jn ◦ gn , f ◦ jn ) < δn

for every n ∈ N. It follows that d Yn (in (f (x)), in (f (x  ))) ≤ d Yn (gn (in (x)), in (f (x))) + d Yn (gn (in (x)), gn (in (x  ))) + d Yn (gn (in (x  )), in (f (x  ))) < 3δn + d X (x, x  ) for all x, x  ∈ X. Replacing here d X (f (x), f (x  )) < δn + d Yn (in (f (x)), in (f (x  ))), we obtain d X (f (x), f (x  )) < 4δn + d X (x, x  ). Letting n → ∞, we get d X (f (x), f (x  )) ≤ d X (x, x  ).

(1.3)

On the other hand, dHX (f (jn (Yn )), X) ≤ dHX (f (jn (Yn )), jn (gn (Yn ))) + dHX (jn (gn (Yn )), X)

(1.4)

for all n ∈ N. But gn (Yn ) = Yn , since gn is an isometry. Hence, dHX (jn (gn (Yn )), X) = dHX (jn (Yn ), X) → 0 as

n → ∞,

since jn is a δn -isometry with δn → 0. As d(jn ◦ gn , f ◦ jn ) < δn → 0, we obtain dHX (f (jn (Yn )), jn (gn (Yn ))) → 0. Letting n → ∞ in (1.4), we see that dHX (f (j (Yn )), X) → 0. This shows that f is onto  and so an isometry by (1.3) and [26]. This completes the proof.

22

1 Gromov-Hausdorff Distances

One more property of the C 0 -Gromov-Hausdorff distance between compact metric spaces is given below. Lemma 1.5 Let X, Y, X , Y  ∈ M and f : X → X, g : Y → Y , f  : X → X and g  : Y  → Y  be continuous. If f ≈ f  and g ≈ g  , then dGH0 (f, g) = dGH0 (f  , g  ). Proof By hypothesis, there is an isometry B : Y → Y  such that g  ◦ B = B ◦ g. Fix δ > 0. Then there are (dGH0 (f, g) + δ)-isometries i : X → Y and j : Y → X such that d(g ◦ i, i ◦ f ) < dGH0 (f, g) + δ and d(g ◦ j, j ◦ f ) < dGH 0 (f, g) + δ. By Lemma 1.1, B ◦ i : X → Y  and j ◦ B −1 : Y  → X are (dGH0 (f, g) + δ)-isometries. Since d(g  ◦ B ◦ i, B ◦ i ◦ f ) = d(B ◦ g ◦ i, B ◦ i ◦ f ) = d(g ◦ i, i ◦ f ) < dGH 0 (f, g) + δ and d(j ◦ B −1 ◦ g  , f ◦ j ◦ B −1 ) = d(j ◦ g ◦ B −1 , f ◦ j ◦ B −1 ) = d(j ◦ g, f ◦ j ) < dGH 0 (f, g) + δ, we get dGH0 (f, g  ) ≤ dGH0 (f, g) + δ. Since δ is arbitrary, dGH0 (f, g  ) ≤ dGH0 (f, g). Reversing the roles of g and g  , we get dGH0 (f, g) ≤ dGH0 (f, g  ). So, dGH0 (f, g) =  dGH0 (f, g  ), which implies that dGH0 (f, g) = dGH0 (f, g  ) = dGH0 (f  , g  ). A lemma closely related to the one below was stated without proof as Lemma 2.5 in [29, p.523]. Lemma 1.6 For any δ > 0 and any δ-isometry j : Y → X of compact metric spaces, there is a 3δ-isometry i : X → Y such that d(j ◦ i, I ) ≤ δ

and

d(i ◦ j, I ) ≤ 2δ.

Proof Since j is a δ-isometry, for each x ∈ X we can select i(x) ∈ Y such that d(j (i(x)), x) ≤ δ. This yields a map i : X → Y such that d(j ◦ i, I ) ≤ δ. Next, we prove that i is a 3δ-isometry. On the one hand, since d(i(x), i(x  )) < δ + d(j (i(x)), j (i(x  ))) ≤ δ + d(j (i(x)), x) + d(j (i(x  )), x  ) + d(x, x  ) ≤ 3δ + d(x, x  )

1.7 C 0 -Gromov-Hausdorff Distance for Dynamical Systems

23

and d(x, x  ) < d(j (i(x)), x) + d(j (i(x)), j (i(x  ))) + d(j (i(x  )), x  ) < 3δ + d(i(x), i(x  )) for every x, x  ∈ X, we have sup |d(i(x), i(x  )) − d(x, x  )| ≤ 3δ.

x,x  ∈X

On the other hand, if y ∈ Y , then by taking x = j (y) we have d(i(x), y) < δ + d(j (ix)), x) ≤ 2δ, implying that dH (i(X), Y ) ≤ 2δ. Therefore, i is a 3δ-isometry. Finally, d(i(j (y)), y) < δ + d(j ◦ i(j (y)), j (y)) ≤ δ + δ = 2δ for all y ∈ Y . So, d(i ◦ j, I ) < 2δ, as needed.



A flow on a metric space X is a continuous map φ : X × R → X such that φ(x, 0) = x and φ(φ(x, s), t) = φ(x, s + t) for any x ∈ X and s, t ∈ R. The set of flows (X, φ) on compact metric spaces will be denoted by CDS. For any (X, φ) and (Y, ψ) in CDS, we say that φ and ψ are isometric if there exists an isometry i : X → Y such that i(φ(x, t)) = ψ(i(x), t) for all x ∈ X and t ∈ [0, 1]. For any ε > 0 and X ∈ M, we denote by RepX (ε) the collection of all continuous maps α : X × R → R such that for each fixed x ∈ X, α(x, ·) : R → R is a homeomorphism satisfying α(x, 0) = 0 for all x ∈ X and |α(x, t) − t| < ε for all t ∈ R; such as α is called a reparanetrifation. In this section, we introduce a topology on CDS based on the following definition. Definition 1.12 We define the Gromov-Hausdorff distance DGH0 (φ, ψ) between (X, φ), (Y, ψ) ∈ CDS as the infimum of the numbers ε > 0 for which there exist two ε-isometries i : X → Y and j : Y → X, and α ∈ RepX (ε) and β ∈ RepY (ε) such that for any x ∈ X, y ∈ Y , and t ∈ [0, 1], d(i(φ(x, α(x, t))), ψ(i(x), t)) < ε

and

d(j (ψ(y, β(y, t))), φ(j (y), t)) < ε.

This definition differs from the definition of the C 0 -Gromov-Hausdorff distance in [21], where continuous ε-isometries (but not representations) were considered. This results in a distance between flows that is greater than the one introduced above. Now we will prove several basic properties extending Theorem 1.9.

24

1 Gromov-Hausdorff Distances

Proposition 1.2 For any (X, φ), (Y, ψ), (Z, ξ ) ∈ CDS, the following properties hold: (1) DGH0 (φ, ψ) ≥ 0; equality holds if and only if φ and ψ are isometric. (2) DGH0 (φ, ψ) = DGH0 (ψ, φ). (3) DGH0 (φ, ψ) ≤ 2(DGH0 (φ, ξ ) + DGH0 (ξ, ψ)). Proof Proof of (1). If φ and ψ are isometric, then clearly DGH0 (φ, ψ) = 0. Suppose DGH0 (φ, ψ) = 0. Then for any n ∈ N, there are n1 -isometries in : X → Y and representations jn : Y → X, and αn ∈ RepX (1/n) and βn ∈ RepY (1/n) such that for any x ∈ X, y ∈ Y , and t ∈ [0, 1], d(in (φ(x, αn (x, t))), ψ(in (x), t))
0, we let δ1 = DGH0 (φ, ξ ) + ε. Then there are δ1 isometries i1 : X → Z and j1 : Z → X, and α1 ∈ RepX (δ1 ) and β1 ∈ RepZ (δ1 ) such that for any x ∈ X, z ∈ Z, and t ∈ [0, 1], d(i1 (φ(x, α1 (x, t))), ξ(i1 (x), t)) < δ1 and d(j1 (ξ(z, β1 (z, t))), φ(j1 (z), t)) < δ1 . Let δ2 = DGH 0 (ξ, ψ) + ε. Then there are δ2 -isometries i2 : Z → Y and j2 : Y → Z, and α2 ∈ RepZ (δ2 ) and β2 ∈ RepY (δ2 ) such that for any y ∈ Y , z ∈ Z, and t ∈ [0, 1], d(i2 (ξ(z, α2 (z, t))), ψ(i2 (z), t)) < δ2 and d(j2 (ψ(y, β(y, t))), ξ(j2 (y), t)) < δ2 . We note that i2 ◦i1 and j1 ◦j2 are 2(δ1 +δ2 )-isometries. Now define α1 ◦α2 ∈ RepX (δ1 + δ2 ) by α1 ◦ α2 (x, t) := α1 (x, α2 (i1 (x), t)) for all (x, t) ∈ X × R. For any x ∈ X and t ∈ [0, 1], we have d(ψ(i2 ◦ i1 (x), t),i2 ◦ i1 (φ(x, α1 ◦ α2 (x, t)))) ≤ d(ψ(i2 ◦ i1 (x), t), i2 (ξ(i1 (x), α2 (i1 (x), t))) + d(i2 (ξ(i1 (x), α2 (i1 (x), t))), i2 ◦ i1 (φ(x, α1 ◦ α2 (x, t)))) ≤ δ2 + δ2 + d(ξ(i1 (x), α2 (i1 (x), t)), i1 (φ(x, α1 ◦ α2 (x, t)))) < 2(δ1 + δ2 ). Define β2 ◦ β1 ∈ RepY (δ1 + δ2 ) by β2 ◦ β1 (y, t) := β2 (y, β1 (j2 (y), t))

26

1 Gromov-Hausdorff Distances

for all (y, t) ∈ Y × R. Then d(φ(j1 ◦ j2 (y), t), j1 ◦ j2 (ψ(y, β2 ◦ β1 (y, t)))) < 2(δ1 + δ2 ), where y ∈ Y and t ∈ [0, 1]. Consequently, DGH0 (φ, ψ) < 2(δ1 + δ2 ) = 2(DGH0 (φ, ξ ) + DGH0 (ξ, ψ)) + 4ε. Since ε is arbitrary, we get DGH0 (φ, ψ) ≤ 2(DGH0 (φ, ξ ) + DGH0 (ξ, ψ)).



Proposition 1.2, slows that the distance DGH 0 is a quasi-metric on CDS. If we use the main result of the paper by Paluszynski and Stempak [59], then we can get a metric d1/2 on CDS such that

1/2 d1/2 (φ, ψ) ≤ DGH0 (φ, ψ) ≤ 4d1/2 (φ, ψ) for any φ, ψ ∈ CDS. The topology on CDS induced by the metric d1/2 will be called the topology of Gromov-Hausdorff convergence, which we denote by TGH . Let G be a group with neutral element e. We say that A ⊂ G generates G if every g ∈ G is the product of finitely many elements of A. Hereafter, generating sets A will be assumed to be symmetric, that is, a −1 ∈ A for every a ∈ A. We say that G is finitely generated if it has a finite generating set. A G-action on X is a continuous map T : G × X → X such that T (e, x) = x and T (g, T (h, x)) = T (gh, x) for all x ∈ X and g, h ∈ G. By writing Tg (x) = T (g, x), we identify a G-action T with a parametrized family of homeomorphisms {Tg : X → X}g∈G such that Te = IdX (the identity of X) and Tgh = Tg ◦ Th for every g, h ∈ G. Denote by Act(G, X) the set of G-actions T on X. Definition 1.13 Given compact metric spaces X and Y , a nonempty subset A ⊂ G, T ∈ Act(G, X), and S ∈ Act(G, Y ), we define dGH0 ,A (T , S) as the infimum of the numbers Δ > 0 for which there are Δ-isometries i : X → Y and j : Y → X such that d Y (Sa ◦ i(x), i ◦ Ta (x)) < Δ

sup a∈A,x∈X

and sup

d X (j ◦ Sa (y), Ta ◦ j (y)) < Δ.

a∈A,y∈Y

This definition is a slight variation of that of the equivariant Hausdorff distance for compact metric spaces equipped with isometric actions introduced by Fukaya [29]. The difference is that, unlike the equivariant distance, the above distance does not consider the

1.7 C 0 -Gromov-Hausdorff Distance for Dynamical Systems

27

whole group G, but only a finite subset (usually a generator) A instead. As a result, we obtain a topology that is stronger than the one induced by the equivariant distance. We will prove in Lemma 1.9 that the topologies furnished by different generators are equivalent. By taking G = Z and A = {1, −1}, we recover the Gromov-Hausdorff distance for homeomorphisms defined before. Fukaya’s distance for not necessarily equivariant group actions was recently considered by Chung [22]. Some of the properties of the Gromov-Hausdorff distance for homeomorphisms can be extended to the context of group actions. It suffices to know only the ones proved below. Lemma 1.7 Let X, Y, Z be compact metric spaces, G be a finitely generated group, and A ⊂ G. (1) If R ∈ Act(G, Z), then dGH0 ,A (T , S) ≤ 2(dGH0 ,A (T , R) + dGH0 ,A (R, S)). (2) If A is a generating set, then dGH0 ,A (T , S) = 0 if and only if T and S are isometrically conjugated. Proof First, we prove Item (1). Fix ε > 0. Take Δ1 and Δ2 such that 0 < Δ1 < dGH0 ,A (T , R) + ε

and

0 < Δ2 < dGH0 ,A (R, S) + ε.

Then there are Δ1 -isometries i : X → Z and j : Z → X, and Δ2 -isometries k : Y → Z and l : Z → Y , such that for every a ∈ A, dCZ0 (i ◦ Ta , Ra ◦ i) < Δ1 , dCX0 (j ◦ Ra , Ta ◦ j ) < Δ1 , dCZ0 (Ra ◦ k, k ◦ Sa ) < Δ2 , and dCY 0 (l ◦ Ra , Sa ◦ l) < Δ2 . It follows that dH (l ◦ i(X), Y ) < Δ1 + 2Δ2 . Indeed, since l is a Δ2 -isometry, if y ∈ Y , then there is z ∈ Z such that d(l(z), y) < Δ2 . Since i is a Δ1 -isometry, there is x ∈ X such that d(i(x), z) < Δ1 . Hence, d(l(i(x)), l(z)) < Δ2 + d(i(x), z) < Δ2 + Δ1 ,

28

1 Gromov-Hausdorff Distances

and consequently d(l(i(x)), y) < d(l(i(x)), l(z)) + d(l(z), y) < Δ2 + Δ1 + Δ2 = Δ1 + 2Δ2 , proving the assertion. On the other hand, |d Y (l ◦ i(x), l ◦ i(x  )) − d X (x, x  )| = |d Y (l ◦ i(x), l ◦ i(x  )) − d Z (i(x), i(x  )) + d Z (i(x), i(x  )) − d X (x, x  )| ≤ |d Y (l ◦ i(x), l ◦ i(x  )) − d Z (i(x), i(x  ))| + |d Z (i(x), i(x  )) − d X (x, x  )| < Δ2 + Δ 1 . Therefore, l ◦ i : X → Y is a (Δ1 + 2Δ2 )-isometry and also a 2(Δ1 + Δ2 )-isometry. Likewise, j ◦ k : Y → X is a 2(Δ1 + Δ2 )-isometry. Now, for all x ∈ X, we have d Y (Sa ◦ l ◦ i(x), l ◦ i ◦ Ta (x)) ≤ d Y (Sa ◦ l ◦ i(x), l ◦ Ra ◦ i(x)) + d Y (l ◦ Ra ◦ i(x), l ◦ i ◦ Ta (x)) < Δ2 + Δ1 + Δ2 < 2(Δ1 + Δ2 ). Hence, dCY 0 (Sa ◦ l ◦ i, l ◦ i ◦ Ta ) ≤ 2(Δ1 + Δ2 ). Likewise, dCX0 (j ◦ k ◦ Sa , Ta ◦ j ◦ k) ≤ 2(Δ1 + Δ2 ). So, dGH0 ,A (T , S) ≤ 2(Δ1 + Δ2 ) < 2(dGH0 ,A (T , R) + ε + dGH0 ,A (R, S) + ε). Since ε is arbitrary, dGH0 ,A (T , S) ≤ 2(dGH0 ,A (T , R) + dGH0 ,A (R, S)), proving Item (1). Next we prove Item (2). Assume that A is a finite generating set of G.

1.7 C 0 -Gromov-Hausdorff Distance for Dynamical Systems

29

First, we prove the sufficiency. Suppose T and S are isometric. Then for any Δ > 0, h : Y → X and h−1 : X → Y are Δ-isometries satisfying dCY 0 (Sa ◦ h−1 , h−1 ◦ Ta ) = dCX0 (Ta ◦ h, h ◦ Sa ) = 0 < Δ. Therefore, dGH0 ,A (T , S) ≤ Δ, and since Δ is arbitrary, dGH0 ,A (T , S) = 0. Next, we prove the necessity. Since dGH0 ,A (T , S) = 0, there are sequences of isometries {in : X → Y }n∈N and {jn : Y → X}n∈N such that

1 n-



1 max sup d (Sa ◦ in (x), in ◦ Ta (x)), sup d (jn ◦ Sa (y), Ta ◦ jn (y)) < n a∈A,x∈X a∈A,y∈Y Y

X

for all n ∈ N. Since X is compact, we can select a countable dense subset B of X and assume Ta (B) ⊂ B for every a ∈ A. By a diagonal argument, since Y is compact, we can choose a subsequence {inl }l∈N of {in }n∈N such that inl (b) converges in Y for all b ∈ B. Let i(b) denote the limit of inl (b). Then we obtain a map i : B → Y rule that i(b) = lim inl (b) for every b ∈ B. l→∞

Since inl is a

1 nl -isometry,

we have

|d Y (inl (x), inl (x  )) − d X (x, x  )|
0 such that if Y is a compact metric space and S ∈ Act(G, Y ) satisfies dGH0 ,B (T , S) < δ, then dGH0 ,A (T , S) < δ  . Proof Fix δ  > 0. Since B is a finite generating set and A is finite, we can also fix m ∈ N such that A ⊂ Bm (take, for instance, m = maxa∈A lB (a), where lB is the word length metric on G induced by B). Since m is fixed, (1.5) implies that there is a δ > 0 satisfying (m)

(m)

max{5δ + VT ,A (2δ), δ + VT ,B (δ + VT ,B (δ + · · · + VT ,B (δ + VT ,B (δ))· · ·)} < δ  . Now, let Y be a compact metric space and take S ∈ Act(G, Y ) such that dGH0 ,B (T , S) < δ. It follows from the definitions that there is a δ-isometry j : Y → X such that Δ1 (T , S, j, B) < δ. Together with Lemma 1.8 and the choice of δ, we have

32

1 Gromov-Hausdorff Distances (m)

(m)

Δm ≤ Δ1 + V (Δ1 + V (Δ1 + · · · + V (Δ1 + V (Δ1 ))· · ·) (m)

(m)

< δ + VT ,B (δ + VT ,B (δ + · · · + VT ,B (δ + VT ,B (δ))· · ·) < δ. But δ < 5δ < δ  . So, j is a δ  -isometry. Also, A ⊂ Bm ; thus, sup

d X (j ◦ Sa (y), Ta ◦ j (y)) ≤

a∈A,y∈Y

sup

bm ∈Bm ,y∈Y

d X (j ◦ Sbm (y), Tbm ◦ j (y)) = Δm < δ  ,

proving that d X (j ◦ Sa (y), Ta ◦ j (y)) < δ  .

sup

(1.7)

a∈A,y∈Y

On the other hand, j is a δ-isometry. Hence, by Lemma 1.6, there is a 5δ-isometry i : X → Y such that d X (j ◦ i(x), x) < 2δ for all x ∈ X. The choice of δ implies 5δ < δ  ; hence, i is also a δ  -isometry. Since

d Y (Sa ◦ i(x), i ◦ Ta (x)) < δ + d X (j ◦ Sa ◦ i(x), j ◦ i ◦ Ta (x)) ≤ δ + d X ((j ◦ Sa )(i(x)), (Ta ◦ j )(i(x))) + d X (Ta ◦ j ◦ i(x), j ◦ i ◦ Ta (x)) < 2δ + d X (Ta (j ◦ i(x)), Ta (x)) + d X (Ta (x), j ◦ i(Ta (x))) < 4δ + d X (Ta (j ◦ i(x)), Ta (x))

for x ∈ X and a ∈ A, we get d Y (Sa ◦ i(x), i ◦ Ta (x)) < 4δ + VT ,A (2δ),

sup a∈A,x∈X

which in view of the choice of δ implies that sup

d Y (Sa ◦ i(x), i ◦ Ta (x)) < δ  .

a∈A,x∈X

Then (1.7) implies dGH0 ,A (T , S) < δ  , proving the result.



1.8 Gromov-Hausdorff Space of Continuous Maps

33

To prove Theorem 2.7, we need the following lemma from [23]. Lemma 1.10 For every compact metric space X, every countable group G, every expansive T ∈ Act(G, X) with expansivity constant η, and every ε > 0, there exists a finite subset F ⊂ G such that x, y ∈ X and sup d(Tg x, Tg y) ≤ η

imply

d(x, y) < ε.

g∈F

Proof By contradiction, assume that there is ε > 0 such that for any non-empty finite set F ⊂ G, there are points xF , yF ∈ X such that sup d(Tg x, Tg y) ≤ η

but

d(xF , yF ) ≥ ε.

g∈F

Since G is countable, there is a sequence of non-empty finite subsets Fn of G such that F1 ⊂ F2 ⊂ · · ·

and

G=

Fn .

n∈N

Then, there are sequences {xn }n∈N and {yn }n∈N of Fn such that sup d(Tg (xn ), Tg (yn )) ≤ η

and

d(xn , yx ) ≥ ε,

g∈Fn

for n ∈ N. By compactness, we can assume that xn → x and yn → y for some x, y ∈ X. It follows that d(x, y) ≥ ε (hence x = y), but d(Tg (x), Tg (y)) ≤ η for all g ∈ G. Since η is an expansivity constant, x = y, a contradiction. This completes the proof. 

1.8

Gromov-Hausdorff Space of Continuous Maps

The Gromov-Hausdorff space is the set of compact metric spaces up to isometry M equipped with the Gromov-Hausdorff metric. It is known that this space is Polish, strictly intrinsic, and not boundedly compact [38, 39]. Also, generic elements in this space are totally discontinuous, totally anisometric, homeomorphic to the Cantor set, have no collinear triples of different points, and cannot be embedded into any Hilbert space [68]. Now consider the set of continuous maps of compact metric spaces up to isometric conjugacy. We will equip this set with the C 0 -Gromov-Hausdorff distance defined in Sect. 1.5. The resulting space will be referred to as the Gromov-Hausdorff space of continuous maps. It is natural to ask whether the aforementioned properties of the GromovHausdorff space also hold in the latter space. Some facts are known; for instance, the transitive elements in this space can be approximated by periodic orbits [24, 76].

34

1 Gromov-Hausdorff Distances

More recently, it was proved that a generic element of the Gromov-Hausdorff space of continuous maps is topologically conjugated to a special homeomorphism defined of the Cantor set [10]. This extends part of the aforementioned results in [68] from the GromovHausdorff space to the Gromov-Hausdorff space of continuous maps. In this section, we will analyze the precompactness of subsets of the Gromov-Hausdorff space of continuous maps. More precisely, we prove that such a subset is precompact if and only if it is equicontinuous in a certain sense. Afterwards, we explain the characterization of continuous maps that can be approximated by periodic orbits. Our approach is new and even improves the one in [76]. Let us state these results in a precise way. Consider a compact metric space X. Whenever necessary, we will denote the metric d of X by d X . Given A ⊂ X and ε ≥ 0, we use the notation Aε = {x ∈ X : there exists a ∈ A such that d(a, x) < ε}. Recall that M denotes the Gromov-Hausdorff space. Define C = {f : X → X : f is continuous and X is compact}. We note that the isometric relation f ≈ g is an equivalence relation in C. We denote by C the corresponding set of equivalence classes. We call C the Gromov-Hausdorff space of continuous maps. Denoting by [f ] ∈ C the equivalence class of f ∈ C, we can define by Lemma 1.5 the following quasi-metric in C for [f ], [g] ∈ C: dGH0 ([f ], [g]) = dGH0 (f, g). In what follows, we will write f instead of [f ], understanding that f is an equivalence class rather than a single map. Now denote by D(f ) = X the map assigning the domain X to a map f : X → X. Since isometrically conjugated homeomorphisms have isometric domains, there is an induced map D : C → M. It follows easily from the definitions that DGH0 (D(f ), D(g)) ≤ dGH0 (f, g),

∀f, g ∈ C.

(1.8)

In particular, D is Lipschitz. The first result of this section is given below. Theorem 1.10 A subset F of C is precompact if and only if (1) D(F) is a precompact subset of M. (2) For every ε > 0, there is δ > 0 such that if f ∈ F, x, x  ∈ D(f ), and d(x, x  ) < δ, then d(f (x), f (x  )) < ε. For simplicity, a subset F of C satisfying (1) and (2) above will be referred to as an equicontinuous family.

1.8 Gromov-Hausdorff Space of Continuous Maps

35

This theorem is a counterpart of the Arzelà-Ascoli theorem [61]. Nonetheless, the convergence in [61] is different from the one obtained from dGH0 (compare with [61, p. 401]). This theorem is false if we consider families satisfying only (2) in the definition of equicontinuous family (take, for instance, the set of isometries). It is also false if we replace C by the set of homeomorphisms of compact metric spaces up to isometric conjugacy. To state our second result, we say that g ∈ C is a periodic orbit if D(g) consists of a single periodic orbit. We also say that g is chain transitive if for every y, y  ∈ D(g) and every δ > 0, there is a sequence y0 = y, y1 , . . . , yr = y  such that d(f k (yk ), yk+1 ) ≤ δ for 0 ≤ k ≤ r − 1. In our second result, we characterize the chain transitive maps as those which can be 0 C -Gromov-Hausdorff approximated by periodic orbits. Theorem 1.11 f ∈ C is chain transitive if and only if there is a sequence of periodic GH0

orbits {gn }n∈N in C such that gn −→ f . This result was originally proved by Jung [40], motivated by the MSc dissertation by Cubas [24]. The proof we will give here is new. In view of this result, it is natural to ask whether every continuous map that can be approximated in the C 0 -Gromov-Hausdorff sense (GH0 -approximated for short) by periodic orbits is transitive, that is, has a dense orbit. However, the following counterexample shows that this is not the case. Example 1.14 The identity map of S 1 is not transitive. However, it can be GH0 approximated by periodic orbits gn : Yn → Yn . Indeed, it suffices to choose gn to be a rational rotation on a finite set of S 1 such that its rotation number goes to 0 as n → ∞. Let us give a condition under which C 0 -Gromov-Hausdorff approximately by periodic orbits implies transitivity. We say that a map f : X → X is Lipschitz if there is a constant K > 1 (called Lipschitz constant) such that d(f (a), f (b)) ≤ Kd(a, b) for all a, b ∈ X. GH

GH0

To simplify the notation, we shall denote by Xn −→ X and fn −→ f (as n → ∞) the convergences in M and C, respectively. The convergence in other spaces will be denoted by an → a. Theorem 1.12 Let f ∈ C be Lipschitz with Lipschitz constant K and let {gn : Yn → Yn }n∈N be a sequence of periodic orbits with periods π(gn ). If GH0

gn −−→ f then f is transitive.

and

π(gn ) K π(gn ) dGH0 (f, gn ) → 0

as

n → ∞,

36

1 Gromov-Hausdorff Distances

To prove Theorem 1.10, we need two lemmas. Lemma 1.11 Every precompact subset of C satisfies (2) in the definition of an equicontinuous family. Proof Suppose by contradiction that there is F ⊆ C which is precompact, but does not satisfy (2) in the definition of an equicontinuous family. Then there are ε > 0 and sequences {fn }n∈N in F, {xn }n∈N and {xn }n∈N in D(fn ) such that d(xn , xn ) < n1 , but GH0

d(f (xn ), f (xn )) ≥ ε for any n ∈ N. Since F is precompact, we can assume that fn −→ f for some f ∈ C. So, there are a sequence {δn }n∈N with δn → 0+ and δn -isometries jn : D(fn ) → D(f ) such that dC 0 (jn ◦ fn , f ◦ jn ) < δn for every n ∈ N. On the other hand, we have d(fn (xn ), fn (xn )) < δn + d(jn (fn (xn )), jn (fn (xn ))) ≤ δn + d(jn (fn (xn )), f (jn (xn ))) + d(f (jn (xn )), f (jn (xn ))) + d(f (jn (xn )), jn (fn (xn ))) ≤ 3δn + d(f (jn (xn )), f (jn (xn ))). Also, d(jn (xn ), jn (xn )) < δn + d(xn , xn ) ≤ δn +

1 . n

So, d(jn (xn ), jn (xn )) → 0 as n → ∞, and consequently d(f (jn (xn )), f (jn (xn ))) → 0, because f is continuous and D(f ) is compact. Therefore, ε ≤ d(fn (xn ), fn (xn )) ≤ 3δn + d(f (jn (xn )), f (jn (xn ))) → 0, 

which is absurd. This completes the proof.

Proof (Of Theorem 1.10) By Lemma 1.11, precompactness of F implies that F satisfies (2) in the definition of an equicontinuous family. But precompactness of F also implies (1) in the definition, namely, D(F) is precompact in M. Indeed, take a sequence {Xn }n∈N in D(F). Then Xn = D(fn ) for some sequence {fn }n∈N in F. Since F is precompact, GH0

there is a convergent subsequence, say fnl −→ f ∈ C. Then,

1.8 Gromov-Hausdorff Space of Continuous Maps

37

dGH (Xnl , D(f )) = dGH (D(fnl ), D(f )) ≤ dGH0 (fnl , f ) → 0. GH0

So, Xnl −→ D(f ); thus, D(F) is precompact. Therefore, F is equicontinuous. Conversely, suppose that F ⊆ C is equicontinuous. Take a sequence {fn }n∈N ⊂ F. We shall prove that fn has a convergent subsequence. For this, we note that D(fn ) ∈ D(F); GH

so we can assume that D(fn ) −→ X for some X ∈ M. From this and Lemma 1.6 we get sequences of δn -isometries {in : X → D(fn )}n∈N and {jn : D(fn ) → X}n∈N with δn → 0 as n → ∞, such that d(jn ◦ in , IX ) → 0 and d(in ◦ jn , ID(fn ) ) → 0 as n → ∞. Consider the composition jn ◦ fn ◦ in : X → X. Take a countable dense subset A ⊆ X. By a diagonal argument, we can assume that the sequence jn ◦ fn ◦ in (a) converges to some f (a) ∈ X for all a ∈ A. This defines a map f : A → X by f (a) = lim jn ◦ fn ◦ in (a). n→∞

We claim that f is uniformly continuous. Indeed, take ε > 0 and let δ > 0 be given by the equicontinuity of F for 2ε . If d(a, a  ) < 2δ for some a, a  ∈ A, then d(in (a), in (a  )) < δn + d(a, a  ) < δn +

δ 0. Since both in and jn are δn -isometries, δn → 0, and f is uniformly continuous, it follows from Item (2) in Theorem 1.10 that there are δ > 0 and n1 ∈ N such that d(jn ◦ fn ◦ in (p), jn ◦ fn ◦ in (q))
0 and x, x  ∈ X. Since jn is a δn -isometry with δn → 0 and each gn is a periodic orbit, there are yn ∈ Yn and 0 ≤ ln ≤ π(gn ) such that d(x, jn (yn )) < ε and d(x  , jn (gnln (yn )) < ε for all n large enough. Now consider the sequence x1 , . . . , xln ∈ X defined by xm = jn (gnm (yn )) for 1 ≤ m ≤ ln . Since

1.8 Gromov-Hausdorff Space of Continuous Maps

41

d(f (xm ), xm+1 ) = d(f (jn (gnm (yn ))), jn (gnm+1 (yn ))) = d(f ◦ jn (gnm (yn )), jn ◦ gn (gnm (y))) < δ, we get that the above sequence is a δn -chain. As δn → 0, f is chain transitive and the result follows.  To prove Theorem 1.12, we will use the following short lemma. Lemma 1.14 Let f ∈ C and {gn }n∈N be a sequence in C. If an > 0 is a sequence such that limn→∞ an dGH 0 (gn , f ) → 0, then there are sequences {kn }n∈N ⊂ N, {δkn }n∈N with δkn → 0+ , and δkn -isometries jkn : Ykn → X such that lim akn δkn = 0.

n→∞

Proof Given n ∈ N, there is kn ∈ N such that akn dGH 0 (gkn , f ) < n1 . Then dGH 0 (gkn , f ) < ak1 n ; so there is dGH 0 (gkn , f ) < δkn < ak1 n . It follows that n n limn→0 akn δkn = 0 and we get a δkn -isometry jkn : Ykn → X from the definition of dGH 0 .  Proof (Of Theorem 1.12) To prove that f is transitive, it suffices to show that for every pair of open sets U, V ⊂ X, there is l ∈ N such that f l (U ) ∩ V = ∅. Given such U and V , we will find l as follows. Put an = π(gn )K π(gn ) as in Lemma 1.14 to get sequences {kn }n∈N ⊂∈ N, {δkn }n∈N → 0, and δkn -isometries jkn : Ykn → X. For simplicity, we assume kn = n for all n ∈ N. We will prove d(f l (jn (yn )), jn (gnl (yn ))) ≤ lK l δn

(1.11)

for all yn ∈ Yn and 1 ≤ l ≤ π(gn ). Indeed, since d(f (jn (yn )), jn (gn (yn ))) = d(f ◦ jn (yn ), jn ◦ gn (yn )) ≤ δn , we have d(f (jn (yn )), jn (gn (yn ))) ≤ δn ≤ Kδn , proving (1.11) for l = 1. Now suppose that 1 ≤ l < π(gn ) satisfies (1.11). Since d(f l+1 (jn (yn )), jn (gnl+1 (yn ))) ≤ d(f l (f ◦ jn (yn )), f l (jn ◦ gn (yn ))) + d(f l (jn (gn (yn ))), jn (gnl (gn (yn ))))

42

1 Gromov-Hausdorff Distances

≤ K l d(f ◦ jn (yn ), jn ◦ gn (yn )) + lK l δn ≤ K l δn + lK l δn = (l + 1)K l δn , (1.11) holds for l + 1. Therefore, (1.11) is proved by induction. Now we use (1.11) to complete the proof of the theorem. Fix an open set V  ⊂ V with closure V  ⊂ V . Since jn : Yn → X is a δn -isometry, dH (jn (Yn ), X) ≤ δn . So, jn (Yn ) ∩ U = ∅ for all large enough n. Hence, we can choose a sequence {yn }n∈N ⊂ Yn such that jn (yn ) ∈ U for all large enough n. Likewise, there is a sequence {yn }n∈N ⊂ Yn with yn ∈ V  for every n large. Since gn is periodic, yn = gnln (yn ) for some sequence {ln }n∈N ⊂ N. Thus, jn (yn ) ∈ U and jn (gnln (yn )) ∈ V  for all enough n. Since K > 1, by applying (1.11), we obtain d(f ln (jn (yn )), jn (gnln (yn ))) ≤ ln K ln δn ≤ π(gn )K π(gn ) δn for n ∈ N. Since an = π(gn )K π(gn ) , we have π(gn )K π(gn ) δn → 0 as n → ∞ by Lemma 1.14. Hence, d(f ln (jn (yn )), jn (gnln (yn ))) → 0 as n → ∞. Since jn (gnln (yn )) ∈ V  and V  ⊂ V , we get f ln (jn (yn )) ∈ V for n large enough. Taking l = ln and x = jn (yn ), we get f l (x) ∈ f l (U ) ∩ V , and so f l (U ) ∩ V = ∅. This completes the proof. 

Exercises Exercise 1.15 Prove that a surjective map i : X → Y of compact metric spaces is an isometry if and only if d(i(x), i(x  )) ≤ d(x, x  ) for every x, x  ∈ X (see [26] for the case X = Y ). Exercise 1.16 Prove that a complete metric space is compact if and only if it has a finite 1 n -net for every n ∈ N. Prove also that a Cauchy sequence in a metric space is convergent if and only if it has a convergent subsequence. Exercise 1.17 Prove that the Gromov-Hausdorff space is noncompact and path connected. Exercise 1.18 (Original Gromov-Hausdorff distance) Let X and Y be compact metric spaces. Define dˆGH (X, Y ) as the infimum of all Δ > 0 such that there exist a metric space Z and isometric immersions i : X → Z and j : Y → Z such that

1.8 Gromov-Hausdorff Space of Continuous Maps

43

d(i(X), j (Y )) ≤ Δ. Prove that dˆGH is equivalent to dGH , namely, that there is K > 1 such that K −1 dGH (X, Y ) ≤ dˆGH (X, Y ) ≤ KdGH (X, Y ) for all X, Y ∈ M. Exercise 1.19 Prove that the finite metric spaces form a dense subset of the GromovHausdorff space. Conclude that the Gromov-Hausdorff space is separable. Exercise 1.20 Prove the necessity in Gromov precompactness theorem, namely, slow that if X ⊂ M is precompact, then 1. supX∈X diam(X) < ∞; 2. for every ε > 0, there is N(ε) ∈ N such that every X ∈ X has an ε-net of at most than N (ε) elements. Exercise 1.21 Find continuous maps f, g : X → X of a compact metric space such that dGH0 (f, g) < d(f, g) (compare with Item (1) of Theorem 1.9). Exercise 1.22 Find continuous maps f : X → X, g : Y → Y and r : Z → Z of compact metric spaces such that dGH0 (f, g) > dGH0 (f, r) + dGH0 (r, g). Exercise 1.23 Prove that if f : X → X and g : Y → Y are continuous maps of compact metric spaces, then dGH0 (f, g) ≤ max{dGH (X, Y ), diam(X), diam(Y )}. Exercise 1.24 Let f : X → X be a continuous map of a compact metric space. Suppose that for every Δ > 0, there is Δ > 0 such that if g : Y → Y is a continuous map of a compact metric space satisfying d(g ◦ i, i ◦ f ) ≤ Δ for some Δ -isometry i : X → Y , then dGH0 (f, g) ≤ Δ. Exercise 1.25 Prove the following properties of the Gromov-Hausdorff distance for group actions dGH0 ,A (T , S), where G is a group with generating set A: 1. DGH0 (X, Y ) ≤ dGH0 ,A (T , S). 2. If X = Y , then dGH0 ,A (T , S) ≤

sup

d(Ta (x), Sa (x)).

a∈A,x∈X

3. dGH0 ,A (T , S) < ∞. 4. If R ∈ Act(G, Z) for a compact metric space Z, then dGH0 ,A (T , S) ≤ 2(dGH0 ,A (T , R) + dGH0 ,A (R, S)). 5. If A is finite, then dGH0 ,A (T , S) = 0 if and only if T and S are isometric. Exercise 1.26 Is the GH-variational principle (Theorem 1.6) valid for every pair of compact metric spaces X and Y ?

44

1 Gromov-Hausdorff Distances

A map i : X → Y of compact metric spaces is δ-continuous if there is β > 0 such that d(x, x  ) ≤ β implies d(i(x), i(x  )) ≤ δ; δ-surjective if dH (i(X), Y ) ≤ δ; and δ-injective if there is β > 0 such that d(i(x), i(x  )) ≤ β implies d(x, x  ) ≤ δ. We say that i is a δ-homeomorphism if it is δ-continuous, δ-surjective and δ-injective. Exercise 1.27 Define D : M × M → [0, ∞[ by setting D(X, Y ) as the infimum of the numbers δ > 0 such that there are δ-homeomorphisms i : X → Y and j : Y → X. Prove that this D is normalized and so if does not satisfy (P3) in Sect. 1.6. Does D(X, Y ) = 0 imply that X and Y are homeomorphic? Exercise 1.28 A continuous map f : X → X of a compact metric space is minimal if the sets {f n (x) : n ∈ N} are dense in X for every x ∈ X. Find an example of a homeomorphism of a compact metric space that is not transitive but can be GH0 approximated by minimal homeomorphisms. GH0

Exercise 1.29 Let {gn }n∈N ⊂ C be a sequence such that gn −→ f for some f ∈ C. Suppose that each gn has a compact invariant set Λn . Is it true that there is a compact GH

invariant set Λ of f such that Λn −→ Λ? Exercise 1.30 A continuous map f : X → X of a compact metric space is equicontinuous if for every ε > 0, there is δ > 0 such that x, x  ∈ X and d(x, x  ) ≤ δ implies d(f n (x), f n (x  )) ≤ ε for every n ∈ N. Is the C 0 -Gromov-Hausdorff limit of a sequence of equicontinuous maps equicontinuous? Exercise 1.31 Is C complete? Try to use Theorem 1.10 to prove it. Exercise 1.32 Let H be the space of isometric classes of homeomorphisms of compact metric spaces. Is H complete? Exercise 1.33 Let D : C → M be the map assigning to each f ∈ C its domain D(f ) (up to isometries). Prove that D is well-defined and DGH0 (D(f ), D(g)) ≤ dGH0 (f, g)

∀f, g ∈ C.

Exercise 1.34 Prove that the map D in Exercise 1.33 has a global cross section, that is, there is C  ⊂ C such that D maps C  homeomorphically onto M. Is D : C → M a covering map or a fibre bundle? Exercise 1.35 Use Lemma 1.13, (1.9), (1.10) and Exercise 1.24 to prove Theorem 1.10. Exercise 1.36 Extend and prove the results in Sect. 1.8 to finitely generated groups actions, flows or semiflows.

2

Stability

2.1

Introduction

In this chapter, we use the C 0 -Gromov-Hausdorff distance between maps of metric spaces to introduce the notion of topological GH-stability. We will prove that there are topologically stable homeomorphisms that are not topologically GH-stable. Also, we show that every topologically GH-stable circle homeomorphism is topologically stable. We then prove that every expansive homeomorphism with the shadowing property of a compact metric space is topologically GH-stable. This is related to Walters’ stability theorem [76]. We extend these results to group actions and to dissipative semiflows on complete metric spaces.

2.2

Definitions and Statement of Main Results

We first recall the classical notion of topological stability introduced by Walters [75]. Recall that Id denotes the identity map of a metric space X (sometimes we write IdX instead to emphasize the dependence on X). Definition 2.1 A homeomorphism f : X → X of a compact metric space X is topologically stable if for every ε > 0, there is δ > 0 such that for every homeomorphism g : X → X with d(f, g) < δ, there is a continuous map h : X → X with d(h, IdX ) < ε such that f ◦ h = h ◦ g. This definition together with the C 0 -Gromov-Hausdorff distance motivates the following notion of stability.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Lee, C. Morales, Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-12031-2_2

45

46

2 Stability

Definition 2.2 A homeomorphism f : X → X of a compact metric space X is topologically GH-stable if for every ε > 0, there is δ > 0 such that for every homeomorphism g : Y → Y of a compact metric space Y satisfying dGH0 (f, g) < δ, there is a continuous ε-isometry h : Y → X such that f ◦ h = h ◦ g. Both definitions can be extended to continuous maps by just replacing homeomorphism by continuous map in their statement. In our first result, we will prove that there are topologically stable homeomorphisms that are not topologically GH-stable. Theorem 2.1 There exist a compact metric space X and a homeomorphism f : X → X that is topologically stable, but not topologically GH-stable. Although we do not know if every topologically GH-stable homeomorphism is topologically stable, we will prove that this is the case for circle homeomorphisms. Theorem 2.2 Every topologically GH-stable circle homeomorphism is topologically stable. Next, we recall that a homeomorphism f : X → X of a metric space X is expansive if there is δ > 0 (called the expansivity constant) such that if x, y ∈ X satisfy d(f n (x), f n (y)) ≤ δ for all n ∈ Z, then x = y. Moreover, f has the shadowing property if for every ε > 0, there is δ > 0 such that for every bi-infinite sequence {xn }n∈Z satisfying d(xn+1 , f (xn )) < δ for every n ∈ Z, there is x ∈ X such that d(f n (x), xn ) < ε for every n ∈ Z. Walters’ stability theorem asserts that every expansive homeomorphism with the shadowing property of a compact metric space is topologically stable [76]. Here, we will prove that all such homeomorphisms are also topologically GH-stable. Theorem 2.3 Every expansive homeomorphism with the shadowing property of a compact metric space is topologically GH-stable.

2.3

Proof of Theorem 2.1

The proof will use the following lemma. Recall that a homeomorphism g : X → X of a metric space X is minimal if Og (x) is dense in X for all x ∈ X, where Og (x) = {g n (x) : n ∈ N} is the g-orbit of x. Lemma 2.1 Let f : X → X be a topologically GH-stable homeomorphism of a compact metric space X. If infz∈X dH (X, Of (z)) > 0, then there is δ > 0 such that

2.3 Proof of Theorem 2.1

47

no homeomorphism g : Y → Y of a compact metric space Y with dGH0 (f, g) < δ is minimal. Proof By the hypothesis, there is ε > 0 such that dH (X, Of (z)) > ε for all z ∈ X. For this ε, we let δ > 0 be given by the topological GH-stability of f . If this δ does not work, then we could select a minimal homeomorphism g : Y → Y of a compact metric space Y with dGH0 (f, g) < δ. Then by topological GH-stability, there is a continuous ε-isometry h : Y → X such that f ◦ h = h ◦ g. Fix y ∈ Y . Since g is minimal, Og (y) is dense in Y and so h(Og (y)) is dense in h(Y ) by the continuity of h. On the other hand, f ◦ h = h ◦ g. So, h(Og (y)) = Of (z), where z = h(y). It follows that Of (z) is dense in h(Y ). But h is an ε-isometry, so dH (X, h(Y )) < ε. Since Of (z) is dense in h(Y ), we conclude that dH (X, Of (z)) < ε, contradicting the choice of ε. This completes the proof.  Proof (Proof of Theorem 2.1) Consider the unforced undamped Duffing oscillator given by the flow Φ in R2 generated by the system ⎧ ⎨x˙ = y, ⎩y˙ = x − x 3 . It has three types of orbits: the outer circles Y , the ones forming the figure eight set X (which contains the equilibrium p), and two inner circles. We define f : X → X as the time-one map Φ1 of Φ restricted to X. Let us prove first that f is topologically stable. Split X = C1 ∪ C2 as the union of two circles C1 , C2 , intersecting at the equilibrium p. Clearly, any homeomorphism g : X → X fixes p, and if g is C 0 -close to f , then it also leaves C1 and C2 invariant. Take ε > 0. It follows that there is δ > 0 such that if dC 0 (f, g) < δ, then there are four fixed points a1 , b1 ∈ C1 and a2 , b2 ∈ C2 satisfying the following properties: • • •

max{d(p, a1 ), d(p, b1 ), d(p, a2 ), d(p, b2 )} < ε. g([p, a1 ] ∪ [p, b1 ] ∪ [p, a2 ] ∪ [p, b2 ]) = [p, a1 ] ∪ [p, b1 ] ∪ [p, a2 ] ∪ [p, b2 ]. If x ∈ C1 \ ([p, a1 ] ∪ [p, b1 ]), then lim g n (x) = b1 and lim g n (x) = a1 .

•

If x ∈ C2 \ ([p, a2 ] ∪ [p, b2 ]), then lim g n (x) = a2 and lim g n (x) = b2 .

n→∞

n→−∞

n→∞

n→−∞

For such maps g, we pick ei ∈ Ci (i = 1, 2) and consider the intervals [ei , g(ei )[ in Ci for i = 1, 2. It follows that for every x ∈ Ci \ ([p, ai ] ∪ [p, bi ]), there is a unique integer ni (x) such that g −ni (x) (x) ∈ [ei , g(ei )[ for i = 1, 2.

48

2 Stability

Making δ smaller if necessary, we obtain diffeomorphisms hi : [ei , g(ei )[→ [ei , f (ei )[ that is C 0 -close to the identity for i = 1, 2. Define h : X → X by ⎧ n (x) −n (x) ⎪ ⎨ f 1 (h1 (g 1 (x))), n (x) h(x) = f 2 (h2 (g −n2 (x) (x))), ⎪ ⎩ p,

if x ∈ C1 \ ([p, a1 ] ∪ [p, b1 ]), if x ∈ C2 \ ([p, a2 ] ∪ [p, b2 ]), if x ∈ [p, a1 ] ∪ [p, b1 ] ∪ [p, a2 ] ∪ [p, b2 ].

Straightforward computations show that h is continuous and dC 0 (h, IdX ) < ε. Moreover, if x ∈ C1 \ ([p, a1 ] ∪ [p, b1 ]), then f (h(x)) = f n1 (x)+1 (h1 (g −n1 (x) (x))). But n1 (g(x)) = n1 (x) + 1; so, f (h(x)) = f n1 (g(x)) (h1 (g −n1 (g(x)) (g(x)))) = h(g(x)). Hence, f (h(x)) = h(g(x)) if x ∈ C1 \ ([p, a1 ] ∪ [p, b1 ]). Similarly, we can prove that f (h(x)) = h(g(x)) if x ∈ C2 \ ([p, a2 ] ∪ [p, b2 ]). Next, if x ∈ [p, a1 ] ∪ [p, b1 ] ∪ [p, a2 ] ∪ [p, b2 ], then f (h(x)) = f (p) = p and h(g(x)) = p, since g([p, a1 ] ∪ [p, b1 ] ∪ [p, a2 ] ∪ [p, b2 ]) = [p, a1 ] ∪ [p, b1 ] ∪ [p, a2 ] ∪ [p, b2 ]. All together, we have f ◦ h = h ◦ g. Therefore, f is topologically stable. Now we prove that f is not topologically GH-stable. Take g : Y → Y as the time-one map Φ1 |Y of Φ restricted to the outer circle Y . One can easily check that dGH0 (f, g) → 0 as Y converges to X. Also, since DΦ1 (p) · Φ(p) = Φ(Φ1 (p)) for p ∈ R2 , the map g is topologically conjugated to a circle rotation. On the other hand, by taking Y close to X, we can see that the period of Y goes to infinity. Since this a period depends continuously on Y , we can choose Y arbitrarily close to X so that g has an irrational rotation number (and so it is minimal). Consequently, f can be C 0 -GH-approximated by minimal homeomorphisms. Since f clearly satisfies the condition infz∈X dH (X, Of (z)) > 0, f is not topologically GH-stable by Lemma 2.1. 

2.4

Isometric Stability: Proof of Theorem 2.4

Item (1) of Theorem 1.9 implies that every topologically GH-stable homeomorphism of a compact metric space satisfies the auxiliary definition below. Definition 2.3 We say that a homeomorphism f : X → X of a compact metric space X is isometrically stable if for every ε > 0, there is δ > 0 such that for every homeomorphism g : X → X with d(f, g) < δ, there is a continuous ε-isometry h : X → X satisfying f ◦ h = h ◦ g. Since being ε-C 0 -close to the identity implies being a 2ε-isometry, every topologically stable homeomorphism is isometrically stable. But it seems that the converse is false in general because an ε-isometry (or even an isometry) need not be ε-C 0 -close to the identity. Nevertheless, we will prove such a converse in the case of the circle S 1 .

2.4 Isometric Stability: Proof of Theorem 2.4

49

Theorem 2.4 A circle homeomorphism is isometrically stable if and only if it is topologically stable. Proof As noted above, every topologically stable homeomorphism is isometrically stable (not only on the circle, but also on every compact metric space). To prove the converse on S 1 , we will follow the arguments in [78]. Recall that p is a periodic point of f if there is a minimal integer n > 0 (called a period) such that f n (p) = p. Denote by Per(f ) the set of periodic points of f . We say that p ∈ Per(f ) is topologically hyperbolic if the map f 2n (x) − x changes its sign at x = p, where n is the period of p. As pointed by Yano [78], to prove that a homeomorphism f : S 1 → S 1 is topologically stable, it suffices to prove that: (a) Per(f ) is non-empty and finite. (b) Every element of Per(f ) is topologically hyperbolic. Hence, it suffices to prove that every isometrically stable homeomorphism f : S 1 → S 1 satisfies (a) and (b). Replacing f by f 2 if necessary, we can assume that f is orientation preserving. Denote by length(I ) the length of an interval I ⊂ S 1 . Fix 0 < ε < 14 such that length(f (B)) < 18 whenever B is an interval of length(B) < 4ε. 1 from the isometric stability of f . For this ε, we fix 0 < δ < 16 It follows from Peixoto’s Theorem [55, p. 51] that there exists a diffeomorphism g with dC 0 (f, g) < δ such that Per(g) is non-empty and finite. Also, by the choice of δ, there is a continuous ε-isometry h : S 1 → S 1 such that f ◦ h = h ◦ g. Since 0 < ε < 14 , we have that h is onto. Indeed, denote by N, S, E, and W the north, south, east, and west poles on S 1 . Denote by [N, E] the interval in S 1 bounded by N and E, containing neither S nor W . Then h([N, E]) does not intersect {h(S), h(W )}. Otherwise, there would exist P ∈ [N, E] with, say h(P ) = h(S), whence 1 1 ≤ d(P , S) = |d(h(P ), h(S)) − d(P , S)| < ε < , 4 4 which is absurd. Similarly, the image of the corresponding intervals [E, S], [S, W ], and [W, N] under h does not intersect {h(N ), h(W )}, {h(E), h(N )}, and {h(S), h(E)}, respectively. All together, we have that h is onto. Next we take x ∈ Per(g) with period n. Since f n (h(x)) = h(g n (x)) = h(x), we have h(x) ∈ Per(f ), and so Per(f ) is non-empty. Take any x ∈ Per(f ) and consider K = h−1 (Of (x)). Since h is onto, K is non-empty. If y ∈ K, then h(y) = f i (x) for some 0 ≤ i ≤ n − 1 it is specified above. Thus, h(g(y)) = f (h(x)) = f i+1 (x) ∈ Of (x), proving g(y) ∈ K whenever y ∈ K. Hence, K is compact, g-invariant, and non-empty.

50

2 Stability

It follows that K contains a periodic point of g. Since the collection {h−1 (Of (x)) : x ∈ Per(f )} is disjoint and Per(g) is finite, we conclude that Per(f ) is finite. This proves (a). The proof of (b) follows from the following Claim: There is Δ > 0 such that for every homeomorphism g with dC 0 (f, g) < Δ the cardinality of Per(g) is not less than that of Per(f ). Indeed, by (a), there is a positive integer n such that Per(f ) = Fix(f n ). Then, without loss of generality, we can assume that Per(f ) = Fix(f ), where Fix(f ) is the set of fixed points of f . Now take Δ = δ (where δ is as above), x ∈ Fix(f ), and a homeomorphism g with dC 0 (f, g) < δ. By the choice of δ, unambiguously g is orientation-preserving. Also, there is a continuous ε-isometry h : S 1 → S 1 with f ◦ g = g ◦ h. It follows that h−1 (x) is a compact g-invariant set that is non-empty since h is onto. If y, y  ∈ h−1 (x), then h(y) = h(y  ) = x. Then d(y, y  ) = |d(h(y), h(y  )) − d(y, y  )| < ε. Hence, h−1 (x) is contained in an interval of length at most 2ε. It follows that, we can define sup h−1 (x) and inf h−1 (x) unambiguously. Putting B = [inf h−1 (x), sup h−1 (x)], we have length(B) ≤ 2ε; so length(f (B)) ≤ 18 by the choice of ε. Further, since length(g(B)) ≤ length(f (B)) + 2dC 0 (f, g), we have length(g(B)) ≤

1 1 + 2δ < . 8 4

On the other hand, h−1 (x) is g-invariant, so both endpoints of g(B) are in B. Since g is a homeomorphism and the total length of S 1 is one, the estimate length(g(B)) < 14 above implies that g(B) ⊂ B. Since B is an interval and g is continuous, this implies that g has a fixed point in B. Thus, we proved that each x ∈ Fix(f ) corresponds to a fixed point of g. So, whenever g satisfies dC 0 (f, g) < δ, the cardinality of Per(g) is at least that of Per(f ). The Claim is proved. Now we use the Claim to prove (b). Suppose by contradiction that f has a periodic point that is not topologically hyperbolic. Then we can eliminate it by a small perturbation g, contradicting the Claim. Hence, (b) holds.  Another consequence of Theorem 2.4 is as follows. A diffeomorphism f : S 1 → S 1 is called Morse-Smale if Per(f ) is non-empty and every element x ∈ Per(f ) satisfies (f n ) (x) = ±1, where n is the period of x. Corollary 2.1 The following properties are equivalent for every homeomorphism f : S1 → S1: (1) f is isometrically stable. (2) f is topologically stable. (3) f is topologically conjugate to a Morse-Smale diffeomorphism.

2.4 Isometric Stability: Proof of Theorem 2.4

51

Proof (1) is equivalent to (2) by Theorem 2.4 and (2) is equivalent to (3) by Yano [78].  One more corollary is that, on in the circle, the isometric stability is invariant under topological equivalence. More precisely, we have the following result. Corollary 2.2 A circle homeomorphism topologically conjugate to an isometrically stable circle homeomorphism is isometrically stable. Proof Let f  be a circle homeomorphism that is topologically conjugate to an isometrically stable circle homeomorphism f . Since f is isometrically stable, by Corollary 2.1, there is a Morse-Smale diffeomorphism g : S 1 → S 1 such that f and g are topologically conjugate. Since f and f  are homeomorphic, f  and g are also topologically conjugate.  Then, f  is isometrically stable by Corollary 2.1. We do not know if the isometric stability is invariant under topological equivalence on general metric spaces (the answer, however, seems to be negative). Meanwhile, let us recall that, in general, topological stability is invariant under topological conjugacy. An analogous result for topological GH-stability reads as follows. Theorem 2.5 Every homeomorphism of a compact metric space that is isometric to a topologically GH-stable homeomorphism is itself topologically GH-stable. Proof Let f : X → X and f  : X → X be homeomorphisms of compact metric spaces X and X . Suppose that f and f  are isometric and that f is topologically GH-stable. Fix an isometry v : X → X such that f = v ◦ f  ◦ v −1 . Take ε > 0 and let δ > 0 be given by the topological GH-stability of f . Let g  : Y  → Y  be a homeomorphism of a compact metric space Y  such that dGH0 (f  , g  ) < 2δ . By Theorem 1.9, dGH0 (f, g  ) ≤ 2(dGH0 (f, f  ) + dGH0 (f  , g  )) < 2 ·

δ = δ. 2

Then, by the choice of δ, there is a continuous ε-isometry h : Y  → X such that f ◦ h = h ◦ g  . So, v ◦ f  ◦ v −1 ◦ h = h ◦ g  . Therefore, setting h = v −1 ◦ h, we get a continuous ε-isometry h : Y  → X satisfying f  ◦ h = h ◦ g  . Then f  is topologically GH-stable.  Proof (Of Theorem 2.2) Apply Theorem 2.4 and the fact that every topologically GHstable homeomorphism is isometrically stable. 

52

2.5

2 Stability

Proof of Theorem 2.5

The proof is inspired by Walters’ stability theorem [76]. Let f : X → X be an expansive homeomorphism with the shadowing property of a compact metric space X. Fix ε > 0 and take 0 < ε < 18 min{ε, e}, where e is the expansivity constant of f . For this ε, we choose δ from the shadowing property. We can assume that δ < ε. Now take a homeomorphism g : Y → Y of a compact metric space Y such that dGH0 (f, g) < δ. Then there are δ-isometries i : X → Y and j : Y → X such that d(g ◦ i, i ◦ f ) < δ and, more importantly, d(j ◦ g, f ◦ j ) < δ. Take y ∈ Y and consider the sequence {xn }n∈Z defined by xn = j (g n (y)) for n ∈ Z. Since

d(xn+1 , f (xn )) = d(j (g(g n (y))), f (j (g n (y)))) = d(j ◦ g(g n (y)), f ◦ j (g n (y))) < δ for n ∈ N, the choice of δ and the shadowing property provide a point x ∈ X such that d(f n (x), xn ) ≤ ε. In particular, d(f n (x), xn ) < 2e for all n ∈ Z. Then such an x is unique (by expansivity). Denoting x = h(y), we obtain a map h : Y → X satisfying d(f n (h(y)), j (g n (y))) ≤ ε for y ∈ Y and n ∈ Z. Taking here n = 0 above, we get d(h(y), j (y)) < ε for every y ∈ Y , and so namely, d(h, j ) ≤ ε. Then we have dH (h(Y ), X) ≤ dH (h(Y ), J (Y )) + dH (j (Y ), X) ≤ ε + δ < ε. Moreover, for all y, y  ∈ Y , we have |d(h(y), h(y  )) − d(y, y  )| ≤ |d(h(y), h(y  )) − d(j (y), j (y  ))| + |d(j (y), j (y  )) − d(y, y  )|

2.5 Proof of Theorem 2.5

53

≤ |d(h(y), h(y  )) − d(h(y), j (y  ))| + |d(h(y), j (y  )) − d(j (y), j (y  ))| + δ ≤ d(h(y  ), j (y  )) + d(h(y), j (y)) + δ < 2ε + δ < ε, i.e., h : Y → X is an ε-isometry. On the other hand, since d(f n (h(g(y)), j (g n (g(y))) ≤ ε

and

d(f n (f (h(y))), j (g n (g(y)))) ≤ ε

for all n ∈ Z, we have f (h(y)) = h(g(y)) for all y ∈ Y . Therefore, f ◦ h = h ◦ g. It remains to prove that h is continuous. Fix Δ > 0. Since e is an expansivity constant of f , there is N ∈ N+ such that d(a, b) ≤ Δ whenever d(f n (a), f n (b)) ≤ e for every −N ≤ n ≤ N (see [76]). Since g is continuous and Y compact, we have that g is uniformly continuous. So, there is γ > 0 such that d(g n (y), g n (y  ) ≤ 8ε for all −N ≤ n ≤ N whenever y, y  ∈ Y satisfy d(y, y  ) < γ . Then, whenever d(y, y  ) < γ , we have d(f n (h(y)), f n (h(y  ))) = d(h(g n (y)), h(g n (y  ))) ≤ d(h(g n (y)), j (g n (y))) + d(j (g n (y)), j (g n (y  ))) + d(h(g n (y  )), j (g n (y  ))) ≤ 2ε + δ + d(g n (y), g n (y  )) ≤ 3ε +

ε 0, there is δ > 0 such that for every compact metric space Y and every S ∈ Act(G, Y ) with dGH0 ,A (T , S) ≤ δ, there is a continuous ε-isometry h : Y → X such that Tg ◦ h = h ◦ Sg for every g ∈ G. We first prove that the above notion of topological GH-stability does not depend on finite generating sets. Theorem 2.6 If T ∈ Act(G, X) is topologically GH-stable with respect to some finite generating set of G, then T is topologically GH-stable with respect to any finite generating set of G. Proof Let A and B be finite generators of G. Suppose that T ∈ Act(G, X) is topologically GH-stable with respect to A and take ε > 0. Let δ  > 0 be given by the topological GH-stability of T with respect to A. For this δ  , we choose δ > 0 as in Lemma 1.9. Then if S ∈ Act(G, Y ) for some compact metric space Y and dGH0 ,B (T , S) < δ, then dGH0 ,A (T , S) < δ  by Lemma 1.9. So, there is a continuous ε-isometry h : Y → X such  that Tg ◦ h = h ◦ Sg for every g ∈ G. This completes the proof. This theorem leads to the following definition. Definition 2.7 We say that T ∈ Act(G, X) is topologically GH-stable if T is topologically GH-stable with respect to some finite generating set of G. Let us present two sufficient conditions for a finitely generated group action of a compact metric space to be topologically GH-stable. Given T ∈ Act(G, X), a finite generating set A, and δ > 0, a δ-pseudo orbit of T with respect to A is a sequence {xg }g∈G in X such that d(Ta (xg ), xag ) < δ for all a ∈ A and g ∈ G. We say that {xg }g∈G can be ε-shadowed if there is x ∈ X such that d(Tg (x), xg ) < ε for all g ∈ G. Definition 2.8 ([57]) We say that T has the shadowing property if there is a finite generating set A of G with the following property (called the shadowing property with respect to A): For every ε > 0, there exists δ > 0 such that any δ-pseudo orbit {xg }g∈G with respect to A can be ε-shadowed.

2.6 Gromov-Hausdorff Stability for Group Actions

55

It is well known that this concept does not depend on finite generating sets [63]. The next is the classical definition of expansivity for group actions. Definition 2.9 We say that T ∈ Act(G, X) is expansive if there is c > 0, called an expansivity constant of T , such that if x, y ∈ X and d(Tg x, Tg y) ≤ c for all g ∈ G, then x = y. With these definitions, we can state the following result, generalizing Theorem 2.3. Theorem 2.7 If T is an expansive action with the shadowing property of a finitely generated group on a compact metric space, then T is topologically GH-stable. Proof Let X be a compact metric space and G be a finitely generated group. Let T ∈ Act(G, X) be expansive with the shadowing property. Let η be an expansivity constant of the action T . Fix ε > 0 and take 0 < ε¯ < 18 min{ε, η}. Let A be a finite generating set of G. Choose δ corresponding to ε¯ as in the definition of shadowing property with respect to A, and assume δ < ε¯ . Let Y be a compact metric space and S ∈ Act(G, Y ) with dGH0 ,A (T , S) < δ. Then there are δ-isometries i : X → Y and j : Y → X such that d Y (Sa ◦ i(x), i ◦ Ta (x)) < δ

sup x∈X,a∈A

and

sup d X (j ◦ Sa (y), Ta ◦ j (y)) < δ. y∈Y,a∈A

Choose y ∈ Y and consider the sequence {xg }g∈G ⊂ X defined by xg = j ◦ Sg (y) for all g ∈ G. Since d X (xag , Ta (xg )) = d X (j ◦ Sag (y), Ta ◦ (j ◦ Sg (y))) = d X ((j ◦ Sa )Sg (y), (Ta ◦ j )Sg (y)) < δ for all a ∈ A and g ∈ G, we have that {xg }g∈G ⊂ X is a δ-pseudo orbit of T with respect to A. By the shadowing property of T , there is x ∈ X such that d(Tg (x), xg ) < ε¯ for all g ∈ G. In particular, d(Tg (x), xg ) < ε¯ < η2 for all g ∈ G. Since η is an expansivity constant, we see that such an x is unique. Then, by denoting x = h(y), we obtain a map h : Y → X satisfying d(Tg (h(y)), j (Sg (y))) < ε¯

(2.1)

for all y ∈ Y and g ∈ G. Taking here g = e above, we get d(h(y), j (y)) < ε¯ for all y ∈ Y . Then dH (h(Y ), X) ≤ dH (h(Y ), j (Y )) + dH (j (Y ), X) ≤ ε¯ + δ < 2¯ε < ε.

56

2 Stability

Moreover, for any y, y  ∈ Y , we have |d X (h(y), f (y  )) − d Y (y, y  )| ≤ |d X (h(y), f (y  )) − d X (j (y), j (y  ))| + |d X (j (y), j (y  )) − d Y (y, y  )| ≤ |d X (h(y), h(y  )) − d X (h(y), j (y  ))| + |d X (h(y), j (y  )) − d X (j (y), j (y  ))| + δ ≤ d X (h(y  ), j (y  )) + d X (h(y), j (y)) + δ < 2¯ε + δ < 3¯ε < ε, so h : Y → X is an ε-isometry. Now, given a ∈ A and replacing y by Sa (y) in (2.1), we get d(Tg (h(Sa (y))), j (Sg (Sa (y)))) < ε¯ , and replacing g by ga again in (2.1), we have d(Tg (Ta (h(y))), j (Sg (Sa (y)))) < ε¯ for all g ∈ G. Therefore, d(Tg (Ta (h(y))), Tg (Ta (h(y)))) ≤ d(Tg (h(Sa (y))), j (Sg (Sa (y)))) + d(Tg (Ta (f (y))), j (Sg (Sa (y)))) < 2¯ε < η for all a ∈ A and g ∈ G. Since η is an expansivity constant, h(Sa (y)) = Ta (h(y)) for all y ∈ Y and a ∈ A. Therefore, Ta ◦ h = h ◦ Sa for all a ∈ A. Since A generates G, we have Tg ◦ h = h ◦ Sg for all g ∈ G. It remains to verity that h is continuous. Fix a number Δ > 0. By Lemma 1.10, there exists a finite subset F ⊂ G such that sup d(Tg (x), Tg (x  )) ≤ η implies d X (x, x  ) < Δ. g∈F

Since Sg is uniformly continuous for all g ∈ F , there is δ1 > 0 such that d Y (Sg (y), Sg (y  ))
0, and let δ > 0 be given by the definition of the topological GH-stability of T with respect to A. Since T and T  are isometrically conjugate, dGH0 ,A (T , T  ) = 0 by Item (2) of Lemma 1.7. Now choose S  ∈ Act(G, Y  ) such that dGH0 ,A (T  , S  ) < 2δ . Then, by Item (1) of Lemma 1.7, dGH0 ,A (T , S  ) ≤ 2(dGH0 ,A (T , T  ) + dGH0 ,A (T  , S  )) < δ. So, by the definition of topological GH-stability of T , there is a continuous ε-isometry h : Y  → X such that Tg ◦ h = h ◦ Sg for every g ∈ G. Since T and T  are isometrically

58

2 Stability

conjugate, there is an isometry j : X → X such that Tg = j ◦ Tg ◦ j −1 for every g ∈ G. Replacing above, we get j ◦ Tg ◦ j −1 ◦ h = h ◦ Sg for all g ∈ G. Define l = j −1 ◦ h : Y  → X . Then j ◦ Tg ◦ l = j ◦ l ◦ Sg , so Tg ◦ l = l ◦ Sg for all g ∈ G. Since j is an isometry and h is a continuous ε-isometry, we easily get that l is a continuous  ε-isometry. Therefore, T  is topologically GH-stable with respect to A, as desired.

2.7

Gromov-Hausdorff Stability of Global Attractors

The results of this section will be applied to the Gromov-Hausdorff stability of partial differential equations (Chaps. 5, 6, and 7). Let X be a metric space and S : X × R+ 0 →X be a continuous map. We say that S(t) is a semidynamical system (or a semiflow) on X if S(x, 0) = x and S(S(x, t1 ), t2 ) = S(x, t1 + t2 ) for x ∈ X and t1 , t2 ≥ 0. For simplicity, we denote S(·, t) by S(t). Definition 2.11 We say that a compact invariant set A is the global attractor of S(t) if it attracts all bounded sets, that is, distX (S(t)B, A) → 0 as t → ∞ for any bounded set B ⊂ X, where distX (A, B) = sup inf dX (a, b) a∈A b∈B

for all A, B ⊂ X. Definition 2.12 Let H be a Hilbert space with a norm  · . We say that M is an inertial manifold if it is a finite-dimensional Lipschitz manifold that is invariant under S(t) and attracts all trajectories exponentially, that is dist(S(t)u0 , M) ≤ C(u0 )e−kt for all u0 ∈ H . Let Λ be a topological space and {Sλ : Xλ × R+ 0 → Xλ }λ∈Λ be a family of semidynamical systems on metric spaces {Xλ }λ∈Λ . For each λ0 ∈ Λ, let dλ be a metric on Xλ . Suppose that each system Sλ has a global attractor Aλ . Definition 2.13 We say that the global attractor Aλ (λ ∈ Λ) of the semi-dynamical system Sλ is Gromov-Hausdorff stable with respect to λ if for each λ0 ∈ Λ and ε > 0, there exists a neighborhood Uλ0 of λ0 such that if λ ∈ Uλ0 , then there are ε-isometries i : Aλ → Aλ0 and j : Aλ0 → Aλ , and α ∈ RepAλ (ε) and β ∈ RepAλ (ε) such that for any uλ ∈ Aλ , 0 uλ0 ∈ Aλ0 , and t ∈ [0, 1],

2.7 Gromov-Hausdorff Stability of Global Attractors

59

dλ0 (i(Sλ (uλ , α(uλ , t))) − Sλ0 (i(uλ ), t))) < ε, and dλ (j (Sλ0 (uλ0 , β(uλ0 , t))) − Sλ (j (uλ0 ), t))) < ε. Note that if a semidynamical system S(t) on a metric space X is injective and has the global attractor A, then the restriction of S(t) to A is a dynamical system on A, which will be denoted by (A, φ). The following theorem gives sufficient conditions for the global attractor Aλ of the semidynamical system Sλ to be GH-stable with respect to λ ∈ Λ. Theorem 2.9 Let Λ be a topological space and {Sλ : Xλ × R+ 0 → Xλ }λ∈Λ be a family of semidynamical systems on metric spaces {Xλ }λ∈Λ . Suppose that for each λ ∈ Λ, Sλ has the global attractor Aλ and the restriction on Aλ induces the dynamical systems (φλ , Aλ ) and satisfies the following: (1) Aλ varies continuously with respect to λ ∈ Λ in the Gromov-Hausdorff distance DGH0 , that is, for λ0 ∈ Λ and ε > 0, there exists a neighborhood Uλ0 of λ0 such that for any λ ∈ Uλ0 , DGH0 (Aλ , Aλ0 ) < ε; (2) for any λ0 ∈ Λ, there is a neighborhood Uλ0 of λ0 such that {Sλ }λ∈Uλ0 is equicontinuous on Aλ × [0, 1], that is, for any ε > 0, there exists δ > 0 such that if dXλ (x, y) < δ and |t − s| < δ, then dXλ (Sλ (x, t), Sλ (y, s)) < ε for any λ ∈ Uλ0 , x, y ∈ Aλ , and t, s ∈ [0, 1]; (3) for the ε-isometries i : Aλ → Aλ0 and j : Aλ0 → Aλ obtained in (1), dC 0 (j ◦ i, idAλ ), dC 0 (i ◦ j, idAλ0 ) < ε, and dλ0 (iSλ j (x, t), Sλ0 (x, t)) < ε uniformly for (x, t) ∈ Aλ0 × [0, 1] and λ ∈ Uλ0 . Then the global attractor Aλ of the semidynamical system Sλ is GH-stable with respect to λ ∈ Λ. Proof For any ε > 0, by the assumption (2), we can choose a constant 0 < δ < ˜ < δ and |t − s| < δ, then that if u, u˜ ∈ Aλ and t, s ∈ [0, 1] with dλ (u, u)

ε such 3

dλ (φλ (u, t), φλ (u, ˜ s)) < ε. For λ0 ∈ Λ, by the assumptions (1) and (3), we see that there is a neighborhood Uλ0 of λ0 such that for any λ ∈ Uλ0 , there exist δ-isometries iˆ : Aλ → Aλ0 and jˆ : Aλ0 → Aλ such that

60

2 Stability

ˆ λ jˆ(u, t)), φλ0 (u, t)) < δ dC 0 (j ◦ i, idAλ ), dC 0 (i ◦ j, idAλ0 ) < δ and dλ0 (i(φ for every (u, t) ∈ Aλ0 × [0, 1]. Then we have dλ (φλ (jˆ(u), t), jˆ(φλ0 (u, t))) ˆ λ (jˆ(u), t)), iˆ ◦ jˆ(φλ0 (u, t))) + ≤ dλ0 (i(φ

ε 3

ˆ λ (jˆ(u), t)), φλ0 (u, t)) + dλ0 (φλ0 (u, t), iˆ ◦ jˆ(φλ0 (u, t))) + ≤ dλ (i(φ

ε < ε. 3

Moreover, ˆ λ (u, t)), φλ0 (i(u), ˆ dλ0 (i(φ t)) ˆ λ (u, t)), i(φ ˆ λ (jˆ ◦ i(u), ˆ ˆ λ (jˆ ◦ i(u), ˆ ˆ t))) + dλ0 (i(φ t)), φλ0 (i(u), t)) ≤ dλ0 (i(φ 2ε ˆ t)) + ≤ dλ (φλ (u, t), φλ (jˆ ◦ i(u), < ε. 3

This shows that DGH0 (φλ , φλ0 ) < ε for all λ ∈ Uλ0 , and completes the proof of the theorem.



Remark 2.1 We can derive that if X, Y are compact metric spaces and i : X → Y is an ε-isometry, then there exists a 5ε-isometry j : Y → X such that dC 0 (j ◦ i, idX ), dC 0 (i ◦ j, idY ) < 5ε. We will need the following proposition, which can be considered as a generalization of Theorem 5.2 in [37]. Proposition 2.1 Let Λ be a topological space and {Sλ : Xλ × R+ 0 → Xλ }λ∈Λ be a family of semidynamical systems on metric spaces {Xλ }λ∈Λ . Suppose that for each λ ∈ Λ, (1) Sλ has the global attractor Aλ ; (2) there is a bounded open neighborhood Dλ of Aλ such that for any t ∈ R+ 0 , λ0 ∈ Λ, and ε > 0, there exists a neighborhood W of λ0 such that for any λ ∈ W , there is an ε-immersion iλ : Dλ → Xλ0 such that (i) iλ (Dλ ) ⊂ B(Dλ0 , ε) and (ii) d(iλ (Sλ (x, t)), Sλ0 (iλ (x), t)) < ε for all x ∈ Aλ .

2.7 Gromov-Hausdorff Stability of Global Attractors

61

Then, the map A : Λ → M given by A(λ) = Aλ is residually continuous. Proof Step 1. We first show that for any λ0 ∈ Λ and ε > 0, there exists a neighborhood W of λ0 such that for any λ ∈ W , there is an ε-immersion iλ : Dλ → Xλ0 such that iλ (B(Aλ , ε/4) ∩ Dλ ) ⊂ B(Aλ0 , ε). Since Aλ0 is the global attractor of Sλ0 (t), there is T > 0 such that Sλ0 (B(Dλ0 , ε), T ) ⊂ B(Aλ0 , ε/4). By the assumption (2), there exists a neighborhood W of λ0 such that for λ ∈ W , there exists an ε/4-immersion iλ : Dλ → Xλ0 such that iλ (Dλ ) ⊂ B(Dλ0 , ε/4) and d(iλ (Sλ (x, T )), Sλ0 (iλ (x), T ))
0, we denote by E(ε) the collection of λ0 ∈ Λ such that there is a neighborhood W of λ0 with the following property: for any λ, λ ∈ W , DGH0 (Aλ , Aλ ) <  ε. We observe that n∈N E(1/n) consists of all points of continuity of the map A : Λ → M. To prove that the map A is residually continuous, we show that E(ε) is open and dense in Λ for any ε > 0. It is clear that E(ε) is open in Λ, and so it is enough to show that E(4ε) is dense in Λ for any ε > 0. Let ε > 0 and let U be a nonempty open set in Λ. Fix λ0 ∈ U , and take a finite open cover {Bi }i∈K of Aλ0 in Xλ0 such that diam(Bi ) < ε and Aλ0 ∩ Bi = ∅ for all i ∈ K. Let I be the collection of all subsets J of K such that there are λ1 ∈ U and a δ-immersion iλ1 : Dλ1 → Xλ0 for some δ < ε/2 satisfying iλ1 (Aλ1 ) ∩ Bi = ∅ for all i ∈ J, and

iλ1 (Uλ1 ) ⊂ Bi for an open neighborhood Uλ1 of Aλ1 . i∈J

62

2 Stability

It is clear that I is nonempty since K ∈ I. Since K is finite, we can choose a minimal element J of I. Take λ1 ∈ U and a δ-immersion iλ1 corresponding to J by the definition of I. Let W ⊂ U be a neighborhood of λ1 chosen as in Step 1. For any λ ∈ W , there is an (ε/4 − δ/2)-immersion iλ : Dλ → Xλ1 such that iλ (Uλ ) ⊂ Uλ1 for a neighborhood Uλ of Aλ . It is clear that iλ1 ◦ iλ is an (ε/4 + δ/2)-immersion from Dλ to Xλ1 such that iλ1 ◦ iλ (Uλ ) ⊂ iλ1 (Uλ1 ) ⊂

Bi .

i∈J

By the minimality of J , we have iλ1 ◦ iλ (Aλ ) ∩ Bi = ∅ for all i ∈ J . This implies that DGH 0 (Aλ , Aλ1 ) < 2ε for any λ ∈ W , and so λ1 ∈ E(4ε). This completes the proof of the proposition. 

Exercises Exercise 2.14 Use Exercise 7.4.7 in [14, p. 261] to deduce directly from the definition that every homeomorphism of a finite metric space is topologically GH-stable. Recall that x ∈ X is a periodic point of a map f : X → X if there is a positive integer n such that f n (x) = x. Exercise 2.15 Prove that if a transitive homeomorphism of a compact metric space f : X → X is topologically GH-stable, then the periodic orbits of f are dense in X. Exercise 2.16 Is every topologically stable circle homeomorphism topologically GHstable? Recall that a continuous map f : X → X of a compact metric space is positively expansive if there is ε > 0 such that if x, x  ∈ X and d(f n (x), f n (x  )) ≤ ε for all n ∈ N, then x = y. Moreover, f has the shadowing property if for every ε > 0, there is δ > 0 such that for all sequence {xn }n∈N with d(f (xn ), xn+1 ) ≤ δ for all n ≥ N, there is x ∈ X such that d(f n (x), xn ) ≤ ε for all n ∈ N. Exercise 2.17 Taking into account the remark after Definition 5.2, prove that every positively expansive continuous map with the shadowing property of a compact metric space is topologically GH-stable. Given a map f : X → X of a metric space, a δ-chain is a finite set x0 , . . . , xk such that d (f (xi ), xi+1 ) < δ for all 0 ≤ i ≤ k − 1. A δ-chain from x to y is a δ-chain with x0 = x and xk = y. We write x ∼ y if for any δ > 0 there are δ-chains from x to y and from y to

2.7 Gromov-Hausdorff Stability of Global Attractors

63

x. Define CR(f ), the chain recurrent set, of f , by the rule: as x ∈ CR(f ) if and only if x ∼ x. Recall that Per(f ) denotes the set of periodic points of f . Exercise 2.18 Is CR(f ) = Per(f ) true for every topologically GH-stable homeomorphism of a compact metric space f : X → X?

3

Continuity of the Shift Operator

3.1

Introduction

The relationships between continuous maps f : X → X and their corresponding limit inverse representation σf : Xf → Xf have been studied in the literature. For example, Chen and Li [17] proved that f have the shadowing or asymptotically shadowing properties if and only if σf does. In [12], the author showed equivalences between the ergodic shadowing property of f and of σf . Several other properties were investigated in [43]. More recently, Tsegmid [74] proved that if f has the average shadowing property, then so does σf . The stability properties of f and those of σf have also been investigated. In [78], the orbit shift structural stability of hyperbolic self covering was obtained. Sun [71] defined orbit shift topological stability and proved that this property holds for all surjective Anosov maps (see also [69]). Related results are given in [19] and [18]. In this chapter, we will investigate how σf : Xf → Xf varies with respect to GromovHausdorff perturbations of f : X → X. Indeed, we show that the map f → σf (called the shift operator) is continuous (with respect to such perturbations) at every topologically Anosov map f . We apply this result to the stability theory of topological dynamical systems. Let us present these results in detail. Let Csur denote the set of surjective continuous maps of compact metric spaces f : X → X. A special subset of Csur is the set of homeomorphisms f : X → X, denoted by H . Let Csur denote the corresponding set of equivalence classes [f ] for f ∈ Csur and, correspondingly, let H ⊂ Csur denote the set of equivalence classes corresponding to elements f ∈ H . Then dGH0 induces a quasi-metric (and thus a topology) on Csur . All continuity properties for maps defined in Csur will refer to this topology.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Lee, C. Morales, Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-12031-2_3

65

66

3 Continuity of the Shift Operator

Let us define the shift operator σ : Csur → H. Let X be a compact metric space. Denote by XZ the set of bi-sequences ξ = {ξi }i∈Z with ξi ∈ X for every i ∈ Z. It follows that XZ is compact if endowed with the metric D(ξ, ξ¯ ) =



2−|i| d(ξi , ξ¯i ).

i∈Z

Any surjective map f : X → X induces a map f˜ : XZ → XZ given by f˜(ξ ) = (f (ξi ))i∈Z . We will see in Remark 3.1 that the map T : Csur → C defined by T ([f ]) = [f˜] is well-defined and continuous with respect to the topology induced by the GromovHausdorff distance. Now define the inverse limit space of f by Xf = {ξ ∈ XZ : f (ξi ) = ξi+1 for all i ∈ Z}. It follows that Xf is a compact subset of XZ and also invariant, that is, f˜(Xf ) = Xf . The restriction σf = f˜|Xf : Xf → Xf is called the shift map associated to f . Notice that if f is continuous, then σf : Xf → Xf is a homeomorphism [3]. In particular, σf ∈ H for all f ∈ Csur . We will see in the next section that the map σ : Csur → H defined by σ ([f ]) = [σf ] for all [f ] ∈ Csur is well-defined. We call this map the shift operator. In the sequel, we give sufficient conditions for its continuity at [f ] ∈ Csur . For simplicity, we write f instead of [f ]. A continuous surjective map f : X → X of a compact metric space is c-expansive if there is δ > 0 such that if ξ, ξ  ∈ Xf and d(ξi , ξi ) ≤ δ for every i ∈ Z, then ξ = ξ  . Given δ > 0, a δ-pseudo orbit of f is a sequence {xi }∞ i=1 such that d(f (xi ), xi+1 ) ≤ δ for every non-negative integer i. If ε > 0, then we say that (xi )∞ i−0 can be ε-shadowed if there is x ∈ X such that d(f i (x), xi ) ≤ ε for every non-negative integer i. We say that f has the shadowing property if for every ε > 0, there is δ > 0 such that every δ-pseudo orbit can be ε-shadowed. Definition 3.1 ([3]) A surjective map f : X → X is topologically Anosov if it is cexpansive and has the shadowing property. It is well known that the property of being topologically Anosov is invariant under topological (and so under isometric) conjugacy [3]. Consequently the property of being topologically Anosov is well-defined in Csur . Our main result is the following.

3.2 Preliminary Facts

67

Theorem 3.1 The shift operator is continuous at every topologically Anosov map of a compact metric space with positive diameter. The rest of the chapter is organized as follows. In Sect. 3.2, we give some preliminaries. In Sect. 3.3, we prove Theorem 3.1. In Sect. 3.4, we apply this theorem to the stability theory of dynamical systems.

3.2

Preliminary Facts

To start, we prove that the shift operator is well defined. Lemma 3.1 If f, g ∈ Csur are isometrically conjugate, then so are σf and σg . Proof Let h : Y → X be an isometry such that f ◦ h = h ◦ g. Define H : Y Z → XZ by H (ξ )i = h(ξi ) for ξ ∈ Y Z and i ∈ Z. Since h is an isometry, D(H (ξ ), H (ξ  )) =



2−|i| d(h(ξi ), h(ξi )) =

i∈Z



d(ξi , ξi ) = D(ξ, ξ  ),

i∈Z

for all ξ, ξ  ∈ Y Z . Then H is an isometric immersion (hence, continuous). On the other hand, if ξ ∈ Yg , that is, g(ξi ) = ξi+1 for i ∈ Z, then f h(ξi ) = hg(ξi ) = h(ξi+1 ). Thus, H (ξ ) ∈ Xf , yielding an isometric immersion H : Yg → Xf . Given η ∈ Xf , we can define ξ ∈ Y Z by ξi = h−1 (ηi ) for i ∈ Z. Clearly, H (ξ ) = η. Since g(ξi ) = gh−1 (ηi ) = h−1 f (ηi ) = h−1 (ηi+1 ), we have ξ ∈ Yg , proving that H is onto (hence, an isometry from Yg to Xf ).

68

3 Continuity of the Shift Operator

Finally, for every ξ ∈ Yg and i ∈ Z, we have σf (H (ξ ))i = h(ξi+1 ) = H (σg (ξ ))i , proving σf ◦ H = H ◦ σg . This completes the proof.



Part of the above proof can be used to prove the following remark. Remark 3.1 The map T : Csur → C defined by T (f ) = f˜ is both well defined and continuous with respect to the topology induced by the Gromov-Hausdorff distance. Proof We can see as above that if f, g ∈ C are isometrically conjugate, then so are f˜ and g. ˜ This implies that T is well defined. Now fix ε. For a suitable δ > 0, we assume that f, g ∈ Csur satisfy dGH0 (f, g) < δ. Then there are δ-isometries i : X → Y and j : Y → X such that d(g ◦ i, i ◦ f ) < δ

and

d(j ◦ g, f ◦ j ) < δ.

Define the maps I : XZ → Y Z and J : Y Z → XZ by I (ξ )k = i(ξk )

and

J (η)k = j (ηk )

for (k, ξ, η) ∈ Z × XZ × Y Z . Following the proof of the lemma above, we can see that for δ small, the maps I and J are ε-isometries satisfying D(g˜ ◦ I, I ◦ f˜) < ε

and

D(J ◦ g, ˜ f˜ ◦ J ) < ε.

Hence, (f˜, g) ˜ < ε and the proof follows.



Remark 3.2 The above remark suggests at once that the shift operator σ : Csur → H at every f ∈ Csur should be continuous (this obviously implies Theorem 3.1). However, the proof above breaks in particular when trying to prove the inclusions I (Xf ) ⊂ Yg and J (Yg ) ⊂ Xf . We define δ-isometries in a different manner when f is topologically Anosov (see Eq. (3.1) in the proof of Theorem 3.1). The following lemma permits a reduction of the topology induced by dGH0 in C. Lemma 3.2 Let f : X → X be a continuous map of a compact metric space. Then for every Δ > 0, there exists Δ > 0 such that if g : Y → Y is a continuous map of a compact metric space satisfying d(j ◦ g, f ◦ j ) ≤ Δ for some Δ -isometry j : Y → X, then dGH0 (f, g) ≤ Δ.

3.3 Proof of Theorem 3.1

69

Proof Fix Δ > 0. Take β > 0 such that d(a, b) ≤ β for a, b ∈ X implies d(f (a), f (b)) ≤ Δ4 . Now fix 0 < Δ < min{ Δ4 , β}. Let g : Y → Y be a continuous map of a compact metric space such that d(j ◦ g, f ◦ j ) ≤ Δ for some Δ -isometry j : Y → X. Then, by Lemma 1.6, there is a 3Δ -isometry i : X → Y such that d(j ◦ i(x), x) < Δ , for all x ∈ X. Since Δ < Δ, i is a Δ-isometry. Then, since d(g ◦ i(x), i ◦ f (x)) < Δ + d(j ◦ g ◦ i(x), j ◦ i ◦ f (x)) ≤ Δ + d((j ◦ g)(i(x)), (f ◦ j )(i(x))) + d(f ◦ j ◦ i(x), j ◦ i ◦ f (x)) < 2Δ + d(f (j ◦ i(x)), f (x)) + d(f (x), j ◦ i(f (x))) < 3Δ + d(f (j ◦ i(x)), f (x)) and d(j ◦ i(x), x) < Δ < β for every x ∈ X, we get d(g ◦ i(x), i ◦ f (x)) ≤ 3Δ +

3Δ Δ Δ < + =Δ 4 4 4

for all x ∈ X, proving d(g ◦ i, i ◦ f ) ≤ Δ. Again, since Δ < Δ, j is a Δ-isometry and  d(j ◦ g, f ◦ j ) ≤ Δ. Then dGH0 (f, g) ≤ Δ, completing the proof. The last lemma is Theorem 2.3.7 in [3, p. 81], restated here for the reader’s convenience. Lemma 3.3 A continuous surjection f : X → X of a compact metric space has the shadowing property if and only if for every ε > 0, there is δ > 0 such that for every ξˆ ∈ XZ with d(ξˆi , ξˆi+1 ) < δ (for i ∈ Z), there is ξ ∈ Xf such that d(ξi , ξˆi ) < ε for every i ∈ Z.

3.3

Proof of Theorem 3.1

In this section, we will prove Theorem 3.1. We have to show that if f : X → X is a surjective c-expansive map with the shadowing property of a compact metric space with diameter diam(X) > 0, then for every Δ > 0, there is δ > 0 such that if a surjective continuous map g : Y → Y of a compact metric space satisfies dGH0 (f, g) ≤ δ, then dGH0 (σf , σg ) ≤ Δ.

70

3 Continuity of the Shift Operator

By Lemma 3.2, it suffices to show that for every Δ > 0, there is δ > 0 such that if dGH0 (f, g) ≤ δ, then there is a Δ -isometry J : Yg → Xf such that D(σf ◦J, J ◦σg ) ≤ Δ . We proceed as follows. Let e be a c-expansivity constant of f , fix Δ > 0, and take 0 < ε
0, there is δ > 0 such that for every surjective continuous map g : Y → Y of a compact metric space satisfying dGH0 (f, g) < δ, there is a continuous ε-isometry H : Yg → Xf such that σf ◦ H = H ◦ σg . With this definition, we obtain the following result. Theorem 3.2 Every topologically Anosov map of a compact metric space with positive diameter is orbit shift topologically GH-stable. Proof Let f be a topologically Anosov map of a compact metric space with positive diameter. It follows from Theorems 2.2.29 and 2.3.8 in [3] that σf : Xf → Xf is expansive and has the shadowing property. Then σf is topologically GH-stable by Theorem 2.3. Now take ε > 0 and let Δ > 0 be given by the topological GH-stability of σf . For this Δ, we choose δ from Theorem 3.1. Then if g : Y → Y is a surjective continuous map of a compact metric space and dGH0 (f, g) ≤ δ, then dGH0 (σf , σg ) ≤ Δ. Thus, by the topological GH-stability of f , there is a continuous ε-isometry H : Yg → Xf such that  σf ◦ H = H ◦ σg . This completes the proof.

3.4 Application to Stability Theory

75

As applications, we will present two examples. The first one is based on the following definition. Given a continuous map f : X → X, ε > 0, and ξ ∈ Xf , we define Wεs (ξ ) = {z0 ∈ X : d(f n (ξ0 ), f n (z0 )) = d(ξn , f n (z0 )) ≤ ε for all n ≥ 0} and Wεu (ξ ) = {z0 ∈ X : there exists η ∈ Xf such that π(η) = z0 and d(x−n , η−n ) ≤ ε for all n ≥ 0}. Definition 3.4 A continuous map f : X → X is Anosov [69] if there is c > 0 such that for every 0 < ε < c, there is δ > 0 such that the intersection Wεs (ξ ) ∩ Wεu (η) consists of exactly one point for every ξ, η ∈ Xf with d(ξ0 , η0 ) ≤ δ. Sun [71] proved that every surjective Anosov map of a compact metric space is orbit shift topologically stable. This motivates the following result. Theorem 3.3 Every surjective Anosov map of a compact metric space is orbit shift topologically GH-stable. Proof Let f : X → X be a surjective continuous map of a compact metric space. If f is Anosov, then f is surjective and c-expansive with the shadowing property by Lemma 1 in [72, p. 170]. Therefore, f is topologically Anosov and thus is orbit shift topologically  GH-stable by Theorem 3.2. For the next result, we need the following lemma. Lemma 3.4 If h : Y → X is a continuous map of compact metric spaces, then the map H : Y Z → XZ defined by H (ξ )i = h(ξi ) for i ∈ Z and ξ ∈ Y Z is continuous. Proof Fix ρ > 0 and take N ≥ 0 such that ∞ 

2−i ≤

i=N

ρ . 4 diam(X)

Fix γ > 0 such that if ξ−N +1 , ξ−N +2 , ξ1 , . . . , ξN −1 , η−N +1 , η−N +2 , y1 , . . . , ηN −1 ∈ Y and N −1  i=−N +1

2−|i| d(ξi , ηi ) ≤ γ ,

76

3 Continuity of the Shift Operator

then N −1 

2−|i| d(h(ξi ), h(ηi )) ≤

i=−N +1

ρ . 2

Now, suppose that ξ, η ∈ Y Z satisfy D(ξ, η) ≤ γ . Then N −1 

2−|i| d(ξi , ηi ) ≤ D(ξ, η) ≤ ρ,

i=−N +1

and so, N −1 

2−|i| d(h(ξi ), h(ηi )) ≤

i=−N +1

ρ . 2

It follows that D(H (ξ ), H (η)) =

N −1 

2−|i| d(h(ξi ), h(ηi )) +

i=−N +1

+

N 

∞ 

2−i d(h(ξi ), h(ηi ))

i=N

2i d(h(ξi ), h(ηi ))

i=−∞ ∞

≤

 ρ + 2 diam(X) 2−(i+1) = ρ. 2 i=N



Hence, H is continuous. We can now prove.

Theorem 3.4 Let f : X → X be a surjective continuous map of a compact metric space. If f is topologically GH-stable, then f is orbit shift topologically GH-stable. Proof Recall that by the diameter of a subset A ⊂ X is diam(A) = sup{d(a, a  ) : a, a  ∈ A}. Since X has more than one point, diam(X) > 0. Fix ε > 0 and take N ∈ N such that  |i|≥N +1

2−|i| ≤

ε . 3 diam(X)

3.4 Application to Stability Theory

For this N , we choose 0 < β < d(a, b) ≤ β for a, b ∈ X

ε 3

77

such that implies

d(f k (a), f k (b)) ≤

2ε for 0 ≤ k ≤ 2N. 9

For this β, we fix δ > 0 from the topological GH-stability of f . Now take a surjective continuous map g : Y → Y of a compact metric space such that dGH0 (f, g) ≤ δ. Then there is a continuous β-isometry h : Y → X such that f ◦ h = h ◦ g. Take H : Y Z → XZ as in Lemma 3.4 for this h. Then H is continuous. If ξ ∈ Yg , then f (h(ξi )) = h(g(ξi )) = h(ξi+1 ) for every i ∈ Z. So, H (ξ ) ∈ Xf , proving H (Yg ) ⊂ Xf . Therefore, we obtain a continuous map H : Yg → Xf . Since σf (H (ξ ))i = H (ξ )i+1 = h(ξi+1 ) = h(σg (ξ )i ) = H (σg (ξ ))i for i ∈ Z and ξ ∈ Yg , one has that σ (H (ξ )) = H (σg (ξ )) for ξ ∈ Yg , proving σf ◦ H = H ◦ σg . It remains to verify that H is an ε-isometry. If ξ, ξ  ∈ Yg , then D(H (ξ ), H (ξ  )) =



2−|i| d(H (ξ )i , H (ξ  )i )

i∈Z

=



2−|i| d(h(ξi ), h(ξi ))

i∈Z

≤ 3β + D(ξ, ξ  ) ≤ ε + D(ξ, ξ  ), proving that D(H (ξ ), H (ξ  )) − D(ξ, ξ  ) ≤ ε. Interchanging the roles of H (ξ ) (resp., H (ξ  )) and ξ (resp., ξ  ) in this argument, we get D(ξ, ξ  ) − D(H (ξ ), H (ξ  )) ≤ ε (see the proof of Lemma 3.1). Therefore, sup |D(H (ξ ), H (ξ  )) − D(ξ, ξ  )| ≤ ε.

(3.4)

ξ,ξ  ∈Yg

Now take η ∈ Xf . Since h is a β-isometry, there is y ∈ Y such that d(h(y), η−N ) ≤ β. Define ξ−N +k = g k (y) for k ≥ 0. This yields a sequence {ξi }i≥−N . Since g is surjective, we can extend this sequence for i < −N to get ξ ∈ Yg . It follows that d(h(ξi ), ηi ) = d(f i+N (h(y)), f i+N (η−N )) ≤

2ε 9

78

3 Continuity of the Shift Operator

for any −N ≤ i ≤ N. Then D(H (ξ ), η) =

 i∈Z

2−|i| d(H (ξ )i , ηi ) ⎛

≤ diam(X) ⎝

 |i|≥N +1

≤

⎞ 2−|i| ⎠ +

N 

2−|i| d(h(ξi ), ηi )

i=−N

2ε ε + · 3 = ε. 3 9

Therefore, DH (H (Yg ), Xf ) ≤ ε, which, together with (3.4), proves that H is an εisometry, as needed. 

Exercises Exercise 3.5 Prove that the tent map f (x) = max{2x, 1 − 2x} for x ∈ [0, 1] is neither topologically GH-stable, nor orbit shift topologically GH-stable. Exercise 3.6 Is it true that if f : X → X is a homeomorphism of a compact metric space, then f is orbit shift topologically GH-stable if and only if it is topologically GH-stable? Exercise 3.7 Is Theorem 3.1 true when diam(X) = 0, i.e., when X is a one-point set?

Shadowing from the Gromov-Hausdorff Viewpoint

4.1

Introduction

When simulating a given system, it is important to know under which conditions approximated trajectories may be tracked by real ones. If this is the case for all approximated trajectories, then we say that the system has the shadowing property. This property was discovered by Bowen [62], who proved it for hyperbolic systems. Since then, several generalizations were proposed. In this section, we introduce one more scale generalization treat takes into account Gromov-Hausdorff perturbations of the underlying systems. This leads two kinds of shadowing, namely, the Gromov-Hausdorff and the weak Gromov-Hausdorff shadowing properties. As the names indicate, every system with the Gromov-Hausdorff shadowing property has the weak Gromov-Hausdorff shadowing property. We show that every topologically GH-stable system has the weak Gromov-Hausdorff shadowing property, that the shadowing property implies the Gromov-Hausdorff shadowing property, and finally that every expansive system with the weak Gromov-Hausdorff shadowing property is topologically GH-stable.

4.2

Definitions, Statement of Main Results, and Proofs

Hereafter, X will denote a compact metric space with more than one point. The definition of shadowing property motivates the following one. We say that a homeomorphism f : X → X of a compact metric space has the Gromov-Hausdorff shadowing property (or GH- shadowing property for short) if for every ε > 0, there is δ∗ > 0 such that for every 0 < δ < δ∗ and every homeomorphism g : Y → Y of a compact metric space with

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Lee, C. Morales, Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-12031-2_4

79

4

80

4 Shadowing from the Gromov-Hausdorff Viewpoint

dGH0 (f, g) ≤ δ, there is a δ-isometry j : Y → X such that for every δ-pseudo orbit {yn }n∈Z of g, there is x ∈ X such that d(f n (x), j (yn )) ≤ ε for every n ∈ Z. In the recent paper [9], the following variation of shadowing was proposed: We say that f has the weak shadowing property if for every ε > 0, there is δ > 0 such that for every homeomorphism g : X → X with d(f, g) ≤ δ and every y ∈ X, there is x ∈ X such that d(f n (x), g n (y)) ≤ ε for every n ∈ Z. This definition motivates the following concept. Definition 4.1 We say that a homeomorphism f : X → X of a compact metric space has the weak Gromov-Hausdorff shadowing property (or the weak GH-shadowing property for short) if for every ε > 0, there is δ∗ > 0 such that for every 0 < δ < δ∗ and every homeomorphism g : Y → Y of a compact metric space with dGH0 (f, g) ≤ δ, there is a δ-isometry j : Y → X such that for every y ∈ Y , there is x ∈ X such that d(f n (x), j (g n (y))) ≤ ε for every n ∈ Z. The difference between GH-shadowing and weak GH-shadowing is that the former deals with general δ-pseudo orbits (yn )n∈Z , while the latter deals with true orbits (g n (y))n∈Z . In particular, every homeomorphism with the GH-shadowing property has the weak GH-shadowing property. We shall prove the following facts for every homeomorphism f : X → X of a compact metric space. Proposition 4.1 If f has the shadowing property, then f has the GH-shadowing property. 

Proof Let ε > 0 and δ  > 0 be given by the shadowing property of f for this ε. Set δ∗ = δ3 and take 0 < δ < δ∗ . If g : Y → Y is a homeomorphism of a compact metric space such that dGH0 (f, g) ≤ δ, then there is a δ-isometry j : Y → X such that d(f ◦ j, j ◦ g) ≤ δ. Let (yn )n∈Z be any δ-pseudo orbit of g. Since for every n ∈ Z, d(f (j (yn )), j (yn+1 )) ≤ d(f ◦ j (yn ), j ◦ g(yn )) + d(j (g(yn )), j (yn+1 )) ≤ d(f ◦ j, j ◦ g) + δ + d(g(yn ), yn+1 ) ≤ 3δ < 3δ∗ = δ  , (j (yn ))n∈Z is a δ  -pseudo orbit of f . Then there is x ∈ X such that d(f n (x), j (yn )) ≤ ε  for every n ∈ Z. This ends the proof. It is natural to ask if the converse of Proposition 4.1 holds. Question 4.1 Does the GH-shadowing property imply the shadowing property? A partial answer we provide uses the following auxiliary shadowing concept:

4.2 Definitions, Statement of Main Results, and Proofs

81

Definition 4.2 We say that f has the isometric shadowing property if for every ε > 0, there is δ∗ > 0 such that for every 0 < δ < δ∗ , there is a δ-isometry j : X → X such that for every δ-pseudo orbit (xn )n∈Z of f , there is x ∈ X such that d(f n (x), j (xn )) ≤ ε for every n ∈ Z. Proposition 4.2 If f has the GH-shadowing property, then f has the isometric shadowing property. Proof Given ε > 0, we take δ∗ as provided by the GH-shadowing property. If 0 < δ < δ∗ , then we can take g = f to get dGH0 (f, g) = 0 < δ. So, there is a δ-isometry j : X → X, as desired.  The following fact is easy to prove. Proposition 4.3 If f has the shadowing property, then f has the isometric shadowing property. Recall that a homeomorphism f : X → X is expansive if there is e > 0 (called an expansivity constant) such that if x, y ∈ X and d(f n (x), f n (y)) ≤ e for every n ∈ Z, then x = y. Proposition 4.4 If f is expansive and has the weak GH-shadowing property, then f is topologically GH-stable. Proof Let e be an expansivity constant of f . Take ε > 0 and let 0 < ε < 18 min{e, ε}. For this ε , we let δ∗ be given by the weak GH-shadowing property of f . Take 0 < δ < 1  3 min{δ∗ , ε } and a homeomorphism g : Y → Y of a compact metric space such that dGH0 (f, g) ≤ δ. Let j : Y → X be the δ-isometry given by the weak GH-shadowing property for this g. Then for y ∈ Y , there is x ∈ X such that d(f n (x), j (g n (y))) ≤ ε for all n ∈ Z. This x is unique: if there is an x  different from x, then d(f n (x), f n (x  )) ≤ d(f n (x), j (g n (y))) + d(f n (x  ), j (g n (y))) ≤ 2ε < e, and so x = x  by expansivity. From this uniqueness, we get a map h : Y → X such that d(f n (h(y)), j (g n (y))) ≤ ε

(4.1)

82

4 Shadowing from the Gromov-Hausdorff Viewpoint

for y ∈ Y and n ∈ Z. Taking here n = 1, we get d(h(y), j (y)) ≤ ε . Therefore, d(h(y), h(y  )) ≤ d(h(y), j (y)) + δ + d(y, y  ) + d(j (y  ), h(y  )) < ε + d(y, y  ) and also, d(y, y  ) < δ + d(j (y), h(y)) + d(h(y), h(y  )) + d(j (y  ), h(y  )) < ε + d(h(y), h(y  )) for all y, y  ∈ Y . This implies that sup |d(h(y), h(y  )) − d(y, y  )| < ε.

y,y  ∈Y

Moreover, for x ∈ X there is y ∈ Y such that d(j (y), x) < δ. So, d(h(y), x) ≤ d(h(y), j (y)) + d(j (y), x) ≤ ε + δ < 2ε < ε. Hence, dH (h(Y ), X) < ε, proving that h is an ε-isometry. By replacing n by n + 1 and y by g(y) in (4.1), we obtain d(f n (f (h(y)), j (g n+1 (y)) ≤ ε and d(f n (g(h(y)), j (g n+1 (y))) ≤ ε for all n ∈ Z, respectively. It follows that d(f n (f (h(y)), f n (h(g(y))) ≤ 2ε < e for all n ∈ Z. Since e is an expansivity constant of f , we get f (h(y)) = g(h(y)) for every y ∈ Y , proving f ◦ h = h ◦ g. Finally, we show that h is continuous. Fix Δ > 0. Since e is an expansivity constant of f , there is N ∈ N such that a, b ∈ X and d(f n (a), f n (b)) ≤ e for − N ≤ n ≤ N

implies

d(a, b) ≤ Δ

4.2 Definitions, Statement of Main Results, and Proofs

83

(cf. [76]). Since g is continuous and Y compact, there is γ > 0 such that d(y, y  ) ≤ γ for y, y  ∈ Y

implies

d(g n (y), g n (y  )) ≤

e for − N ≤ n ≤ N. 3

Then if y, y  ∈ Y and d(y, y  ) ≤ γ , we have d(f n (h(y)), f n (h(y  ))) = d(h(g n (y)), h(g n (y  ))) ≤ d(h(g n (y)), j (g n (y))) + d(j (g n (y)), j (g n (y  )))+ d(h(g n (y  )), j (g n (y  ))) < 2ε + δ + d(g n (y), g n (y  )) < 3ε +

10e e < 0. Take δ∗ from the isometric shadowing property of f for ε = diam(F 4 1 0 < δ < 4 min{δ∗ , ε}. For this δ, take the δ-isometry j : X → X from the isometric shadowing property of Id. Since F is compact, there are points x, y ∈ F such that d(x, y) = diam(F ). Since F is connected, there exists a sequence x = x0 , x1 , . . . , xN = y such that d(xn , xn+1 ) < δ for every 0 ≤ n ≤ N − 1. By setting xn = x for n ≤ 0 and xn = y for n ≥ N, we get a δ-pseudo orbit (xn )n∈Z of Id. It follows that there is x ∈ X such that d(x, j (xn )) ≤ ε for every n ∈ Z. Then, diam(F ) = d(x, y) < δ + d(j (x0 ), j (xN )) ≤ δ + d(x, j (x0 )) + d(x, j (xN ))

84

4 Shadowing from the Gromov-Hausdorff Viewpoint

diam(F ) + 2ε 4 3 diam(F ) ≤ < diam(F ), 4

≤

a contradiction, which completes the proof.



From this fact, we get the following. Proposition 4.6 The identity Id : X → X has the GH-shadowing property if and only if X is totally disconnected. Recall that a homeomorphism f : X → X is minimal if the orbit Of (x) = (f n (x))n∈Z is dense in X for every x ∈ X. Proposition 4.7 If f : X → X is a minimal homeomorphism of a compact connected metric space with more than one point, then f has neither the GH-shadowing property, nor the isometric shadowing property. Proof It suffices to show that f does not have the isometric shadowing property. Suppose, by contradiction, that it does. Since X is not a point, diam(X) > 0. Let δ∗ be given by the isometric shadowing property of f for ε = diam(X) . Take 0 < δ < 4 1 min{δ , ε} and the corresponding δ-isometry j : X → X. ∗ 4 Take any x0 ∈ X. Since f is minimal, there is k ∈ N such that f k (x0 ) ∈ B(x0 , δ). Defining xmk+i = f i (x0 ) for m ∈ Z and 0 ≤ i < k, we obtain a δ-pseudo orbit (xn )n∈Z of f . Then, by isometric shadowing, there is x ∈ X such that d(f n (x), j (xn )) ≤ ε for all n ∈ Z. Taking here n = mk for m ∈ Z, we obtain d((f k )m (x), j (x0 )) ≤ ε for every m ∈ Z. So, Of k (x) ⊂ B[x0 , ε].

4.2 Definitions, Statement of Main Results, and Proofs

85

But f is minimal and X is connected, so f k is minimal. Then we have X = Of k (x) and X ⊂ B[x0 , ε]. Thus, diam(X) ≤ 2ε =

diam(X) 2

a contradiction, the proof in complete.



The following property can be proved as in the shadowing property case [3]. Proposition 4.8 f has the GH-shadowing (resp., isometric shadowing) property if and only if f k has such a property for some k ∈ N. We use it in the following example. Example 4.2 Proposition 4.7 implies that irrational rotations of the circle have neither the GH-shadowing property, nor the isometric shadowing property. Since f k = I for some k ∈ N when f is a rational rotation of the circle. Propositions 1.4 and 4.8 imply that rational rotations of the circle do not have these properties.

Exercises The following exercise is motivated by the notion of persistent homeomorphism introduced by Lewowicz [50]. Exercise 4.3 Find an example of a GH-persistent homeomorphism, that is, a homeomorphism f : X → X of a compact metric space such that for every ε > 0, there is δ∗ > 0 such that for every homeomorphism g : X → X of a compact metric space with dGH0 (f, g) ≤ δ for some 0 < δ < δ∗ , there is a δ-isometry j : Y → X such that for every x ∈ X, there is y ∈ Y such that d(f n (x), j (g n (y))) ≤ ε for every n ∈ Z. Exercise 4.4 (True or false) An equicontinuous (Exercise 1.30) homeomorphism of a compact metric space X has the GH-shadowing property if and only if X is totally disconnected.

Part II Applications to PDEs

Introduction

In the theory as well as in the applications of differential equations in nonlinear science, and especially in engineering and mathematical physics, finding exact solutions used to be one of the major concerns. Exact solutions can be used to analyze the qualitative properties of the equations encountered in such applications. However, in general, finding exact solutions of equations that describes complicated natural phenomena is difficult and often impossible. Even when exact solutions can be found, extracting from them qualitative information on solution curves, such as their asymptotic behavior or the existence of invariant manifolds, etc, is not an easy task. In the geometric or qualitative theory of differential equations, the main goal is to describe the geometry of solution curves. At the end of the nineteenth century, Poincaré introduced a new method to understand the long-time behavior of solution curves of ordinary differential equations without the need to find exact solutions of the equations under study. Precisely, he used one map to express the collection of all solutions of an equation, map called the dynamical system or the flow generated by the given equation, and investigated the existence of special solutions such as equilibrium points, periodic solutions, recurrent solutions. Since the early 1980s and to the 2000s, many authors, among them Chepyzhov, Glendinning, Hale, Henry, Sell, Robinson, Temam, Vishik, Bo You, initiated and deepened the study of the qualitative properties of solutions of partial differential equations, in a setting that today is called the geometric theory of partial differential equations [20, 34, 64, 65, 73]. In this theory questions concerning stability play an important role. There are several ways to discuss and characterize the stability of solutions of differential equations. The most useful tools in this respect employ objects called global attractors and inertial manifolds. A global attractor is a compact invariant set which attracts all solution curves; an inertial manifold is an invariant manifold (non-compact, in general) which attracts all solution curves exponentially. Often a given partial differential equation can be transformed into an ordinary differential equation in a Banach space. The 89

90

Introduction

importance of global attractors and inertial manifolds is that the study of the asymptotic behaviors of solutions of an ordinary differential equation in an infinite-dimensional space can be reduced to a study in a finite-dimensional setting. Here we are interested in studying the behavior of global attractor and inertial manifold of reaction-diffusion equations with respect to perturbations of the domain and of the equation (compare with perturbation theory of linear operators [41]). Recently a lot of interesting results in this direction have been obtained by many researchers (see for example [4, 6, 11, 51]). Let us consider a reaction-diffusion equation of the form ⎧ ⎨∂ u − Δu = f (u) t ⎩u = 0

in Ω × (0, ∞), on ∂Ω × (0, ∞).

(1)

For the problem of domain perturbation, one of the difficulties is that the phase space of the induced dynamical system changes as we change the domain. In fact, the phase spaces H01 (Ω) and H01 (Ωε ) which contain global attractors A and Aε , respectively, can be disjoint even if Ωε is a small perturbation of Ω ⊂ Rn . To overcome this difficulty, many people used the technique adopted their by Henry in [34, 36] which makes it possible to consider the problem of continuity of the attractors in the fixed phase space H 1 (Ω0 ) as Ωε → Ω0 (e.g., see [11, 60]). More precisely, they followed the general approach, which basically consists in “pull-backing” the perturbed problems to the fixed domain Ω0 and then considering the family of abstract semilinear problems thus generated. Arrieta and Carvalho [6] applied another method, which uses extended phase spaces to compare attractors living in different phase spaces. More precisely, they considered the extended phase space Hε1 := H 1 (Ω ∩ Ωε ) ⊕ H 1 (Ω \ Ωε ) ⊕ H 1 (Ωε \ Ω) with the norm  · Hε1 . However, the norms  · Hε1 and  · H 1 on the extended spaces Hε1 δ

and Hδ1 , respectively, are in general not comparable. In this part, we will use the Gromov-Hausdorff distance between two global attractors and between two inertial manifolds which belong to disjoint phase spaces to derive their continuity and the Gromov-Hausdorff stability under perturbations of the domain and equation. The notion of Gromov-Hausdorff distance defined on the collection of metric spaces was first introduced by Edwards [27] and was developed by Fukaya et al. [16, 30] to study the convergence and collapsing of Riemannian manifolds. Arbieto and Morales [5] investigated the stability of expansive maps with the shadowing property under the Gromov-Hausdorff topology. We note that pull-backing or domain extension methods do not consider an appropriate topology to compare two global attractors in different phase spaces. Moreover, these methods do not provide information on the change of the dynamics of solution curves

Introduction

91

inside the global attractor under the perturbations being considered. The GromovHausdorff distance induces a topology on the collection of all metric spaces up to isometry, and as such it gives us a strong tool to compare our objects in different phase spaces. Moreover, this distance applied to dynamical systems allows us to get information on the internal dynamics under perturbations of the domain. To study the problem of perturbation of the equation, many researchers transform Eq. (1) into an ordinary differential equation on a Banach space, e.g., Lebesgue space Lp , Sobolev space like W 1,p or fractional Sobolev space. For example, consider the transformed equation on Banach space L2 (Ω) ut + Au = F (u) for u ∈ L2 (Ω)

(2)

where A is −Δ with Dirichlet boundary condition, and F is the Nemytskii operator of f . Many researchers assumed that Eq. (2) induces the C 1 dynamical systems and applied the methods which have been developed in the theory of differential dynamical systems such as hyperbolicity, transversality and Morse-Smale systems, to study the internal structure of solutions of Eq. (2). However when dealing with evolution equations in Lp spaces, there are no differentiable functions of Nemytskii type from Lp into itself other than affine functions. We note that the theory of stability so far only allows us to consider differentiable dynamical systems with C 1 perturbations which is far from being a reality in Lp spaces. The content of this part is as follows. In Chap. 5, we study the residual continuity of the global attractors with respect to the Gromov-Hausdorff topology and their dynamics for reaction-diffusion equations on finite-dimensional domains. In Chap. 6, we analyze the Gromov-Hausdorff stability and continuous dependence of the inertial manifolds under perturbations of the domain and of the equation. Finally, in Chap. 7, we investigate the geometric theory of the Chafee-Infante equations under Lipschitz perturbations.

5

GH-Stability of Reaction-Diffusion Equations

5.1

Introduction

In this chapter, we use the Gromov-Hausdorff distances between two global attractors (that belong to disjoint phase spaces) and two dynamical systems to consider the continuous dependence of the global attractors and the stability of the dynamical systems on global attractors induced by a reaction-diffusion equation under perturbations of the domain. The novelty of the method is that one compares any two systems in different phase spaces without the need to “pull-back” the perturbed systems to the original domain. Let Ω be an open bounded domain in RN (N ≥ 2) with smooth boundary. Consider the following (initial-boundary value problem for a) reaction-diffusion equation: ⎧ ⎪ ⎪ ⎨ut − Δu + f (u) = g u=0 ⎪ ⎪ ⎩ u(x, 0) = u0

in Ω × (0, ∞), on ∂Ω × (0, ∞),

(5.1)

in Ω.

Assume that u0 ∈ L2 (Ω), g ∈ L2 (RN ) ∩ L∞ (RN ), and f : R → R is a C 1 function satisfying the following condition: there exist four positive constants c0 , c1 , c2 ,  such that

c1 |s|p − c0 ≤ f (s)s ≤ c2 |s|p + c0

with p ≥ 2 for all s ∈ R,

f  (s) ≥ −.

(5.2) (5.3)

It is well known that the problem (5.1) is well posed in various function spaces and has a global attractor in L2 (Ω) (see, for example, [54, 80]).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Lee, C. Morales, Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-12031-2_5

93

94

5 GH-Stability of Reaction-Diffusion Equations

We let Diff(Ω) denote the space of diffeomorphisms h from Ω onto its image Ωh := h(Ω) ⊂ RN with the C 1 topology. For any h ∈ Diff(Ω), let us consider the following reaction-diffusion equation: ⎧ ⎪ ⎪ ⎨ut − Δh u + f (u) = g u=0 ⎪ ⎪ ⎩ u(x, 0) = u0

in Ωh × (0, ∞), on ∂Ωh × (0, ∞),

(5.4)

in Ωh .

Here, Δh denotes the Laplacian operator on Ωh with homogeneous Dirichlet boundary condition. 2 Let Sh : L2 (Ωh ) × R+ 0 → L (Ωh ) be the semidynamical system induced by (5.4), + where R0 = [0, ∞). Denote by Ah the global attractor of Sh . Note that Sh is injective on Ah when g ∈ L∞ (Ωh ). Hence, we can see that the restriction of the semidynamical system Sh to Ah gives rise to a dynamical system, which will be denoted by φh : Ah × R → Ah . A subset R of a topological space X is said to be residual if R contains a countable intersection of open and dense subsets of X. We say that a map f : X → Y between topological spaces is residually continuous if there exists a residual subset R of X such that f is continuous at every point of R. In this section, we prove the following two theorems. Theorem 5.1 The map A : Diff() → M given by A(h) = Ah is residually continuous, where Ah is the global attractor of the semidynamical system Sh induced by (5.4). Theorem 5.2 The global attractor Ah of the semidynamical system Sh induced by (5.4) is residually GH-stable with respect to h ∈ Diff(Ω). More precisely, the map φ : Diff() → CDS given by φ(h) = φh , where φh is the dynamical system on Ah , is residually continuous.

5.2

Proof of Theorem 5.1

For any h ∈ Diff(), we let Ωh := h(Ω) and Δh be the Laplacian operator on Ωh . If we denote by λ1 (Ωh ) the first eigenvalue of Δh with the domain H 2 (Ωh ) ∩ H01 (Ωh ), then the Poincaré inequality ∇h u2L2 (Ω ) ≥ λ1 (Ωh )u2L2 (Ω h

h)

(5.5)

holds for all u ∈ H01 (Ωh ), where ∇h denotes the gradient on Ωh . Note that λ1 (Ωh ) ≥ λ1 (Ωh˜ ) if Ωh is a subset of Ωh˜ , for h, h˜ ∈ Diff(Ω). In view of the inequality (5.5), we can consider the space H01 (Ωh ) with the norm uH 1 (Ωh ) := ∇h uL2 (Ωh ) . 0

5.2 Proof of Theorem 5.1

95

We first recall the well-posedness of the problem (5.4) in the such weak solutions, which will be used in the proof of Theorem 5.1. Lemma 5.1 ([54, 80]) The problem (5.4) has a unique weak solution u which depends continuously on the initial datum u0 ∈ L2 (Ωh ) such that for any T > 0, u ∈ C([0, T ]; L2 (Ωh )) ∩ L2 (0, T ; H01 (Ωh )) ∩ Lp (0, T ; Lp (Ωh )), ∂u ∈ L2 (0, T ; H −1 (Ωh )) + Lq (0, T ; Lq (Ωh )), ∂t where q = p/(p − 1) is the conjugate index of p ≥ 2. Moreover, every weak solution u of (5.4) satisfies u(t)2L2 (Ω ) h

  g2L2 (Ω ) 1 h ≤ 2c0 |Ωh | + · (1 − e−2λ1 (Ωh )t ) λ1 (Ωh ) λ1 (Ωh )

+ e−2λ1 (Ωh )t u0 2L2 (Ω u(t)2L2 (0,T ;H 1 (Ω 0

h)

for all t ≥ 0,

p

h ))

+ 2c1 uLp (0,T ;Lp (Ωh ))

≤ 2c0 |Ωh |T +

T g2L2 (Ω λ1 (Ωh )

(5.6)

h)

(5.7)

+ u0 2L2 (Ω ) , h

for all t ≥ 0

and   q p f (u)Lq (0,T ;Lq (Ωh )) ≤ c˜2 |Ωh |T + uLp (0,T ;Lp (Ωh )) .

(5.8)

2 Furthermore, we have a semidynamical system Sh : L2 (Ωh ) × R+ 0 → L (Ωh ) defined by Sh (u0 , t) := u(t), where u(t) is the unique weak solution of (5.4) with initial datum u0 , and Sh has a compact absorbing set Bh in L2 (Ωh ):

Bh := {u ∈ H01 (Ωh ) : uH 1 (Ωh ) ≤ rh } ⊂ B(0, Rh ), 0

(5.9)

where rh and Rh are sufficiently large constants that depend on Ωh and the constants in (5.2) and (5.3). For any h0 , hn ∈ Diff() with n ∈ N, we consider the map in : L2 (Ωhn ) → L2 (Ωh0 ) defined by 2 in (u) = u ◦ (hn ◦ h−1 0 ) for u ∈ L (Ωhn ),

96

5 GH-Stability of Reaction-Diffusion Equations

and its inverse in−1 : L2 (Ωh0 ) → L2 (Ωhn ), given by 2 in−1 (v) = v ◦ (h0 ◦ h−1 n ) for v ∈ L (Ωh0 ).

Then we see that in is a bounded isomorphism and the restricted map in : H01 (Ωhn ) → H01 (Ωh0 ) is also a bounded isomorphism. We need the following lemma, which will allow us to apply Proposition 2.1 in the proof of Theorem 5.1. Lemma 5.2 Let {hn }n∈N be a sequence in Diff() and {vn,0 }n∈N be a sequence in L2 (Ωh0 ) such that hn → h0 and vn,0 → v0 as n → ∞. Then for any T > 0, we have in (Shn (in−1 (vn,0 ), t)) − Sh0 (v0 , t)L2 (Ωh

0

)

→ 0 for all t ∈ [0, T ].

Proof Take T > 0 and let un,0 = in−1 (vn,0 ) for each n ∈ N. Then un (t) = Shn (un,0 , t) is a weak solution of the problem ⎧ ⎪ ⎪ ⎨∂t un − Δhn un + f (un ) = g ⎪ ⎪ ⎩

in Ωhn × (0, T ],

un = 0

on ∂Ωhn × (0, T ],

un (x, 0) = un,0

in Ωhn ,

(5.10)

and v(t) = Sh0 (v0 , t) is a weak solution of the problem ⎧ ⎪ ⎪ ⎨∂t v − Δh0 v + f (v) = g ⎪ ⎪ ⎩

in Ωh0 × (0, T ],

v=0

on ∂Ωh0 × (0, T ],

v(x, 0) = v0

in Ωh0 .

(5.11)

Let vn = in (un ) and zn = vn − v. We have un ∈ L2 (0, T ; H01 (Ωhn )), vn ∈ L2 (0, T ; H01 (Ωh0 )), and v ∈ L2 (0, T ; H01 (Ωh0 )). Consequently, in−1 (zn ) ∈ L2 (0, T ; H01 (Ωhn )). Hence, we can multiply the first equation in (5.10) by in−1 (zn ), integrate over Ωhn , and integrate by parts to deduce that  Ω hn

(∂t un )(in−1 (zn ))dx +  =− Ω hn

 Ω hn

∇hn un · ∇hn (in−1 (zn ))dx

f (un )in−1 (zn )dx +

 Ω hn

gin−1 (zn )dx.

(5.12)

5.2 Proof of Theorem 5.1

97

Let y = h0 ◦ h−1 n (x) for x ∈ Ωhn . By changing variables and using the notation x instead of y for convenience, we get from (5.12) that 

 (vn )t zn | det Hn |dx + Ω h0

Ω h0

 =−

H n ∇h0 vn · H n ∇h0 zn | det Hn |dx 

f (vn )zn | det Hn |dx +

zn in (g)| det Hn |dx,

Ω h0

(5.13)

Ω h0

where  Hn := Hn (x) =

D(hn ◦ h−1 0 (x))

=

∂(hn ◦ h−1 0 (x))i ∂xj

 N ×N

for x ∈ Ωh0 , and H n = (Hn−1 )T is the transpose of the inverse Hn−1 . Since zn ∈ L2 (0, T ; H01 (Ωh0 )), we can multiply the first equation in (5.11) by zn and integrate over Ωh0 to obtain 





vt zn dx + Ω h0

Ω h0

∇h0 v · ∇h0 zn dx = −

 f (v)zn dx + Ω h0

gzn dx.

(5.14)

Ω h0

By combining (5.13) and (5.14), we get 

 (zn )t zn | det Hn |dx + Ωh0

 (| det Hn | − 1)vt zn dx +

Ωh0





= Ωh0

Ωh0

I − H n ∇h0 v · H n ∇h0 zn | det Hn |dx



+ 

Ωh0

+ 

(5.15) 

∇h0 v · (I − H n )∇h0 zn | det Hn |dx +  (1 − |det Hn |)f (v)zn dx −

Ωh0

Ωh0

(1 − |det Hn |)∇h0 v · ∇h0 zn dx

(f (vn ) − f (v))zn | det Hn |dx Ωh0

 (in (g) − g)zn | det Hn |dx +

+

|H n ∇h0 zn |2 | det Hn |dx

Ωh0

gzn (| det Hn | − 1)dx, Ωh0

where the following identity (and other simple identities) is used: 

 Ω h0

H n ∇h0 vn · H n ∇h0 zn | det Hn |dx −



=

 |H n ∇h0 zn | | det Hn |dx − 2

Ω h0

Ω h0



Ω h0

∇h0 v · ∇h0 zn dx

I − H n ∇h0 v · Hn−1 ∇h0 zn | det Hn |dx

98

5 GH-Stability of Reaction-Diffusion Equations





− Ω h0

∇h0 v · (I − H n )∇h0 zn |det Hn |dx −

Ω h0

(1 − | det Hn |)∇h0 v · ∇hn zn dx.

Here, I denotes the identity matrix. We now estimate each term in the equality (5.15). For the left-hand side of the equality (5.15), we have 

1 d (zn )t zn | det Hn |dx = 2 dt Ω h0

 |zn |2 | det Hn |dx,

(5.16)

Ω h0

and using the Hölder inequality, we get  Ω h0

 (| det Hn | − 1)vt zn dx ≤ | det Hn | − 1∞ vH 1 (Ωh ) zn H 1 (Ωh 0

+ f (v)Lq (Ωh0 ) zn Lp (Ωh0 )

≤ | det Hn | − 1∞

2

0

)



+ | det Hn | − 1∞ gL2 (Ωh ) zn L2 (Ωh 1

0

0

0

v2H 1 (Ω 0

h0 )

0

)

1 + zn 2H 1 (Ω ) h0 2 0 

(5.17)

1 1 q p f (v)Lq (Ωh ) + zn Lp (Ωh ) 0 0 q p 1  1 + | det Hn | − 1∞ gL2 (Ωh ) + zn L2 (Ωh ) , 0 0 2 2

+

where  · ∞ denotes the norm in L∞ (Ωh0 ). For the right-hand side of Eq. (5.15), the first three terms are estimated as  Ω h0



I − H n ∇h0 v · H n ∇h0 zn | det Hn |dx 



+ Ω h0

∇h0 v · (I − H n )∇h0 zn | det Hn |dx +

Ω h0

(1 − |det Hn |)∇h0 v · ∇h0 zn dx

≤ I − H n ∞  det Hn ∞ H n ∞ vH 1 (Ωh ) zn H 1 (Ωh 0

0

0

+ I − H n ∞  det Hn ∞ vH 1 (Ωh ) zn H 1 (Ωh 0

0

0

+ | det Hn | − 1∞ vH 1 (Ωh ) zn H 1 (Ωh 0

1

0

0

0

0

0

)

)

)



I − H n ∞  det Hn ∞ H n ∞ + 1 + | det Hn | − 1∞ ≤ 2   2 2 · vH 1 (Ω ) + zn H 1 (Ω ) , 0

h0

0

h0

(5.18)

5.2 Proof of Theorem 5.1

where A∞ =

99

sup {|aij (x)|, i, j = 1, . . . , N } for any all N × N matrix A = x∈Ω h0

(aij (x))N i,j =1 . The fourth term is estimated as  Ω h0

(1 − | det Hn |)f (v)zn dx ≤ | det Hn | − 1∞ f (v)Lq (Ωh ) zn Lp (Ωh ) 0 0  ≤ | det Hn | − 1∞

 1 1 q p f (v)Lq (Ω ) + zn Lp (Ω ) , h0 h0 q p

(5.19) and the fifth term is estimated as   − (f (vn ) − f (v))zn | det Hn |dx ≤  Ω h0

|zn |2 | det Hn |dx,

(5.20)

Ω h0

where the condition (5.3) is used. Finally, the sixth and seventh terms obey the estimates  Ω h0

(in (g) − g)zn | det Hn |dx ≤ in (g) − gL2 (Ωh ) zn L2 (Ωh )  det Hn ∞

(5.21)

gzn (| det Hn | − 1)dx ≤ gL2 (Ωh ) zn L2 (Ωh ) | det Hn | − 1∞ .

(5.22)

0

0

and  Ω h0

0

0

By substituting the estimates from (5.16) to (5.22) into (5.15), we deduce that 1 d 2 dt



 |zn | | det Hn |dx + 2

Ω h0

Ω h0



≤ Ω h0

|H n ∇h0 zn |2 | det Hn |dx

|zn |2 | det Hn |dx + | det Hn | − 1∞ K(t)



1 I − H n ∞  det Hn ∞ H n ∞ + 1 + | det Hn | − 1∞ + 2   · v2H 1 (Ω ) + zn 2H 1 (Ω ) 0

h0

0

h0

+ in (g) − gL2 (Ωh ) zn L2 (Ωh )  det Hn ∞ , 0

where

0

(5.23)

100

5 GH-Stability of Reaction-Diffusion Equations

    1 1 1 q p 2 2 vH 1 (Ω ) + zn H 1 (Ω ) + 2 f (v)Lq (Ωh ) + zn Lp (Ωh ) K(t) = h0 h0 0 0 2 q p 0 0 + gL2 (Ωh ) + zn L2 (Ωh ) . 0

0

This implies that 

d  −2t e dt

|zn |2 | det Hn |dx



Ω h0

 ≤ 2e−2t | det Hn | − 1∞ K(t) + e−2t v2H 1 (Ω 0

 h0

+ zn 2H 1 (Ω ) 0

h0 )

(5.24)

 

× I − H n ∞  det Hn ∞ H n ∞ + 1 + | det Hn | − 1∞ + 2e−2t in (g) − gL2 (Ωh ) zn L2 (Ωh )  det Hn ∞ . 0

0

By integrating (5.24) from 0 to t ∈ [0, T ], we derive that  |zn |2 | det Hn |dx Ω h0

≤e



 |zn (0)| | det Hn |dx + 2e

2t

2

Ω h0

2t

| det Hn | − 1∞

T

K(t)dt

(5.25)

0

 

+ e2t I − H n ∞  det Hn ∞ H n ∞ + 1 + | det Hn | − 1∞ 

T

× 0

+

 v(t)2H 1 (Ω 0

 + zn (t)2H 1 (Ω ) h0 ) h0 0

dt

e2t  det Hn ∞ zn L∞ (0,T ;L2 (Ωh )) in (g) − gL2 (Ωh ) . 0 0 

The estimates (5.6), (5.7), and (5.8) imply that there is a constant C > 0 (independent of n) such that  0



T 0

T

K(t)dt ≤ C, zn L∞ (0,T ;L2 (Ωh

 v(t)2H 1 (Ω 0

0

))

≤ C,

 + zn (t)2H 1 (Ω ) h0 ) h0 0

dt ≤ C.

Since hn → h0 in Diff(0 ) and det is a continuous operator,  det Hn ∞ and H n ∞ are uniformly bounded with respect to n. Therefore, from (5.25), we conclude that

5.2 Proof of Theorem 5.1

101



 |zn | | det Hn |dx ≤ e 2

Ω h0

2t Ω h0

|zn (0)|2 | det Hn |dx + Ce2t in (g) − gL2 (Ωh

0

)



+ Ce2t | det Hn | − 1∞ + I − H n ∞ .

Finally, we deduce that inf | det Hn (x)|zn (t)2L2 (Ω

x∈Ωh0

h0 )

≤ e2t  det Hn ∞ zn (0)2L2 (Ω ) (5.26) h0   + Ce2t in (g) − gL2 (Ωh ) + | det Hn | − 1∞ + I − H n ∞ . 0

Note that inf | det Hn (x)| ≥ x∈Ωh0

1 2

for all sufficiently large n ∈ N. Moreover, we get

| det Hn | − 1∞ → 0 and I − H n ∞ → 0 as n → ∞. Since Cc∞ (RN ) is dense in L2 (RN ) and g ∈ L2 (RN ), we can choose ξm ∈ Cc∞ (RN ) such that ξm → g in L2 (RN ). Hence, we have in (g) − gL2 (Ωh

0

)

≤  det Hn ∞ g − ξm L2 (RN ) + in (ξm ) − ξm L2 (Ωh

0

)

+ ξm − gL2 (Ωh ) . 0

This implies that in (g) − gL2 (Ωh ) → 0 as n → ∞, and now completes (5.26) the proof 0 of the lemma.  With the above lemma and proposition at hand, we are ready to prove Theorem 5.1. End of Proof of Theorem 5.1 We prove the theorem by showing that the assumptions of Proposition 2.1 are satisfied. To this and, we let Λ = Diff() and Dh = B(0, Rh ) for each h ∈ Diff(), where Rh is a constant given in (5.9). Then it is enough to show that for any T > 0, ε > 0, and h0 ∈ Diff(), there exists a neighborhood W of h0 such that for any h ∈ W, there is an ε-immersion i : Dh → L2 (Ωh0 ) satisfying i(Dh ) ⊂ B(Dh0 , ε), and i(Sh (u0 , t)) − Sh0 (i(u0 ), t)L2 (Ωh

0

)

0, ε > 0, and a sequence {hn }n∈N converging to h0 in Diff() such that for any ε-immersion in : Dhn → L2 (Ωh0 ) satisfying in (Dhn ) ⊂ B(Dh0 , ε), there are un,0 ∈ Ahn and tn ∈ [0, T ] such that

102

5 GH-Stability of Reaction-Diffusion Equations

in (Sh (un,0 , tn )) − Sh0 (in (un,0 ), tn )L2 (Ωh

0

)

> ε.

(5.28)

In particular, take in = ihn , where ihn is given as before, that is, ihn (u) = u ◦ (hn ◦ h−1 0 ) for u ∈ L2 (Ωhn ). Let vn,0 = in (un,0 ) for each n ∈ N. Since un,0 ∈ H01 (Ωhn ) and in  → 1 as n → ∞, {vn,0 } is uniformly bounded in H01 (Ωh0 ). By the compactness of the embedding H01 (Ωh0 ) → L2 (Ωh0 ), {vn,0 } has a convergent subsequence in L2 (Ωh0 ), say vn,0 → v0 ∈ L2 (Ωh0 ) as n → ∞. Then (5.28) can be rewritten as in (Shn (in−1 (vn,0 ), tn )) − Sh0 (v0 , tn )L2 (Ωh

0

)

≥

ε 2

for all sufficiently large n. This is a contradiction by Lemma 5.2, and so the proof of  Theorem 5.1 is complete.

5.3

Proof of Theorem 5.2

In that follows, we prove that, residually, the dynamical system on the global attractor induced by the problem (5.1) varies continuously in the topology of Gromov-Hausdorff convergence with respect to perturbations of the domain (Theorem 5.2). As a consequence, we can derive the main result (Theorem 2.9) for the semidynamical systems Sh induced by the equations in (5.4) residually with respect to h. Lemma 5.3 For any h0 ∈ Diff(), there is a neighborhood U of h0 such that {Sh }h∈U is equicontinuous on Ah × [0, 1], that is, for any ε > 0, there exists δ > 0 such that if u0 − u˜ 0 L2 (Ωh ) < δ and |t − s| < δ, then Sh (u0 , t) − Sh (u˜ 0 , s)L2 (Ωh ) < ε for all h ∈ U , u0 , u˜ 0 ∈ Ah , and t, s ∈ [0, 1]. Proof Step 1. We show that for any ε > 0 and h ∈ Diff(), there is δ1 > 0 such that if u0 − u˜ 0 L2 (Ωh ) < δ1 , then Sh (u0 , t) − Sh (u˜ 0 , t)L2 (Ωh )
0 be a constant satisfying (5.3), and take a constant δ1 with δ1 < 2eε  . For any two points u0 , u˜ 0 ∈ L2 (Ωh ) with u0 − u˜ 0 L2 (Ωh ) < δ1 , we let u(t) = Sh (u0 , t), u(t) ˜ = Sh (u˜ 0 , t) and z(t) = u(t) − u(t), ˜ for all t ∈ R+ 0.

5.3 Proof of Theorem 5.2

103

Then z is a weak solution of ⎧ ⎪ ˜ =0 ⎪ ⎨∂t z − Δh z + f (u) − f (u) z=0 ⎪ ⎪ ⎩ z(x, 0) = z0 := u0 − u˜ 0

in Ωh × (0, ∞), on ∂Ωh × (0, ∞), in Ωh .

Hence, we have 1 d z2L2 (Ω ) + ∇h z2L2 (Ω ) = − h h 2 dt

 (f (u) − f (u))zdx. ˜ Ωh

Using (5.3), we obtain the estimate d z2L2 (Ω ) + 2∇h z2L2 (Ω ) ≤ 2z2L2 (Ω ) . h h h dt In particular, d z2L2 (Ω ) ≤ 2z2L2 (Ω ) . h h dt By the Gronwall inequality, we have ε2 , for all t ∈ [0, 1]. 4

z2L2 (Ω ) ≤ e2t z0 2L2 (Ω ) ≤ e2 u0 − u˜ 0 2L2 (Ω ) < h

h

h

Step 2. We claim that for any h0 ∈ Diff() and ε > 0, there exist a neighborhood U of h0 and δ2 > 0 such that for any h ∈ U , ε 2

Sh (u0 , s) − u0 L2 (Ωh )
N1 ,   ε . max Sh0 (v0 , sn ) − Sh0 (ih (un,0 ), sn )L2 (Ωh ) , ih (un,0 ) − v0 L2 (Ωh ) < 0 0 18 Note that sn → 0 as n → ∞. By the continuity of Sh0 (v0 , ·), we can choose N2 > 0 such that Sh0 (v0 , sn ) − v0 L2 (Ωh

0

)


N2 . Then for any n > max{N1 , N2 }, we have 0 = Sh0 (v0 , 0) − v0  ≥ Sh0 (ih (un,0 ), sn ) − ih (un,0 )L2 (Ωh

0

)

− Sh0 (ih (un,0 ), sn ) − Sh0 (v0 , sn )L2 (Ωh − Sh0 (v0 , sn ) − Sh0 (v0 , 0)L2 (Ωh − ih (un,0 ) − v0 L2 (Ωh

0

)

0

0

)

)

> 0.

This contradiction proves (5.31). It follows that for any h ∈ U , u0 ∈ Ah , and s ∈ [0, δ2 ], ε + ih (Sh (u0 , s)) − ih (u0 )L2 (Ωh ) 0 6 ε ≤ + ih (Sh (u0 , s)) − Sh0 (ih (u0 ), s)L2 (Ωh ) 0 6 ε + Sh0 (ih (u0 ), s) − ih (u0 )L2 (Ωh ) ≤ . 0 2

Sh (u0 , s) − u0 L2 (Ωh ) ≤

Step 3. For any h0 ∈ Diff() and ε > 0, let U be a neighborhood of h0 as in Step 2 and let δ = min{δ1 , δ2 }, where δ1 and δ2 are taken from Steps 1 and 2, respectively. Let h ∈ U , u0 , u˜ 0 ∈ Ah , and t, s ∈ [0, 1] be such that u0 − u˜ 0 L2 (Ωh ) < δ and |t − s| < δ. A applying inequalities (5.29) and (5.30), we have

5.3 Proof of Theorem 5.2

105

Sh (u0 , t) − Sh (u˜ 0 , s)L2 (Ωh ) ≤ Sh (u0 , t) − Sh (u˜ 0 , t)L2 (Ωh ) + Sh (u0 , t) − Sh (u˜ 0 , s)L2 (Ωh ) ≤

ε + Sh (Sh (u0 , s), t − s) − Sh (u˜ 0 , s)L2 (Ωh ) 2

≤ ε. This completes the proof of Lemma 5.3.



Remark 5.1 Lemma 5.3 implies that for any h0 ∈ Diff() and T > 0, there is a neighborhood U of h0 such that {Sh }h∈U is equicontinuous on Ah × [0, T ]. End of Proof of Theorem 5.2 For any h, k ∈ Diff(), we define an isomorphism ihk : L2 (Ωh ) → L2 (Ωk ) by ihk (u) = u ◦ (h ◦ k −1 ). We observe that for any ε > 0, there is δ > 0 such that if dC 1 (h, k) < δ, then ihk and ikh are ε-immersions, where dC 1 denotes the C 1 -metric on Diff(Ω). Step 1. We prove that there is a residual subset R of Diff(Ω) such that for any k ∈ R and ε > 0, there is a neighborhood W of k such that for any h ∈ W, the ε-immersion ihk satisfies ihk (Ah ) ⊂ B(Ak , ε) and Ak ⊂ B(ihk (Ah ), ε). For any ε > 0, we denote by F (ε) the collection of all k ∈ Diff(Ω) such that there is a neighborhood W of k with the following property: for any h, h˜ ∈ W, the ε-immersion ihh˜ satisfies ihh˜ (Ah ) ⊂ B(Ah˜ , ε). It is clear that F (ε) is open in Diff(Ω) for any ε > 0.  Let R = n∈N F (1/n). To prove that R is a residual subset of Diff(Ω), it is sufficient to show that F (8ε) is dense in Diff(Ω). Let U be a nonempty open set in Diff(Ω), and h0 ∈ U . Let {Bi = B(ui , ε/3), ui ∈ L2 (Ωh0 )}i∈K be a finite open cover of Ah0 such that Ah0 ∩ Bi = ∅ for any i ∈ K. Let I be the collection of subsets J ⊂ K such that there are k ∈ U and an injective δ-immersion jk : Dk → L2 (Ωh0 ) for some δ < ε/2 satisfying jk (Ak ) ∩ Bi = ∅ for all i ∈ J, and

jk (Uk ) ⊂ Bi for a neighborhood Uk of Ak .

(5.32)

i∈J

We note that I is nonempty since it contains K. Choose a minimal element J of I. Then, by the definition of I, there are k ∈ U and an injective δ-immersion jk : Dk → L2 (Ωh0 ) for some δ < ε/2 satisfying (5.32). By Step 1 in Proposition 2.1, there is a neighborhood W ⊂ U of k such that for any h ∈ W, ihk an the (ε/4 − δ/2)-immersion satisfying ikh  < 2,

and ihk (Uh ) ⊂ Uk for a neighborhood Uh of Ah .

106

5 GH-Stability of Reaction-Diffusion Equations

We can check that jk ◦ ihk is an injective (ε/4 + δ/2)-immersion from Dh to L2 (Ωh0 ) such that jk ◦ ihk (Uh ) ⊂ jk (Uk ) ⊂

Bi .

i∈J

Since J is a minimal element in I, we have jk ◦ihk (Ah )∩Bi = ∅ for all i ∈ J . We observe that Ak ⊂

B(jk−1 (ui ), ε), Ak ∩ B(jk−1 (ui ), ε) = ∅,

i∈J

ihk (Ah ) ⊂

B(jk−1 (ui ), ε), and ihk (Ah ) ∩ B(jk−1 (ui ), ε) = ∅

i∈J

for all i ∈ J . Therefore, Ak ⊂ B(ihk (Ah ), 2ε) and ihk (Ah ) ⊂ B(Ak , 2ε) for all h ∈ W, and so ihk (Ah ) ⊂ B(ihk ˜ (Ah˜ ), 4ε) for all h, h˜ ∈ W. We conclude that ihh˜ (Ah ) = ik h˜ ◦ ihk (Ah ) ⊂ ik h˜ (B(ihk ˜ (Ah˜ ), 4ε)) ⊂ B(Ah˜ , 8ε). Consequently, we have proved k ∈ U ∩ F (8ε). So, F (8ε) is dense in Diff(Ω). Finally, for any k ∈ R and ε > 0, choose n ∈ N with 1/n < ε/2. Since k ∈ F (1/n), there is a neighborhood W of k such that for any h, h˜ ∈ W, the (1/n)-immersion ihh˜ ˜ we obtain satisfies ihh˜ (Ah ) ⊂ B(Ah , 1/n). In particular, letting k = h, ihk (Ah ) ⊂ B(Ak , ε) and Ak ⊂ B(ihk (Ah ), ε) for all h ∈ W. Step 2. We prove that the map φ : Diff(Ω) → CDS defined by φ(h) = φh is continuous on R. Fix k ∈ R and ε > 0. By Lemma 5.2, there exists 0 < δ < ε/3 such that if u − u ˆ L2 (Ωk ) < δ for u ∈ L2 (Ωk ) and uˆ ∈ Ak , then Sk (u, t) − Sk (u, ˆ t)L2 (Ωk )
0 corresponding to ε/2 by Lemma 5.3. Then for any (u, t), (u, ˆ tˆ) ∈ A × ([0, 1] ∩ Q) with u − u ˆ L2 (Ωk ) + |t − tˆ| < δ, we have φ∗ (u, t) − φ∗ (u, ˆ tˆ)L2 (Ωk ) = lim iˆn (φhn (jˆn (u), t)) − iˆn (φhn (jˆn (u), ˆ tˆ))L2 (Ωk ) n→∞   1 ˆ ˆ ˆ + φhn (jn (u), t) − φhn (jn (u), ≤ lim ˆ t )L2 (Ωk ) n→∞ n < ε, as needed. Now the uniformly continuous map φ∗ of A × ([0, 1] ∩ Q) into Ak can be extended to a uniformly continuous map (still denoted by φ∗ ) of Ak × [0, 1] into Ak . For any (u, t) ∈ A × ([0, 1] ∩ Q), we have φ∗ (u, t) − φk (u, t)L2 (Ωk ) = lim iˆn (φhn (jˆn (u), t)) − φk (u, t)L2 (Ωk ) n→∞

≤ lim iˆn (φhn (jˆn (u), t)) − φk (iˆn ◦ jˆn (u), t)L2 (Ωk ) n→∞

+ φk (iˆn ◦ jˆn (u), t) − φk (u, t)L2 (Ωk )   1 ˆ ˆ + φk (in ◦ jn (u), t) − φk (u, t)L2 (Ωk ) ≤ lim n→∞ n = 0. This implies that φ∗ = φk on Ak × [0, 1]. Next, let us show that if (u, t) ∈ Ak × [0, 1], then the sequence {iˆn (φhn (jˆn (u), t))}n∈N converges to φk (u, t). In fact, for any (u, t) ∈ Ak × [0, 1], we can take a sequence {(ul , tl )}l∈N in A × ([0, 1] ∩ Q) converging to (u, t). For any ε > 0, choose a constant δ > 0 (δ < ε) corresponding to ε/4 by Lemma 5.3. Choose l ∈ N satisfying |tl − t| ≤

δ δ and ul − uL2 (Ωk ) ≤ . 4 4

5.3 Proof of Theorem 5.2

109

By the definition of iˆn and the continuity of φk , there exists N3 > 0 such that 1/N3 < δ/4 and iˆn (φhn (jˆn (ul ), tl )) − φk (ul , tl )L2 (Ωk )
N3 . Since jˆn (ul ) − jˆn (u)L2 (Ωhn ) < 1/n + ul − uL2 (Ωk ) < δ/2 for all n > N3 , we have ε φhn (jˆn (ul ), tl ) − φhn (jˆn (u), t)L2 (Ωk ) < . 4 It follows that iˆn (φhn (jˆn (u), t))−φk (u, t)L2 (Ωk ) ≤ iˆn (φhn (jˆn (u), t)) − iˆn (φhn (jˆn (ul ), tl ))L2 (Ωk ) + iˆn (φhn (jˆn (ul ), tl )) − φk (ul , tl )L2 (Ωk ) + φk (ul , tl ) − φk (u, t)L2 (Ωk ) ≤ ε for all n > N3 . Thus, we have checked assumptions (1), (2), and (3) of Theorem 2.9 for the semidynamical system Sh induced by the reaction-diffusion equations (5.4) residually.  This completes the proof of Theorem 5.2. Remark 5.2 Let Ah be the global attractor of the semidynamical system Sh induced by Eq. (5.4). Then one can show that Ah is GH-stable under perturbations of the domain if ˜ < δ, then for and only if, for any T > 0 and ε > 0, there is δ > 0 such that if dC 1 (h, h) any u ∈ Ah , u˜ ∈ Ah˜ and t ∈ [0, T ], iˆhh˜ (φh (u, t) − φh˜ (iˆhh˜ (u), t))L2 (Ω ˜ ) < ε, and h

jˆhh˜ (φh˜ (u, ˜ t) − φh (jˆhh˜ (u), ˜ t))L2 (Ωh ) < ε, where iˆhh˜ and jˆhh˜ are defined as before. We say that an equilibrium u0 of the equation (5.4) is hyperbolic if 0 is not in the spectrum of the linear operator Δ − F  (u0 )I : L2 (Ωh ) → L2 (Ωh ), where I denotes the identity map on L2 (Ωh ) and F  is the Nemytskii operator of f  . Remark 5.3 In this direction, one we can prove the continuity of global attractors and the Gromov-Hausdorff stability of reaction-diffusion equations with Neumann or Robin boundary conditions under perturbations of the domain and equation if every equilibrium of the unperturbed equation is hyperbolic. For more details, see [44, 48].

6

Stability of Inertial Manifolds

6.1

Introduction

In this chapter, we study the Gromov-Hausdorff stability and the continuous dependence of the inertial manifolds under perturbations of the domain and equation. More precisely, we use the Gromov-Hausdorff distances between two inertial manifolds and two dynamical systems to consider the continuous dependence of the inertial manifolds and the stability of the dynamical systems on inertial manifolds induced by reaction-diffusion equations under perturbations of the domain and equation. Let Ω0 be an open bounded domain in RN with smooth boundary. We consider the dissipativity reaction-diffusion equation ⎧ ⎨∂ u − Δu = f (u) t 0 ⎩u = 0

in Ω0 × (0, ∞), on ∂Ω0 × (0, ∞),

(6.1)

where f0 : R → R is a C 1 function such that f0 and f0 are bounded, and f0 satisfies the dissipativity condition, lim sup |s|→∞

f0 (s) < 0. s

It shown in [7] that the problem (6.1) is well posed in various function spaces. Let F0 : L2 (Ω0 ) → L2 (Ω0 ) be the Nemytskii operator generated by f0 . It is clear that F0 is Lipschitz since f0 is bounded, and we may assume Lip F0 > 1.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Lee, C. Morales, Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-12031-2_6

111

112

6 Stability of Inertial Manifolds

To simplify the notation, we write h → id instead of dC 1 (h, id) → 0. Let F be the collection of C 1 functions fh : R → R (h ∈ Diff(0 )) with the dissipativity condition such that d C 1 (fh , f0 ) ≤ dC 1 (h, id), where the metric d C 1 on F is given by d C 1 (fh , fh˜ ) := min{dC 1 (fh , fh˜ ), 1}

for h, h˜ ∈ Diff(Ω0 ),

where id denotes the identity map on Ω0 . For each h ∈ Diff(Ω0 ) and fh ∈ F, we consider the following perturbation of Eq. (6.1): ⎧ ⎨∂ u − Δu = f (u) t h ⎩u = 0

in Ωh × (0, ∞), on ∂Ωh × (0, ∞).

(6.2)

Then the problem (6.2) is well posed and the Nemytskii operator Fh : L2 (Ωh ) → L2 (Ωh ) of fh is Lipschitz. For any h ∈ Diff(Ω0 ) C 1 -close to id, we consider the equation ut + Ah u = Fh (u)

(6.3)

for u ∈ L2 (Ωh ), where Ah denotes the operator −Δ on Ωh with Dirichlet boundary condition. For simplicity, we write Aid = A0 and Fid = F0 . We know that Ah has a sequence of eigenvalues {λhi }∞ i=1 such that 0 < λh1 ≤ λh2 ≤ · · · → ∞ and a sequence of corresponding eigenfunctions {φih }∞ i=1 , which is an orthonormal basis 1 2 in L (Ωh ) and orthogonal in H0 (Ωh ). We denote by Sh (t) the semidynamical system induced by Eq. (6.3), defined by Sh (t) : L2 (Ωh ) → L2 (Ωh ), Sh (t)(u0 ) = uh (t) for any t ≥ 0, where uh (t) is the unique solution of (6.3) with uh (0) = u0 . For any h ∈ Diff(Ω0 ), fh ∈ F, and m ∈ N, let Pmh be the projection of L2 (Ωh ) onto h } and Qh be the orthogonal complement of P h . For simplicity, we will span{φ1h , . . . , φm m m 0 write Pmid := Pm0 and Qid m := Qm . Definition 6.1 We say that M ⊂ L2 (Ω0 ) is an m-dimensional inertial manifold of the semidynamical system S0 (t) induced by Eq. (6.1) if it is the graph of a Lipschitz map Φ : Pm0 L2 (Ω0 ) → Q0m L2 (Ω0 ) such that (i) M is invariant, that is, S0 (t)M = M for all t ∈ R;

6.1 Introduction

113

(ii) M attracts all trajectories of S0 (t) exponentially, that is, there are C > 0 and k > 0 such that for any u0 ∈ L2 (Ω0 ), there is v0 ∈ M satisfying S0 (t)u0 − S0 (t)v0 L2 (Ω0 ) ≤ Ce−kt u0 − v0 L2 (Ω0 )

for all t > 0.

We are interested in the behavior of the inertial manifolds (which belong to disjoint phase spaces) of Eq. (6.3) with respect to perturbations of the domain Ω0 . To study this, we first need to establish the existence of an inertial manifold of Eq. (6.3) when h is C 1 close enough to id. Theorem 6.1 Let the above assumptions on the operator Ah and the nonlinearity Fh hold. In addition, let the following spectral gap condition hold: √ λ0m+1 − λ0m > 2 2L0

for some m ∈ N,

(6.4)

where L0 is a Lipschitz constant of the nonlinearity F0 and λ0n is the nth eigenvalue of A0 for n ∈ N. Then there exists δ > 0 such that if dC 1 (h, id) < δ, then Eq. (6.3) admits an m-dimensional inertial manifold Mh . 

Proof See Theorem 1.1 in [46].

Remark 6.1 To the best of our knowledge, the existence of an inertial manifold for (6.1) was first proved by Foias et al. [28] with a non-optimal constant C on the right-hand side of assumption (6.4) (for more details, see Theorem 2.1 in [28]). Moreover, Romanov [66] proved the existence of an inertial manifold of (6.1) under the spectral gap condition (6.4) using the Lyapunov-Perron method in [70]. For a detailed exposition of the classical theory of inertial manifolds, refer to the papers by Zelik [79] and Kostianko and Zelik [42] for a sharp gap condition. To study how the asymptotic dynamics of evolutionary equation (6.3) changes when we vary the domain Ωh , our first task is to find a way to compare the inertial manifolds of the equations in different domains. One of the difficulties in this direction is that the phase space L2 (Ω0 ) of the induced semidynamical system changes as we change the domain Ω0 . In fact, the phase spaces L2 (Ω0 ) and L2 (Ωh ) which contain inertial manifolds M0 and Mh , respectively, can be disjoint even if Ωh is a small perturbation of Ω0 . In this direction, Arrieta and Santamaria [8] estimated the distance of inertial manifolds Ms of a satisfies evolutionary problem of the form ut + Aε u = Fε (u)

(6.5)

for ε ∈ [0, ε0 ] on a Hilbert space Xε . For this purpose, they first assumed that the operator A0 satisfies the following spectral gap condition

114

6 Stability of Inertial Manifolds

λ0m+1 − λ0m ≥ 18L0 and λ0m ≥ 18L0 for some m ∈ N to use the Lyapunov-Perron method for the existence of inertial manifolds (see Proposition 2.1 in [8]). They also assumed that the nonlinear terms Fε have a uniformly bounded support, that is, there exists R > 0 such that supp Fε ⊂ DR = {u ∈ Xε : uXε ≤ R} for ε ∈ [0, ε0 ]. This assumption implies that the inertial manifold Mε of (6.5) does not leave the ball DR when ε varies. In fact, we have Mε ∩ (Xε \ DR ) = Pmε (Xε ) ∩ (Xε \ DR ) for ε ∈ [0, ε0 ]. Note that the inertial manifold Mε (or M0 ) of (6.5) is described by the graph of a Lipschitz map Φε (or Φ0 ). Under the above assumptions, they proved that Φε − Eε Φ0 L∞ (Rm ,Xε ) → 0

as ε → 0,

where Eε is an isomorphism from X0 to Xε (for more details, see Theorem 2.3 in [8]). Note that the norms  · L∞ (Rm ,Xε ) and  · L∞ (Rm ,Xε ) are not comparable in general if ε = ε . For any ε ∈ [0, ε0 ], we take hε ∈ Diff(Ω0 ) satisfying dC 1 (hε , id) = ε. Then the perturbed phase space Xε in [8] can be considered as the space L2 (Ωhε ). In this chapter, we do not assume that the nonlinear terms Fh (h ∈ Diff(Ω0 )) have a uniformly bounded support. Recently, Lee et al. [49] introduced the Gromov-Hausdorff distance between two dynamical systems on compact metric spaces to analyze how the asymptotic dynamics of the global attractors of (6.1) changes when we vary the domain Ω0 . To compare the asymptotic behavior of the dynamics on inertial manifolds, we first need to introduce the notion of Gromov-Hausdorff distance between two dynamical systems on noncompact metric spaces. Let (X, dX ) and (Y, dY ) be two metric spaces. For any ε > 0 and a subset B of X, we recall that a map i : X → Y is an into ε-isometry on B if |dY (i(x), i(y)) − dX (x, y)| < ε for all x, y ∈ B. In the case B = X, we say that i : X → Y is an into ε-isometry. An into ε-isometry i : X → Y is called an ε-isometry if Uε (i(X)) = Y , where Uε (i(X)) is the ε-neighborhood of i(X). The Gromov-Hausdorff distance dGH (X, Y ) between X and Y is defined as the infimum of the numbers ε > 0 such that there are ε-isometries i : X → Y and j : Y → X. Let X = {(Xh , dXh ) : h ∈ Diff(Ω0 )} be a collection of metric spaces. For simplicity, we write Xh instead of (Xh , dXh ). Note that the Gromov-Hausdorff distance dGH (X, Y ) here is equivalent to the Gromov-Hausdorff distance introduced in [14, Definition 7.3.10].

6.1 Introduction

115

Definition 6.2 We say that Xh ∈ X converges to Xk in the Gromov-Hausdorff sense as h → k if for any ε > 0 and xk ∈ Xk , there is δ > 0 such that if dC 1 (h, k) < δ, then there is xh ∈ Xh such that for any r > 0, dGH (B(xh , r), B(xk , r)) < ε, where B(x, r) is the closed ball centered at x with radius r. Note that Xh converges to Xk in the Gromov-Hausdorff sense if dGH (Xh , Xk ) → 0

and

h → k.

However, the converse is not true in general. Let S be a dynamical system on X, namely, S : X × R → X. For any subset B of X, we denote by S|B the restriction of S to B × R. Definition 6.3 Let S1 and S2 be dynamical systems on metric spaces X and Y , respectively. For any x ∈ X, y ∈ Y , and r > 0, the Gromov-Hausdorff distance T (S | DGH 1 B(x,r) , S2 |B(y,r) ) between S1 |B(x,r) and S2 |B(y,r) with respect to T > 0 is defined as the infimum of ε > 0 such that there are maps i : X → Y and j : Y → X, and reparameterizations α ∈ RepB(x,r) (ε) and β ∈ RepB(y,r) (ε) with the following properties: (i) i and j are into ε-isometries on B(x, r) and B(y, r), respectively, satisfying B(y, r) ⊂ Uε (i(B(x, r))) and B(x, r) ⊂ Uε (j (B(y, r))), (ii) dY (i(S1 (x, α(x, t))), S2 (i(x), t)) < ε for x ∈ B(x, r) and t ∈ [−T , T ], and dX (j (S2 (y, β(y, t))), S1 (j (y), t)) < ε for y ∈ B(y, r) and t ∈ [−T , T ], where RepB (ε) is the collection of continuous maps α : B × R → R such that for each   α(x, t) fixed x ∈ B, α(x, ·) is a homeomorphism on R with  − 1 < ε for t = 0. t Definition 6.4 Let DS = {(Xh , Sh ) : h ∈ Diff(Ω0 )} be a collection of dynamical systems on metric spaces Xh let xk ∈ Xk . We say that the dynamical system (Xk , Sk ) ∈ DS is Gromov-Hausdorff stable if for any ε > 0 and T > 0, there exists δ > 0 such that if T (S | dC 1 (h, k) < δ, then there is xh ∈ Xh such that for any r > 0, DGH h B(xh ,r) , Sk |B(xk ,r) ) < ε. We observe that the Gromov-Hausdorff stability of dynamical systems on the global attractors under perturbations of the domain was first studied in [49]. Throughout this chapter, we assume that the following holds:

116

6 Stability of Inertial Manifolds

√ λ0m+1 − λ0m > 2 2L0 and λ0m > L0 for some m ∈ N.

(6.6)

Moreover, we assume that m is the smallest number satisfying (6.6) and the inertial manifold Mh for Eq. (6.3) means the unique m-dimensional inertial manifold for (6.3). Let us, we state the main results of this section. Theorem 6.2 The inertial manifold Mh of Eq. (6.3) converges to M0 in the GromovHausdorff sense as h → id, that is, for any ε > 0 and u0 ∈ M0 , there exists δ > 0 such that if dC 1 (h, id) < δ, then there is uh ∈ Mh such that for any r > 0, dGH (B(uh , r), B(u0 , r)) < ε. Let Sh (t) be the dynamical system on the inertial manifold Mh induced by Eq. (6.3). Theorem 6.3 The dynamical system S0 (t) is Gromov-Hausdorff stable. For the proofs of Theorems 6.2 and 6.3, we first need the continuity of the spectra of Ah with respect to h. Proposition 6.1 The spectral data of Ah behave continuously as h → id. More precisely, for any fixed  ∈ N and a sequence {hn }n∈N in Diff(Ω0 ) with hn → id, there exist a subsequence {hk := hnk }k∈N of {hn }n∈N and a collection of eigenfunctions {ξ10 , . . . , ξ0 } of A0 with corresponding eigenvalues {λ01 , . . . , λ0 } such that λhi k → λ0i and φihk → ξi0 in L2 (RN ) for all 1 ≤ i ≤ . 

Proof See Proposition 2.1 in [46].

6.2

Proof of Theorem 6.2

For any h ∈ Diff(Ω0 ), we let Lh = sup |fh (s)| and s∈R

L0 = sup |f0 (s)|.

(6.7)

s∈R

It is clear that Lh and L0 are Lipschitz constants of the nonlinear terms Fh and F0 , respectively, such that Lh → L0 as h → id. Define a map jh : L2 (Ω0 ) → L2 (Ωh ) by jh (u) := u ◦ h−1

for all u ∈ L2 (Ω0 ).

6.2 Proof of Theorem 6.2

117

Then clearly jh is an isomorphism, and jh  → 1 as h → id. Here, jh  = jh L∞ (L2 (Ω0 ),L2 (Ωh )) . Hence, we may assume that jh  < 2 for all h ∈ Diff(Ω0 ). Now to prove Theorem 6.2, Suppose its conclusion is not true. Then there are ε > 0 and u0 ∈ M0 such that for any n ∈ N, there is hn ∈ Diff(Ω0 ) with dC 1 (hn , id) < 1/n such that for any uhn ∈ Mhn , there exists rn > 0 such that dGH (B(uhn , rn ), B(u0 , rn )) ≥ ε. hn For each n ∈ N, let {λh1 n , . . . , λhmn } and {φ1hn , . . . , φm } be the first m eigenvalues and corresponding m eigenfunctions of Ahn , respectively. By Proposition 6.1, there 0 }, associated to the first m eigenvalues are m eigenfunctions, denoted by {φ10 , . . . , φm 0 0 {λ1 , . . . , λm } of A0 , and a subsequence of {hn }n∈N , still denoted by {hn }n∈N , such that λhi n → λ0i and φihn → φi0 in L2 (RN ) as n → ∞ for all 1 ≤ i ≤ m. We assume that |λhi n − λ0i | < 1 for all n ∈ N and 1 ≤ i ≤ m. For any h ∈ Diff(Ω0 ), define a map ψh : Pmh L2 (Ωh ) → Rm by

ψh

 m 

 ai φih

= (a1 . . . . , am ),

ai ∈ R.

i=1

For each n ∈ N, we choose uhn ∈ Mhn such that ψhn Pmhn uhn = ψ0 Pm0 u0 . To complete the proof of Theorem 6.2 we will show that dGH (B(uhn , rn ), B(u0 , rn )) < ε for all sufficiently large n ∈ N. For this, we need several lemmas. Lemma 6.1 For any fixed 1 ≤ i ≤ m, we have jhn φi0 − φihn L2 (Ωhn ) → 0 Proof See Lemma 3.1 in [46].

as n → ∞. 

Let Ψh : Pmh L2 (Ωh ) → Qhm L2 (Ωh ) be the Lipschitz map whose graph is the inertial manifold Mh in Theorem 6.1. We may assume that Lip Ψh ≤ 1 (see the proof of Theorem 1 in [66]). If we let Φh = Ψh ◦ ψh−1 , then Mh can be considered as the graph of Φh with Lip Φh ≤ 1. With these notations, we have the following lemma.

118

6 Stability of Inertial Manifolds

Lemma 6.2 For any p0 ∈ Pm0 L2 (Ω0 ) and pn ∈ Pmhn L2 (Ωhn ), m    ψh pn − ψ0 p0  m ≤ α(hn ) |ai | + jhn p0 − pn L2 (Ωhn ) , n R i=1

where α(hn ) = sup{jhn φi0 − φihn L2 (Ωhn ) : i = 1, . . . , m} and p0 =

m

0 i=1 ai φi .



Proof See Lemma 3.2 in [46].

By Proposition 6.1, we can take a constant r > 0 such that λ01 , λh1 n > r for all n ∈ N. For any n ∈ N and T > 0, we define hn

γhn (T ) = sup{|e−λi

t

− e−λi t | : 1 ≤ i ≤ m, −T ≤ t ≤ T } 0

and ρ(hn ) = Fhn (jhn u) − jhn F0 (u)L∞ (L2 (Ω0 ),L2 (Ωhn )) . Then we observe that γhn (T ) → 0 and ρ(hn ) → 0 as n → ∞. For any p ∈ Rm and a bounded set B ⊂ Rm , we define βhn (p) = Φhn (p) − jhn Φ0 (p)L2 (Ωhn ) and βhn (B) = sup{βhn (p) : p ∈ B}. Let p0 (t) and pn (t) be the solutions of the equations dp0 + A0 p0 = Pm0 F0 (p0 + Φ0 (ψ0 p0 )) dt

(6.8)

dpn + Ahn pn = Pmhn Fhn (pn + Φhn (ψhn pn )) dt

(6.9)

and

with initial conditions p0 (0) = ψ0−1 p and pn (0) = ψh−1 p, respectively, for some p ∈ Rm . n With these notations, we have the following estimates. Lemma 6.3 For any T > 0 and a bounded subset B of M0 , there exists C > 0 such that for any n ∈ N and t ∈ [−T , 0],

6.2 Proof of Theorem 6.2

119

 0 pn (t) − jhn p0 (t)L2 (Ωhn ) ≤ e(λm +1)t Cγhn (T ) + Cα(hn ) +

λ0m

1 1 Lhn βhn (ψ0 B−T ) + 0 ρ(hn ) +1 λm + 1

+ 2T e(λm +1)t Cγhn (T )  C(2 + Lhn ) (2Lhn −λ0m −1)t + α(h ) n e λ0m + 1 0

and for any n ∈ N and t ∈ [0, T ],  Lh pn (t) − jhn p0 (t)L2 (Ωhn ) ≤ ert Cγhn (T ) + Cα(hn ) + ert n βhn (ψ0 BT )+ r ert

ρ(hn ) r

+ 2T ert Cγhn (T ) + ert

 C(2 + Lhn ) α(hn ) e(2Lhn −r)t , r

 pn (t) − jhn p0 (t)L2 (Ωhn ) ≤ ert Cγhn (T ) + Cα(hn ) Lh n ρ(hn ) βhn (ψ0 BT ) + ert + 2T ert Cγhn (T ) r r  rt C(2 + Lhn ) α(hn ) e(2Lhn −r)t , +e r

+ ert

where p0 (t) and pn (t) are the solutions of (6.8) and (6.9), respectively, such that p0 (0) ∈ Pm0 B and ψ0 p0 (0) = ψhn pn (0), and B−T = {p0 (t) : t ∈ [−T , 0]}, BT = {p0 (t) : t ∈ [0, T ]}. Proof Let T > 0 be arbitrary, let B be a bounded subset of M0 , and denote Bˆ −T = S0 (B, [−T , 0]) and Bˆ T = S0 (B, [0, T ]). Let p0 (t) and pn (t) be the solutions of (6.8) and (6.9), respectively, such that p0 (0) ∈ Pm0 B and ψ0 p0 (0) = ψhn pn (0). By the variation of constants formula for (6.8) and (6.9), we have pn (t) − jhn p0 (t) = e−Ahn t pn (0) − jhn e−A0 t p0 (0)  t + e−Ahn (t−s) (Pmhn Fhn (pn + Φhn (ψhn pn )) 0

− Pmhn jhn F0 (p0 + Φ0 (ψ0 p0 )))ds

(6.10)

120

6 Stability of Inertial Manifolds



t

+ 0

(e−Ahn (t−s) Pmhn jhn − jhn e−A0 (t−s) Pm0 )

F0 (p0 + Φ0 (ψ0 p0 ))ds := I + I I + I I I,

for all t ∈ [−T , T ].

(6.11)

Since F0 (Bˆ −T ) and F0 (Bˆ T ) are bounded in L2 (Ω0 ), there exists a constant C > 0 such   0 0 ˆ ai φi0 in Pm0 Bˆ −T ∪ Pm0 Bˆ T and v = m that for any u = m i=1 i=1 bi φi in Pm F0 (B−T ) ∪   m m 0 Pm F0 (Bˆ T ), we have i=1 |ai | < C and i=1 |bi | < C.  0 Step 1. We first estimate I for t ∈ [−T , 0]. We write p0 (0) = m i=1 ai φi . Since I = e−Ahn t

m 

m 

ai φihn − jhn e−A0 t

i=1

=

m 

ai (e

ai φi0

i=1

−λhi n t

−e

−λ0i t

)φihn

+

m 

i=1

ai e−λi t (φihn − jhn φi0 ), 0

i=1

we have I L2 (Ωhn )

  m    hn  −λhi n t −λ0i t ≤  (e −e )ai φi    i=1

+

m 

L2 (Ωhn )

|ai |e−λi t φihn − jhn φi0 L2 (Ωhn ) 0

i=1

≤ γhn (T )

m 

|ai | + e

−(λ0m +1)t

α(hn )

i=1

 m 

 |ai |

i=1

≤ Cγhn (T ) + Ce−(λm +1)t α(hn ). 0

(6.12)

Step 2. We estimate I I for t ∈ [−T , 0]. For this, we first consider the following expression: Fhn (pn +Φhn ψhn pn ) − jhn F0 (p0 + Φ0 ψ0 p0 ) = Fhn (pn + Φhn ψhn pn ) − Fhn (jhn p0 + Φhn ψhn pn ) + Fhn (jhn p0 + Φhn ψhn pn ) − Fhn (jhn p0 + Φhn ψ0 p0 ) + Fhn (jhn p0 + Φhn ψ0 p0 ) − Fhn (jhn p0 + jhn Φ0 ψ0 p0 ) + Fhn (jhn p0 + jhn Φ0 ψ0 p0 ) − jhn F0 (p0 + Φ0 ψ0 p0 ).

6.2 Proof of Theorem 6.2

121

By Lemma 6.2, we have Fhn (pn +Φhn ψhn pn ) − jhn F0 (p0 + Φ0 ψ0 p0 )L2 (Ωhn ) ≤ 2Lhn pn (s) − jhn p0 (s)L2 (Ωhn ) + Lhn βhn (ψ0 B−T ) + CLhn α(hn ) + ρ(hn ). Hence, we get  I I L2 (Ωhn ) ≤ 2Lhn

0 t

 + Lh n

0

t



+ CLhn 0 t

 + Lh n

t

+ CLhn 

0

+

e−Ahn (t−s) βhn (ψ0 B−T )ds 0

e−Ahn (t−s) α(hn )ds +

t

 ≤ 2Lhn

e−Ahn (t−s) pn (s) − jhn p0 (s)L2 (Ωhn ) ds



0

e−Ahn (t−s) ρ(hn )ds

t

e−(λm +1)(t−s) pn (s) − jhn p0 (s)L2 (Ωhn ) ds 0

0



e−(λm +1)(t−s) βhn (ψ0 B−T )ds 0

0

e−(λm +1)(t−s) α(hn )ds 0

t

e−(λm +1)(t−s) ρ(hn )ds 0

t

≤ 2Lhn e−(λm +1)t 0



0

t

e(λm +1)s pn (s) − jhn p0 (s)L2 (Ωhn ) ds 0

e−(λm +1)t e−(λm +1)t βhn (ψ0 B−T ) + CLhn 0 α(hn ) 0 λm + 1 λm + 1 0

+ Lh n

(6.13)

0

e−(λm +1)t ρ(hn ), λ0m + 1 0

+

0 0 where t e(λm +1)s < λ0 1+1 was used in deriving the last inequality. m Step 3. We estimate I I I for t ∈ [−T , 0]. To this end, we first consider the following: (jhn e−A0 (t−s) Pm0 − e−Ahn (t−s) Pmhn jhn )(v0 ) = jhn

m  i=1

hn

(e−λi (t−s) − e−λi 0

(t−s)

)bi φi0 +

m  i=1

hn

e−λi

(t−s)

bi (jhn φi0 − φihn )

122

6 Stability of Inertial Manifolds

+ e−Ahn (t−s) (Pmhn vn − Pmhn jhn v0 ) := I I I1 + I I I2 + I I I3 , where v0 = we have

∞

∈ F0 (Bˆ −T ) and vn =

0 i=1 bi φi

I I I1 L2 (Ωhn )

∞

hn i=1 bi φi

∈ L2 (Ωhn ). For any s ∈ (t, 0],

  m     −λ0i (t−s) −λhi n (t−s) 0 = jhn (e −e )bi φi    i=1

  m    −λ0i (t−s) −λhi n (t−s) 0 ≤ 2  (e −e )bi φi    i=1

≤ 2γhn (T )

L2 (Ωhn )

L2 (Ω0 )

m 

|bi | ≤ 2Cγhn (T ),

i=1

I I I2 L2 (Ωhn )

 m     hn  −λhi n (t−s) 0 = e bi (jhn φi − φi )   i=1

L2 (Ωhn )

≤ e−(λm +1)(t−s) α(hn ) 0

m 

|bi | ≤ Ce−(λm +1)(t−s) α(hn ), 0

i=1

and I I I3 2L2 (Ω

hn )

=

m 

hn

e−2λi

(t−s)

i=1

≤e

|(Pmhn vn − Pmhn jhn v0 , φihn )|2



−2(λ0m +1)(t−s)

α(hn )

m 

2 |bi |

.

i=1

It follows that I I I3 L2 (Ωhn ) ≤ e−(λm +1)(t−s) α(hn ) 0

m  i=1

Since

0 t

e(λm +1)s ds < 0

1 , λ0m +1

we have

|bi | ≤ Ce−(λm +1)(t−s) α(hn ). 0

6.2 Proof of Theorem 6.2

123

 I I I L2 (Ωhn ) ≤

t

0

 2Cγhn (T )ds +

0

2Ce−(λm +1)(t−s) α(hn )ds 0

t

e−(λm +1)t . λ0m + 1 0

≤ 2T Cγhn (T ) + 2Cα(hn )

(6.14)

Step 4. We estimate pn (t)−jhn p0 (t)L2 (Ωhn ) for t ∈ [−T , 0]. By putting (6.12), (6.13), and (6.14) together into (6.11), we get pn (t) − jhn p0 (t)L2 (Ωhn ) ≤ Cγhn (T ) + Ce−(λm +1)t α(hn )  0 0 0 + 2e−(λm +1)t Lhn · e(λm +1)s pn (s) − jhn p0 (s)L2 (Ωhn ) ds 0

(6.15)

t

+

e

−(λ0m +1)t

λ0m + 1

e−(λm +1)t λ0m + 1 0

Lhn βhn (ψ0 B−T ) + ρ(hn )

e−(λm +1)t . λ0m + 1 0

+ 2T Cγhn (T ) + C(2 + Lhn )α(hn )

(6.16)

Let g(t) = e(λm +1)t phn (t) − jhn p0 (t)L2 (Ωhn ) . Multiply both sides of inequality (6.16) 0

by e(λm +1)t to get 0

g(t) ≤ e

(λ0m +1)t

+ +

λ0m

 Cγhn (T ) + Cα(hn ) + 2Lhn

0

g(s)ds t

1 1 0 Lhn βhn (ψ0 B−T ) + 0 ρ(hn ) + 2T e(λm +1)t Cγhn (T ) +1 λm + 1

C(2 + Lhn ) α(hn ). λ0m + 1

Applying Gronwall’s inequality, we derive that  0 g(t) ≤ e(λm +1)t Cγhn (T ) + Cα(hn ) + +

λ0m

1 Lh βh (ψ0 B−T ) +1 n n

 C(2 + Lhn ) 1 (λ0m +1)t 2Lhn t ρ(h α(h ) + 2T e Cγ (T ) + ) . n hn n e λ0m + 1 λ0m + 1

Consequently, for any t ∈ [−T , 0], we have pn (t) − jhn p0 (t)L2 (Ωhn )

 0 ≤ e(λm +1)t Cγhn (T ) + Cα(hn )

124

6 Stability of Inertial Manifolds

1 1 0 Lh βh (ψ0 B−T ) + 0 ρ(hn ) + 2T e(λm +1)t Cγhn (T ) λ0m + 1 n n λm + 1  C(2 + Lhn ) 0 α(hn ) e(−2Lhn −λm −1)t . + λ0m + 1

+

Step 5. Finally, we estimate pn (t) − jhn p0 (t)L2 (Ωhn ) for t ∈ [0, T ]. By the same techniques as in Step 1, we have I L2 (Ωhn ) ≤ Cγhn (T ) + e−rt Cα(hn ). Furthermore, we obtain  I I L2 (Ωhn ) ≤ 2Lhn

t 0

e−Ahn (t−s) pn (s) − jhn p0 (s)L2 (Ωhn ) ds 

+ Lhn βhn (ψ0 BT )  + CLhn α(hn ) ≤ 2Lhn e−rt



t

0

t

e−Ahn (t−s) ds

0

t

e

−Ahn (t−s)

≤ 2Lhn e

t

0

e−Ahn (t−s) ρ(hn )ds

0

ers pn (s) − jhn p0 (s)L2 (Ωhn ) ds

+ CLhn α(hn )e 

t

ds +

0

+ Lhn βhn (ψ0 BT )e−rt

−rt



−rt



t



t

ers ds

0

e ds + ρ(hn )e rs

−rt

0



t

ers ds

0

ers pn (s) − jhn p0 (s)L2 (Ωhn ) ds

CLhn ρ(hn ) Lh α(hn ) + , + n β(hn ) + r r r t where the fact that 0 ers ds ≤ ert /r was is used for the last inequality. To estimate I I I , we consider separately I I I1 L2 (Ωhn ) ≤ 2γhn (T )

m 

|bi | ≤ 2Cγhn (T ),

i=1

I I I2 L2 (Ωhn )

 m     hn  −λhi (t−s) 0 ≤ e bi (jhn φi − φi )   i=1

≤ e−r(t−s) α(hn )

L2 (Ωhn )

m  i=1

|bi | ≤ e−r(t−s) Cα(hn ),

6.2 Proof of Theorem 6.2

125

and I I I3 L2 (Ωhn ) ≤ e−r(t−s) α(hn )

m 

|bi | ≤ e−r(t−s) Cα(hn ).

i=1

Then we get  I I I L2 (Ωhn ) ≤

0

t

 2Cγhn (T )ds +

≤ 2T Cγhn (T ) +

t

2e−r(t−s) Cα(hn )ds

0

2C α(hn ). r

Combining all the estimates above we have, pn (t) − jhn p0 (t)L2 (Ωhn ) ≤ 2e−rt Lhn + Cγhn (T ) + e−rt Cα(hn ) + + 2T Cγhn (T ) +

 0

t

ers pn (s) − jhn p0 (s)L2 (Ωhn ) ds

Lh n ρ(hn ) βhn (ψ0 BT ) + r r

C(2 + Lhn ) α(hn ). r

(6.17)

Denote g(t) = ert pn (t) − jhn p0 (t)L2 (Ωhn ) . Multiplying both sides of (6.17) by ert we obtain  g(t) ≤ 2Lhn

0

t

g(s)ds + ert Cγhn (T ) + Cα(hn ) + ert

Lh n βhn (ψ0 BT ) r

ρ(hn ) C(2 + Lhn ) + 2T ert Cγhn (T ) + ert α(hn ). + ert r r Next, by Gronwall’s inequality, 

Lh n ρ(hn ) βhn (ψ0 BT ) + ert r r  C(2 + Lhn ) α(hn ) e2Lhn t . + 2T ert Cγhn (T ) + ert r

g(t) ≤ ert Cγhn (T ) + Cα(hn ) + ert

Finally, we deduce that  Lh pn (t) − jhn p0 (t)L2 (Ωhn ) ≤ ert Cγhn (T ) + Cα(hn ) + ert n βhn (ψ0 BT ) r  rt ρ(hn ) rt rt C(2 + Lhn ) +e + 2T e Cγhn (T ) + e α(hn ) · e(2Lhn −r)t . r r



126

6 Stability of Inertial Manifolds

In the following lemma, we estimate the linear semigroups on orthogonal complements. Lemma 6.4 For any ε > 0, T > 0, and a bounded subset B of L2 (Ω0 ), there is K > 0 such that for any u ∈ B and n ≥ K, 

T 0

e−Ahn t Qhmn jhn u − jhn e−A0 t Q0m uL2 (Ωhn ) dt < ε.

Proof Since B is bounded, we can choose δ > 0 and k ∈ N (k > m) such that 4δuL2 (Ω0 ) < ε/2

and

2e−(λk+1 −1)δ uL2 (Ω0 ) < ε/6(T − δ) 0

for all u ∈ B. By Proposition 6.1 and Lemma 6.1, we can take a subsequence of {hn }n∈N , still denoted by {hn }n∈N , and the first k eigenfunctions, denoted by {φ10 , . . . , φk0 }, with corresponding eigenvalues {λ01 , . . . , λ0k }, such that hn

γk (hn ) := sup{|e−λi

t

− e−λi t | : 1 ≤ i ≤ k, 0 ≤ t ≤ T } → 0, 0

and αk (hn ) := sup{φihn − jhn φi0 L2 (Ωhn ) : 1 ≤ i ≤ k} → 0 as n → ∞. For any u =

∞

0 i=1 ai φi

in B and t ∈ [0, δ], we have

e−Ahn t Qhmn jhn u−jhn e−A0 t Q0m uL2 (Ωhn ) ≤ e−Ahn t Qhmn jhn uL2 (Ωhn ) + 2e−A0 t Q0m uL2 (Ω0 ) ≤ 4uL2 (Ω0 ) . This implies that 

δ 0

e−Ahn t Qhmn jhn u − jhn e−A0 t Q0m uL2 (Ωhn ) dt
0 such that for any m   0 0 ˆ u= m a φ 0 in B−T and v = m i=1 bi φi in Pm F0 (B−T ), we have i=1 |ai | < C and m i=1 i i |b | < C. i i=1 Step 1. There is N1 > 0 such that for any n ≥ N1 , βhn (ψ0 Pm0 B) ≤ η βhn (ψ0 B−T ) +

ε . 4η0

For any p ∈ B, we have Φhn (ψ0 p) − jhn Φ0 (ψ0 p)L2 (Ωhn )  0 ≤ eAhn s Qhmn Fhn − jhn eA0 s Q0m F0 L2 (Ωhn ) ds −∞

 =

−T −∞



+ ≤

eAhn s Qhmn Fhn − jhn eA0 s Q0m F0 L2 (Ωhn ) ds

0 −T

eAhn s Qhmn Fhn − jhn eA0 s Q0m F0 L2 (Ωhn ) ds

 0 ε + eAhn s Qhmn (Fhn − jhn F0 )L2 (Ωhn ) ds 8η0 −T  0 + (eAhn s Qhmn jhn − jhn eA0 s Q0m )F0 L2 (Ωhn ) ds −T

ε + I + I I. := 8η0 By Lemma 6.3, we obtain hn

eAhn s Qhmn (Fhn − jhn F0 )L2 (Ωhn ) ≤ eλm+1 s (Fhn − jhn F0 )L2 (Ωhn )

130

6 Stability of Inertial Manifolds hn

≤ eλm+1 s (2Lhn pn (s) − jhn p0 (s)L2 (Ωhn ) + Lhn βhn (ψ0 B−T ) + CLhn α(hn ) + ρ(hn ))  0 ≤ 2e(λm +1)s Lhn Cγhn (T ) + 2Lhn Cα(hn ) +

2 L2 βh (ψ0 B−T ) λ0m + 1 hn n

2 0 Lhn ρ(hn ) + 4T e(λm +1)s Lhn Cγhn (T ) +1  hn 0 2C(2 + Lhn ) L + α(h ) e(2Lhn +λm+1 −λm −1)s h n n 0 λm + 1 +

λ0m

hn

hn

hn

+ eλm+1 s Lhn βhn (ψ0 B−T ) + eλm+1 s CLhn α(hn ) + eλm+1 s ρ(hn ). Consequently,  I= ≤

0

−T

eAhn s Qhmn (Fhn − jhn F0 )L2 (Ωhn ) ds

2Lhn Cγhn (T ) n 2Lhn + λhm+1

+

2Lhn Cα(hn ) n 2Lhn + λhm+1 − λ0m − 1

2L2hn βhn (ψ0 B−T ) (λ0m

n + 1)(2Lhn + λhm+1

+

4Lhn T Cγhn (T )

+

Lhn βhn (ψ0 B−T )

n 2Lhn + λhm+1

n λhm+1

 ≤

+

+

− λ0m

+ +

n 2Lhn + λhm+1

+

+

CLhn α(hn ) n λhm+1

+

+

ρ(hn ) n λhm+1

Lh n n λhm+1

 βhn (ψ0 B−T )

2Lhn Cα(hn ) n 2Lhn + λhm+1 − λ0m − 1

2Lhn ρ(hn ) n 2Lhn + λhm+1

n 2Lhn + λhm+1 − λ0m − 1

n + 1)(2Lhn + λhm+1 − λ0m − 1)

2L2hn 2CLhn γhn (T )

2Lhn ρ(hn )

2C(2 + Lhn )Lhn α(hn ) (λ0m

n (λ0m + 1)(2Lhn + λhm+1 − λ0m − 1)

+

− 1)

+

− λ0m

−1

+

4Lhn CT γhn (T ) n 2Lhn + λhm+1

2C(2 + Lhn )Lhn α(hn ) n (λ0m + 1)(2Lhn + λhm+1

− λ0m − 1)

+

CLhn α(hn ) n λhm+1

+

ρ(hn ) n λhm+1

6.2 Proof of Theorem 6.2

 :=

131

2L2hn n (λ0m + 1)(2Lhn + λhm+1 − λ0m − 1)

+

Lh n n λhm+1

 βhn (ψ0 B−T ) + I˜.

(6.22)

By Proposition 6.1, we can take N1 > 0 such that for any n ≥ N1 , λhmn > λ0m − δ

and

λhmn > Lhn − δ.

Note that λ0m + 1 > Lhn and n n 2Lhn + λhm+1 − λ0m − 1 = 2Lhn + (λhm+1 − λhmn ) + (λhmn − λ0m ) − 1 √ > 2Lhn + 2 2Lhn − 1 − δ.

Thus, we have 2L2hn (λ0m

n + 1)(2Lhn + λhm+1

− λ0m

2 ≤ √ 2 2 + 1−δ − 1)

and Lh n n λhm+1

1 . < √ 2 2+1−δ

Since γhn (T ), α(hn ) and ρ(hn ) converge to 0 as n → ∞, we can choose N1 > 0 such ε for all n ≥ N1 . Consequently, that I˜ < 16η0 I < η βhn (ψ0 B−T ) +

ε . 16η0

(6.23)

On the other hand, by Lemma 6.4, we can take N > 0 such that for any u ∈ L2 (Ω0 ) with uL2 (Ω0 ) ≤ C and n ≥ N,  II =

0

−T

(eAhn s Qhmn jhn − jhn eA0 s Q0m )F0 L2 (Ωhn ) ds
0 such that βhn (ψ0 B) < ε for all n ≥ N . Arguing as in Step 1, we conclude that for each k ∈ N, there is Nk > Nk−1 such that for any n ≥ Nk , βhn (ψ0 B−(k−1)T ) < η βhn (ψ0 B−kT ) +

ε . 4η0

Hence, we have βhn (ψ0 Pm0 B) < ηk βhn (ψ0 B−kT ) +

k−1 ε  i ε η < ηk M + . 4η0 4 i=0

Take k > 0 such that ηk M < ε/2 and N > Nk . Then for any n > N, we have βhn (ψ0 Pm0 B) < ε. This completes the proof.  For each n ∈ N, we define a map j˜hn : M0 → Mhn by ψ0 p0 + Φhn (ψ0 p0 ), j˜hn (p0 + Φ0 (ψ0 p0 )) = ψh−1 n for p0 ∈ Pm0 M0 . Note that j˜hn is continuous. We also define jˆhn : B(u0 , rn ) → B(uhn , rn ) by ⎧ ⎨j˜ u if j˜ u ∈ B(u , r ), hn hn hn n jˆhn u = ⎩j˜h v if j˜h u ∈ / B(uhn , rn ), n n where v ∈ B(u0 , rn ) is chosen so that j˜hn u − j˜hn vL2 (Ωhn ) = dist (j˜hn u, ∂B(uhn , rn )). Similarly, we define i˜hn : Mhn → M0 by i˜hn (pn + Φhn (ψhn pn )) = ψ0−1 ψhn pn + Φ0 (ψhn pn ) for pn ∈ Pmhn Mhn and define iˆhn : B(uhn , rn ) → B(u0 , rn ) by

(6.25)

6.2 Proof of Theorem 6.2

133

⎧ ⎨i˜ u if i˜ u ∈ B(u , r ), hn hn 0 n iˆhn u = ⎩i˜h v if i˜h u ∈ / B(u0 , rn ), n n where v ∈ B(uhn , rn ) is chosen so that i˜hn u − i˜hn vL2 (Ω0 ) = dist(i˜hn u, ∂B(u0 , rn )).

(6.26)

Note that i˜hn = j˜h−1 and i˜hn is also continuous. n Lemma 6.6 Let B be a bounded subset of M0 . Then jhn (u) − j˜hn (u)L2 (Ωhn ) → 0

as n → ∞,

uniformly for u ∈ B. Proof Let B be a bounded subset of M0 . For any u ∈ B, there exists ai ∈ R (1 ≤ i ≤ m) such that u = p0 + Φ0 (ψ0 p0 ),

with p0 =

m 

ai φi0 ∈ Pm0 M0 .

i=1

Since ψh−1  ∞ m 0 2 = 1, implies that Lemma 6.2, we have n L (R ,Pm L (Ωhn )) jhn (u) − j˜hn (u)L2 (Ωhn ) = jhn (p0 + Φ0 (ψ0 p0 )) − j˜hn (p0 + Φ0 (ψ0 p0 ))L2 (Ωhn ) ≤ jhn p0 − ψh−1 ψ0 p0 L2 (Ωhn ) + jhn Φ0 (ψ0 p0 ) n − Φhn (ψ0 p0 )L2 (Ωhn ) ≤ ψh−1 |ψhn jhn p0 − ψ0 p0 |Rm + βhn (ψ0 Pm0 B) n ≤ α(hn )

m 

|ai | + βhn (ψ0 Pm0 B).

i=1

By Lemmas 6.1 and 6.5, we see that α(hn ) and β(hn ) converge to 0 as n → ∞. Hence,  we derive that jhn (u) − j˜hn (u)L2 (Ωhn ) → 0 as n → ∞. Corollary 6.1 For n ∈ N, let Bn be a bounded subset of Mhn . Then ihn (u) − i˜hn (u)L2 (Ω0 ) → 0 as n → ∞ uniformly for u ∈ Bn .

134

6 Stability of Inertial Manifolds

Lemma 6.7 For any n ∈ N, let Bn ⊂ Mhn be a bounded set. Then for any T > 0 and ε > 0, there exist δ > 0 and N0 ∈ N such that for all n ≥ N0 , Shn (u, t) − Shn (u, ˜ t)L2 (Ωhn ) < ε, uniformly for u, u˜ ∈ Bn with u − u ˜ L2 (Ωhn ) < δ and t ∈ [−T , T ]. Proof For a given n ∈ N, let Bn ⊂ Mhn be a bounded set and u, u˜ ∈ Bn . Let pn (t) and p˜ n (t) be the solutions of (6.9) with pn (0) = Pmhn u and p˜ n (0) = Pmhn u, ˜ respectively. Then Shn (u, t) = pn (t) + Φhn (pn (t)) and Shn (u, ˜ t) = p˜ n (t) + Φhn (p˜ n (t)).

(6.27)

Let p0 (t) and p˜ 0 (t) be the solutions of Eq. (6.8) with p0 (0) = ψ0−1 ψhn pn (0) and p˜ 0 (0) = ψ0−1 ψhn p˜ n (0), respectively. Then we obtain pn (t) − p˜ n (t)L2 (Ωhn ) ≤ pn (t) − jhn p0 (t)L2 (Ωhn ) + jhn p0 (t) − jhn p˜ 0 (t)L2 (Ωhn ) + jhn p˜ 0 (t) − p˜ n (t)L2 (Ωhn ) := I + I I + I I I.

(6.28)

By Lemma 6.3, we can choose N0 ∈ N such that if n ≥ N0 , then I + III ≤

ε . 4

(6.29)

Moreover, we have I I ≤ 2p0 (t) − p˜ 0 (t)L2 (Ω0 ) . Since p0 is Lipschitz on ψ0−1 ψhn Pmhn Bn × [−T , T ], there exists δ > 0 such that if u, u˜ ∈ ˜ L2 (Ωhn ) < δ, then Bn with u − u II ≤

ε . 4

(6.30)

Note that Φhn (pn (t)) − Φhn (p˜ n (t))L2 (Ωhn ) ≤ pn (t) − p˜ n (t)L2 (Ωhn ) .

(6.31)

By combining (6.27), (6.28), (6.29), (6.30) and (6.31), we complete the proof of the lemma. 

6.2 Proof of Theorem 6.2

135

Proof (End of Proof of Theorem 6.2) We first show that there is N > 0 such that j˜hn is an into ε-isometry on B(u0 , rn ) for all n ≥ N. Since B(u0 , rn ) is bounded in L2 (Ω0 ), we can take Cn > 0 such that uL2 (Ω0 ) < Cn for all u ∈ B(u0 , rn ). By Lemma 6.6, we can take N > 0 such that if n ≥ N , then   jh  − 1 < n

ε ε and jhn (u) − j˜hn (u)L2 (Ωhn ) < . 18Cn 9

For any u, u˜ ∈ B(u0 , rn ), we let u = p + Φ0 (ψ0 p) and u˜ = p˜ + Φ0 (ψ0 p) ˜ for some p, p˜ ∈ Pm0 M0 . For any n ≥ N, we have j˜hn u − j˜hn u ˜ L2 (Ωhn ) − u − u ˜ L2 (Ω0 ) ≤ j˜hn (p + Φ0 (ψ0 p)) − jhn (p + Φ0 (ψ0 p))L2 (Ωhn ) + jhn (p + Φ0 (ψ0 p)) − jhn (p˜ + Φ0 (ψ0 p)) ˜ L2 (Ωhn ) + jhn (p˜ + Φ0 (ψ0 p)) ˜ − j˜hn (p˜ + Φ0 (ψ0 p)) ˜ L2 (Ωhn ) − u − u ˜ L2 (Ω0 ) ≤

2ε ε + (jhn  − 1)u − u ˜ L2 (Ω0 ) < . 9 3

Similarly, we can show that u0 − u˜ 0 L2 (Ω0 ) − j˜hn u0 − j˜hn u˜ 0 L2 (Ωhn ) < ε/3. This implies that j˜hn is an into ε/3-isometry on B(u0 , rn ). To deduce that i˜hn is an into ε/3-isometry on B(uhn , rn ), we can argue as above and use Corollary 6.1. Now we show that jˆhn is an ε-isometry on B(u0 , rn ). Indeed, note that for any u ∈ B(u0 , rn ), ε ε j˜hn u − uhn L2 (Ωhn ) = j˜hn u − j˜hn u0 L2 (Ωhn ) ≤ u − u0 L2 (Ω) + < rn + . 3 3 This implies that j˜hn B(u0 , rn ) ⊂ B(uhn , rn + ε/3). Hence, for u, u˜ ∈ B(u0 , rn ), it is enough to consider the following two cases: / B(uhn , rn ); Case 1. j˜hn u and j˜hn u˜ ∈ / B(uhn , rn ). Case 2. j˜hn u ∈ B(uhn , rn ) and j˜hn u˜ ∈ For Case 1, by (6.25), we have

136

6 Stability of Inertial Manifolds

jˆhn u − jˆhn u ˜ L2 (Ωhn ) = j˜hn v − j˜hn v ˜ L2 (Ωhn ) ≤ j˜hn v − j˜hn uL2 (Ωhn ) + j˜hn v˜ − j˜hn u ˜ L2 (Ωhn ) + j˜hn u − j˜hn u ˜ L2 (Ωhn ) ≤ ε + u − u ˜ L2 (Ω0 ) , where v and v˜ are chosen as in (6.25) corresponding to u and u, ˜ respectively. For the Case 2, we see that jˆhn u − jˆhn u ˜ L2 (Ωhn ) = j˜hn u − j˜hn v ˜ L2 (Ωhn ) ≤ j˜hn u − j˜hn u ˜ L2 (Ωhn ) + j˜hn v˜ − j˜hn u ˜ L2 (Ωhn ) ≤

2ε + u − u ˜ L2 (Ω0 ) . 3

Arguing in the same way, we can show that in any case, u − u ˜ L2 (Ω0 ) ≤ jˆhn u − jˆhn u ˜ L2 (Ωhn ) + ε. For any un ∈ B(uhn , rn ), let u = iˆhn un . Then we see that u ∈ B(u0 , rn ). If j˜hn u ∈ B(uhn , rn ), then un − jˆhn uL2 (Ωhn ) = un − j˜hn iˆhn un L2 (Ωhn ) ≤ i˜hn un − iˆhn un L2 (Ω0 ) + ≤

ε 3

2ε . 3

If j˜hn u ∈ / B(uhn , rn ), then un − jˆhn uL2 (Ωhn ) = un − j˜hn vL2 (Ωhn ) ≤

ε + i˜hn un − iˆhn un L2 (Ω0 ) + j˜hn iˆhn un − j˜hn vL2 (Ωhn ) 3

≤ ε. Here, v ∈ B(u0 , rn ) is chosen as in (6.25) corresponding to iˆhn un . This shows that jˆhn is an ε-isometry on B(u0 , rn ).

6.3 Proof of Theorem 6.3

137

Similarly, we can show that iˆhn is an ε-isometry on B(uhn , rn ). Consequently, we get dGH (B(uhn , rn ), B(u0 , rn )) < ε for all n ≥ N . The contradiction completes the proof.

6.3



Proof of Theorem 6.3

We are now in a position to prove Theorem 6.3. By the way of contradiction we suppose that the conclusion of the theorem is not true. Then there are ε > 0, T > 0, and u0 ∈ M0 such that for any n ∈ N, there is hn ∈ Diff(Ω0 ) with dC 1 (hn , id) < 1/n such that for any uhn ∈ Mhn , there exists rn > 0 such that T DGH (Shn |B(uhn ,rn ) , S0 |B(u0 ,rn ) ) ≥ ε. hn Let {λh1 n , . . . , λhmn } and {φ1hn , . . . , φm } be the first m eigenvalues and corresponding eigenfunctions of Ahn , respectively. By Proposition 6.1, there are eigenfunctions 0 } with respect to the first m eigenvalues {λ0 , . . . , λ0 } of A and a subsequence {φ10 , . . . , φm 0 m 1 of {hn }n∈N , still denoted by {hn }n∈N , such that

φihn → φi0 in L2 (RN ) as n → ∞ for all 1 ≤ i ≤ m. For each n ∈ N, we choose uhn ∈ Mhn such that ψhn Pmhn uhn = ψ0 Pm0 u0 . T (S | Now, let us show that DGH hn B(uhn ,rn ) , S0 |B(u0 ,rn ) ) < ε for sufficiently large n. Let p(t) and pn (t) be the solutions of (6.8) and (6.9), respectively, such that p(0) ∈ Pm0 B(u0 , rn ) and ψ0 p(0) = ψhn pn (0). By Lemma 6.3, there is N > 0 such that pn (t)L2 (Ωhn ) ≤ jhn p(t)L2 (Ωhn ) + M for t ∈ [−T , T ] and n ≥ N, where M > 0 is given as in (6.21). It follows that i˜hn (pn (t) + Φhn (ψhn pn (t))L2 (Ω0 ) = ψ0−1 ψhn pn (t) + Φ0 (ψhn pn (t))L2 (Ω0 ) ≤ pn (t)L2 (Ωhn ) + M ≤ jhn p(t)L2 (Ωhn ) + 2M ≤ 2Cn + 2M,

138

6 Stability of Inertial Manifolds

where Cn = sup{S0 (u, t)L2 (Ω0 ) : u ∈ B(u0 , rn ), t ∈ [−T , T ]}. Then there is u˜ 0 ∈ M0 and r˜n > 0 such that S0 (B(u0 , rn ), [−T , T ]) ⊂ B(u˜ 0 , r˜n ) and i˜hn (Shn (B(uhn , rn ), [−T , T ])) ⊂ B(u˜ 0 , r˜n ), for all n ≥ N. For each n ∈ N, let u˜ hn ∈ Mhn be such that ψhn Pmhn u˜ hn = ψ0 Pm0 u˜ 0 . Next we choose δ > 0 and N0 ∈ N corresponding to T , ε/4, and B(u˜ hn , r˜n + ε) in Lemma 6.7. As in the proof of Theorem 6.2, we can choose N1 ∈ N such that the maps jˆhn : B(u˜ 0 , r˜n ) → B(u˜ hn , r˜n ) and iˆhn : B(u˜ hn , r˜n ) → B(u˜ 0 , r˜n ) are min{δ, ε/4}isometries for any n ≥ N1 . Also, we can assume that j˜hn and iˆhn are into min{δ, ε/4}isometries on B(u˜ 0 , r˜n ) and B(u˜ hn , r˜n ), respectively, for all n ≥ N1 . For a given T > 0 and u ∈ B(u0 , rn ), let u(t) = S0 (u, t) and un (t) = Shn (jˆhn (u), t) for t ∈ [−T , T ], and denote B0n = {Pm0 u(t) : u ∈ B(u0 , rn ), t ∈ [−T , T ]}. Case 1. Suppose j˜hn S0 (u, t) and j˜hn u ∈ B(u˜ hn , r˜n ) for t ∈ [−T , T ]. Then we have   ˆ  jhn (S0 (u, t)) − Shn (jˆhn (u), t)     ≤ j˜hn (u(t)) − jhn (u(t))

L2 (Ωhn )

L2 (Ω

    ≤ j˜hn (u(t)) − jhn (u(t))

L2 (Ω

hn )

hn )

  + jhn (u(t)) − un (t)L2 (Ω

hn )

  + jhn (p(t)) − pn (t)L2 (Ω

  + jhn Φ0 (ψ0 p(t)) − Φhn (ψhn pn (t))L2 (Ω

hn )

hn )

:= In + I In + I I In . By Lemma 6.6, we choose N2 > N1 such that In < ε/6 for any n ≥ N2 . By Lemma 6.3, we take N3 > N2 such that I In = jhn (p(t)) − pn (t)L2 (Ωhn ) < ε/6 for any n ≥ N3 . Moreover, we have I I In = jhn Φ0 (ψ0 p(t)) − Φhn (ψhn pn (t))L2 (Ωhn ) ≤ jhn Φ0 (ψ0 p(t)) − Φhn (ψ0 p(t))L2 (Ωhn ) + Φhn (ψ0 p(t)) − Φhn (ψhn pn (t))L2 (Ωhn )

6.3 Proof of Theorem 6.3

139

 ≤ βhn (ψ0 B˜ 0 ) + α(hn )



m 

|ai (t)| + jhn p(t) − pn (t)L2 (Ωhn )

i=1

= βhn (ψ0 B˜ 0 ) + α(hn )

m 

|ai (t)| + I In .

i=1

Since α(hn ) and βhn (ψ0 B˜ 0 ) converge to 0 as n → ∞, by Lemma 6.3, there is an N4 > N3 such that I I In < ε/6 for all n ≥ N4 . Consequently,   ˆ  jhn (S0 (u, t)) − Shn (jˆhn (u), t)

L2 (Ωhn )

(6.32)

0 and f : R → R is a C 2 function such that f (0) = 0, f  ≤ l

for some l > 0,

(7.2)

Moreover, we assume here that f satisfies the dissipativity condition, namely, lim sup |s|→∞

f (s) < 0. s

This condition on f easily implies that there exists C0 > 0 such that f (s)s ≤ C0 |s| for all s ∈ R.

(7.3)

Chafee and Infante analyzed the asymptotic behavior of orbits of the semidynamical / {12 , 22 , . . .}, five-one system T0 generated by their equation. Henry [35] proved that if λ ∈ 2 then the time-one map T0 (1) of T0 is C -Morse-Smale, and so it has a finite number of equilibria that are hyperbolic. Throughout the paper, we assume that λ ∈ / {12 , 22 , . . .}.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Lee, C. Morales, Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-12031-2_7

141

142

7 Stability of Chafee-Infante Equations

Very recently, Bortolan et al. [13] dealt with Lipschitz perturbations of Morse-Smale semigroups whose critical elements are hyperbolic. For this, they introduced the notions of L-hyperbolicity, L-transversality, and L-Morse-Smale systems to cases with lack of differentiability. The objective of this chapter is to prove the geometric stability of Chafee-Infante equations under Lipschitz perturbations of the domain and equation. Moreover, we study the geometric equivalence and the continuity of global attractors. The geometric stability here is a variation of the Gromov-Hausdorff stability in which parametrized families of PDE’s are considered. One difficulty one faces when one attacks this problem is that the phase space of the induced semidynamical system changes as we change the domain. In fact, the phase spaces H01 (Ω) and H01 (Ωε ), which contain the global attractors A and Aε , respectively, can be disjoint even if Ωε is only a small perturbation of Ω ⊂ Rn . To overcome this difficulty, many people used the technique adopted by Henry [36], which makes it possible to consider the problem of continuity of the attractors as Ωε → Ω in a fixed phase space H01 (Ω) (see, for example, [11, 60]). More precisely, they followed the general approach, which basically involves “pull-backing” the perturbed problems to the fixed domain Ω and then considering the family of abstract semilinear problems thus generated. Another method is adopted by Arrieta and Carvalho [6], which consider the extended phase space Hε1 := H 1 (Ωε ∩ Ω) ⊕ H 1 (Ωε \ Ω) ⊕ H 1 (Ω \ Ω ε ) to compare the attractors in different phase spaces. Let us make some remarks about these two methods. First, for the pull-backing technique, the authors considered the pulled-back systems on a fixed domain with induced global attractors and then studied the continuity of the induced global attractors. For the domain extension method in [6], the authors used the norms  · Hε1 and  · H 1 on the δ

extended spaces Hε1 and Hδ1 (ε, δ ∈ [0, ε0 ]), respectively, which in general cannot be compared if ε = δ. However, we note that it is not easy to obtain some information on the change of dynamics of trajectories inside the global attractors using pull-backing or domain extension methods. We first perturb the Chafee-Infante Eq. (7.1) as follows. For each η ∈ [0, 1], we consider ⎧ ⎨u − u = λf (u) + ηg(u) in Ω × (0, ∞), t xx η (7.4) ⎩u(x, t) = 0 on ∂Ωη × (0, ∞), where g : R → R is a globally Lipschitz function with Lipschitz constant L, Ωη = (a(η), π + b(η)), and a, b : [0, 1] → R are continuous functions with a(0) = b(0) = 0. Note that the nonlinearity λf + ηg of (7.4) satisfies the dissipativity condition for small η. Then there exists η0 > 0 such that for any η ∈ [0, η0 ], the problem (7.4) is well posed in H01 (Ωη ) and has the global attractor Aη (see, for example, Theorems 2.2 and 2.3 in [7]).

7.2 L-Morse-Smale and Equivalence of Global Attractors

143 1/2

For simplicity, we let Xη = L2 (Ωη ) with the L2 norm  · Xη , and let Xη with the H01 norm  · X1/2 . We define a map iη : η

1/2 Xη

→

1/2 X0

= H01 (Ωη )

by

  1 1 iη u(x) = u x + a(η) (π − x) + b(η) x π π 1/2

and a map jη : X0

1/2

→ Xη

by

 b(η) a(η) (π + b(η) − x)− (x −a(η)) . jη u(x) = u x − π + b(η) − a(η) π + b(η) − a(η) 

By Proposition 3.2 in [7], we see that there exist η0 > 0 and C > 0 such that for any η ∈ [0, η0 ] and β ∈ ( 12 , 34 ), iη uη Xβ ≤ C, η

 1/2 for all uη ∈ Aη . This implies that the set η∈[0,η0 ] iη (Aη ) is precompact in X0 . Moreover, we see that iη and jη are continuous, and iη op ≤ 2 and jη op ≤ 2. Let Aη d2 denote the operator − dx 2 with homogeneous Dirichlet boundary condition on ∂Ωη . Let 1/2

Tη be the semidynamical system on Xη induced by the Lipschitz-perturbed system (7.4). We know that each Aη has a family of eigenvalues {λk,η }∞ k=1 such that λ1,η ≤ λ2,η ≤ · · · → ∞, and a family of corresponding eigenfunctions {φk,η }∞ k=1 contained in Xη , which is an 1/2 orthonormal basis in Xη . Let Fη : Xη → Xη be the Nemytskii operator induced by 1/2 f and Gη : Xη → Xη be the Nemytskii operator induced by g. For η ∈ [0, 1] and β ∈ ( 12 , 1), we let 1/2

Xηβ = {u ∈ Xη : Aβη uXη < ∞},  β β β β where Aη u = ∞ k=1 λk,η (u, φk,η )φk,η . Note that Aη · Xη is a norm on Xη , which will be denoted by  · Xβ . η

7.2

L-Morse-Smale and Equivalence of Global Attractors

The notions of L-Morse-Smale map and geometric equivalence between two global attractors were first introduced by Bortolan et al. in [13] to investigate the internal dynamics of global attractors cases with lack of differentiability.

144

7 Stability of Chafee-Infante Equations

Let X be a Banach space, C(X) the set of continuous maps from X into itself, and L(X) the set of bounded linear operators of X into itself. We say that an equilibrium u∗ of T ∈ C(X) is L-hyperbolic if the map S : X → X defined by S(u) = T (u + u∗ ) − T (u∗ ) has a decomposition of the form S = L + N such that L ∈ L(X) is hyperbolic and there is a neighborhood U of 0 such that N : U → X has a sufficiently small Lipschitz constant. We say that ξ : Z → X is a global solution for a map S ∈ C(X) if S(ξ(n)) = ξ(n + 1) for all n ∈ Z. Denote by GS(S) the set of bounded global solutions for S ∈ C(X). For an L-hyperbolic equilibrium u∗ of S ∈ C(X), let us define W s (u∗ , S) = {u ∈ X : S(n)u → u∗ as n → ∞}, W u (u∗ , S) = {u ∈ X : ∃ ξ ∈ GS(S) s.t. ξ(0) = u and ξ(n) → u∗ as n → −∞}, s Wloc (u∗ , S) = {u − u∗ ∈ U : S(n)u → u∗ as n → ∞}, and u Wloc (u∗ , S) = {u − u∗ ∈ U : ∃ ξ ∈ GS(S) s.t. ξ(0) = u and ξ(n) → u∗

as n → −∞}. Here, W s (u∗ , S) (resp., W u (u∗ , S)) is called the stable (resp., unstable) manifold and s (u∗ , S) (resp., W u (u∗ , S)) is called the local stable (resp., local unstable) manifold. Wloc loc Note that if S ∈ C(X) has two L-hyperbolic equilibria u∗1 and u∗2 , then W u (u∗1 , S) ∩ s (u∗ , S) = ∅ if and only if there exists ξ ∈ GS(S) such that Wloc 2 ξ(n) → u∗1 as n → −∞

ξ(n) → u∗2 as n → ∞.

and

In this case, we say that ξ is a connection between u∗1 and u∗2 . Since T0 (1) is MorseSmale, by the simple application of λ-lemma (see, for example, [58]), we see that for any hyperbolic equilibria u∗1 , u∗2 , and u∗3 of T0 (1), s s if W u (u∗1 , T0 (1)) ∩ Wloc (u∗2 , T0 (1)) and W u (u∗2 , T0 (1)) ∩ Wloc (u∗3 , T0 (1)) = ∅, s then W u (u∗1 , T0 (1)) ∩ Wloc (u∗3 , T0 (1)) = ∅.

(7.5)

Let ES = {u∗1 , . . . , u∗l } be the set of L-hyperbolic equilibria of S ∈ C(X). We say that the map S is dynamically gradient with respect to ES if for any ξ ∈ GS, there exist u∗i , u∗j ∈ ES such that lim ξ(m) − u∗i  = 0 and lim ξ(m) − u∗j  = 0

m→−∞

m→∞

7.2 L-Morse-Smale and Equivalence of Global Attractors

145

and are there is subset {u∗i1 , . . . , u∗ip } of ES and no elements ξ1 , . . . , ξp ∈ GS satisfying lim ξj (m) − u∗ij  = 0,

m→−∞

lim ξj (m) − u∗ij +1  = 0

m→∞

for j = 1, . . . , p, where uip+1 = ui1 . Let X1 and X2 be Banach spaces, and S1 ∈ C(X1 ) and S2 ∈ C(X2 ). For the global attractors Ai ⊂ Xi of Si (i = 1 or 2), we introduce the notion of geometric equivalence as a generalization of the geometric equivalence introduced in [13]. Definition 7.1 Let A1 and A2 be the global attractors of S1 and S2 , respectively. We say that A1 is geometrically equivalent to A2 if Si is dynamically gradient with respect to its family of L-hyperbolic equilibria Ei (i = 1, 2) and there exists a bijection B : E1 → E2 such that s s W u (u∗i , S1 ) ∩ Wloc (u∗j , S1 ) = ∅ iff W u (B(u∗i ), S2 ) ∩ Wloc (B(u∗j ), S2 ) = ∅,

where E1 = {u∗1 , . . . , u∗n } and 1 ≤ i, j ≤ n. For any two subsets M, N of X and x0 ∈ M ∩N, we say that M and N are L-transverse  at x0 if there exist closed subspaces X1 , X2 ⊂ X with X = X1 X2 , a real number r > 0, and two Lipschitz functions θ : BrX1 (0) → X2 and σ : BrX2 (0) → X1 with θ (0) = σ (0) = 0, Lip(θ ) < 1, Lip(σ ) < 1, such that {x0 + ξ + θ (ξ ) : ξ ∈ BrX1 (0)} ⊆ M and {x0 + σ (ξ ) + ξ : ξ ∈ BrX2 (0)} ⊆ N. We denote transversality by M − L,x0 N. Definition 7.2 We say that S ∈ C(X) is L-Morse-Smale if (i) S is dynamically gradient with respect to ES = {u∗1 , . . . , u∗p } for p ∈ N; (ii) there exists a neighborhood U of ES in X such that S : U → S(U ) is bi-Lipschitz; s (u∗ , S) = ∅, then there exist n ∈ N and x ∈ X such that (iii) if W u (u∗i , S) ∩ Wloc 0 j u s T n Wloc (u∗i , S) − L,x0 Wloc (u∗j , S) for u∗i , u∗j ∈ ES and 1 ≤ i, j ≤ p; and

(iv) (7.5) holds. In this section, using Theorem 8.9 in [13], we prove that there exists η0 > 0 such that for any η ∈ [0, η0 ], Tη is L-Morse-Smale and the global attractor Aη of Tη is geometrically equivalent to the global attractor A0 of T0 . For Banach spaces X, Y and a subset U of X, we consider the Lipschitz and sup norms for a map T : U → Y given by

146

7 Stability of Chafee-Infante Equations

T U,Lip = sup x,y∈U x=y

T (x) − T (y) x − y

and

T U,∞ = sup T (x). x∈U

Theorem 7.1 (Theorem 8.9 in [13]) Let {Sη }η∈[0,1] be a family of maps that is collectively asymptotically compact and continuous at η = 0 in C(X). Suppose that  (a) Sη has a global attractor Aη for each η ∈ [0, 1] and η∈[0,1] Aη is precompact in X; (b) there exists p ∈ N such that Eη (:= ESη ) = {u∗1,η , . . . , u∗p,η } for each η ∈ [0, 1] and max u∗i,η − u∗i,0  → 0

i=1,··· ,p

(c) there exists a neighborhood O of



as η → 0

η∈[0,1] Eη

for i = 1, . . . , p;

such that

max{Sη − S0 O,∞ , Sη − S0 O,Lip } → 0

as

η→0

and Sη : O → Sη (O) is bi-Lipschitz for each η ∈ [0, 1]; (d) S0 is a L-Morse-Smale map and its derivative is uniformly continuous on Or (E0 ), where Or (E0 ) is the r-neighborhood of E0 for r > 0. Then there exists η0 > 0 such that Sη is a L-Morse-Smale map with Aη geometrically equivalent to A0 for all η ∈ [0, η0 ]. For any a ∈ R and l > 0, consider the eigenvalue problem ⎧ ⎨−u

xx

⎩u = 0

= λu in (a, a + l), on {a, a + l}.

We know that the eigenvalues and corresponding orthonormal eigenfunctions in L2 (a, a + 2 nπ }n∈N and {sin nπ l) of the this problem are given by { nπ l l (· − a)/sin l (· − a)L2 }n∈N , respectively. Hence, we see that the spectral data of the operator A0 behave continuously, in the sense that is, for each k ∈ N, we have 1/2

λk,η → λk,0 and iη φk,η → φk,0 in X0

as η → 0.

Furthermore, we can choose η0 > 0 and δ > 0 such that λ1,η > δ for all η ∈ [0, η0 ]. We need several lemmas, which are proved as Lemmas 2.2 to 2.9 in [47]. Lemma 7.1 There exist M > 0 (independent of η), γ ∈ ( 12 , 1), and a function θ : [0, 1] → R with θ (η) → 0 as η → 0 such that for uη ∈ L2 (Ωη ) and t > 0,

7.2 L-Morse-Smale and Equivalence of Global Attractors

147

iη e−Aη t uη − e−A0 t iη uη X1/2 ≤ Mθ (η)t −γ e−δt uη Xη . 0

Lemma 7.2 For each η ∈ [0, 1], we have the following estimates: Tη (t)uη,0 X1/2 ≤ η

  ! δ L2 + 1 uη,0 Xη e− 2 (t−1) + D2 2 λl + δ

Tη (t)uη,0 L∞ (Ωη ) ≤ eE1 uη,0 L∞ (Ωη ) + eE1

for t ≥ 1,

for t ∈ [0, 1],

where D2 and E1 are given in the proof. Lemma 7.3 The collection {T˜η (1)}η∈[0,1] is collectively asymptotically compact, that is, 1/2 given sequences ηk → 0, nk → ∞, and {uk } bounded in X0 such that {T˜ηk (nk )uk } is bounded, {T˜ηk (nk )uk } has a convergent subsequence. Lemma 7.4 The family {T˜η (1)}η∈[0,1] is continuous at η = 0, that is, for any N ∈ N and 1/2 a compact subset K of X0 ,   max sup T˜η (n)u − T0 (n)uX1/2 → 0

1≤n≤N u∈K

0

as η → 0. 1/2

Lemma 7.5 There exists η0 > 0 such that for any T > 0 and a bounded set U in X0 , there is MT > 0 such that Tη (t)uη − Tη (t)vη X1/2 ≤ MT uη − vη X1/2 η

η

for all η ∈ [0, η0 ] and uη , vη ∈ jη U . Here, MT is independent of η. 1/2

Lemma 7.6 For any bounded set U in X0 , max{T˜η (1) − T0 (1)U,∞ , T˜η (1) − T0 (1)U,Lip } → 0

as η → 0.

 Lemma 7.7 There exist η0 > 0 and a neighborhood O of η∈[0,η0 ] iη Aη such that T˜η (1) is bi-Lipschitz on O with uniform bi-Lipschitz constants for all η ∈ [0, η0 ]. Lemma 7.8 Let E0 = {u∗1,0 , . . . , u∗p,0 } be the set of equilibria of T0 (1). Then there exists η0 > 0 such that for any η ∈ [0, η0 ], iη Eη consists of p L-hyperbolic equilibria of T˜η (1), say iη Eη = {u˜ ∗1,η , . . . , u˜ ∗p,η }, such that

u˜ ∗i,η − u∗i,0 X1/2 → 0 0

as η → 0 for

1 ≤ i ≤ p.

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7 Stability of Chafee-Infante Equations

With the above lemmas at hand, we are now in a position to prove the main result of this section. Theorem 7.2 There is η0 > 0 such that if η ∈ [0, η0 ], then Tη (1) is L-Morse-Smale and Aη is geometrically equivalent to A0 . Proof Lemmas 7.3 to 7.8, ensure that all assumptions in Theorem 7.1 are satisfied for the collection {T˜η (1)}η∈[0,η0 ] , where η0 > 0 is a small constant. Hence, T˜η (1) is L-MorseSmale and iη Aη is geometrically equivalent to A0 for all η ∈ [0, η0 ]. Let iη Eη = {u˜ ∗1,η , . . . , u˜ ∗p,η } be the set of L-hyperbolic equilibria of T˜η (1). For each 1 ≤ i ≤ p, we let jη u˜ ∗i,η = u∗i,η . Then we see that Eη := {u∗1,η , . . . , u∗p,η } is the set of all equilibria of Tη (1) that are L-hyperbolic. Since T˜η (1) is dynamically gradient with respect to iη Eη , it is clear that Tη (1) is dynamically gradient with respect to Eη . Let O be  1/2 a neighborhood of η∈[0,η0 ] iη Aη in X0 as in Lemma 7.7. Then we see that jη O is a 1/2

neighborhood of Aη in Xη 1 ≤ i, j ≤ p,

such that Tη (1) is bi-Lipschitz on jη O. Note that for each

s W u (u∗i,η , Tη (1)) ∩ Wloc (u∗j,η , Tη (1)) = ∅

implies s W u (u˜ ∗i,η , T˜η (1)) ∩ Wloc (u˜ ∗j,η , T˜η (1)) = ∅. 1/2 Since T˜η (1) is L-Morse-Smale, there exist n ∈ N and u0 ∈ X0 such that u s (u˜ ∗i,η , T˜η (1)) − L,u0 Wloc (u˜ ∗j,η , T˜η (1)). T˜η (n)Wloc

Consequently, u s Tη (n)Wloc (u∗i,η , Tη (1)) − L,jη u0 Wloc (u∗j,η , Tη (1)).

In much the same way one can show that Tη (1) satisfies the condition (iv) in Definition 7.2. On the other hand, since iη Aη is geometrically equivalent to A0 , there is a bijection B˜η : E0 → iη Eη such that s W u (u∗i,0 , T0 (1)) ∩ Wloc (u∗j,0 , T0 (1)) = ∅

if and only if s W u (B˜η (u∗i,0 ), T˜η (1)) ∩ Wloc (B˜η (u∗j,0 ), T˜η (1)) = ∅.

7.3 Continuity of Global Attractors

149

Let Bη = jη ◦ B˜η . Then Bη : E0 → Eη is a bijection such that s W u (u∗i,0 , T0 (1)) ∩ Wloc (u∗j,0 , T0 (1)) = ∅

if and only if s W u (Bη (u∗i,0 ), Tη (1)) ∩ Wloc (Bη (u∗j,0 ), Tη (1)) = ∅.

This shows that Aη is geometrically equivalent to A0 , and so completes the proof.

7.3



Continuity of Global Attractors

We first consider the continuity of local unstable manifolds with respect to the Hausdorff distance. Lemma 7.9 For any L-hyperbolic equilibria u∗η of Tη (1) and u∗0 of T0 (1), if iη u∗η − u∗0 X1/2 → 0 as η → 0, then 0

u u dH (Wloc (iη u∗η , T˜η (1)), Wloc (u∗0 , T0 (1))) → 0 as η → 0, 1/2

where dH (A, B) denotes the Hausdorff distance between A and B for A, B ⊂ X0 . Proof T the C 2 differentiability of T0 (1) and Lemma 7.6, ensure that all the assumptions  of Theorem 5.4 in [13] are satisfied. This completes the proof. Theorem 7.3 The global attractor Aη converges continuously to A0 in the GromovHausdorff sense, that is, DGH0 (Aη , A0 ) → 0 as η → 0. Proof Choose η1 > 0, ε0 > 0, and a neighborhood O of A0 such that for all η ∈ [0, η1 ], B(A0 , ε0 ) ⊂ O and iη Aη ⊂ O, 1/2

where B(A0 , ε0 ) denotes the ε0 -neighborhood of A0 in X0 . Step 1. We first show that for any 0 < ε < ε0 , there is η2 > 0 such that iη Aη ⊂ B(A0 , ε/4) for all η ∈ [0, η2 ]. For any 0 < ε < ε0 , choose T > 0 such that T0 (t)O ⊂ B(A0 , ε/8) for all t ≥ T . By Lemma 7.4, there is 0 < η2 < η1 such that if η ∈ [0, η2 ], then ε sup T˜η (T )u − T0 (T )uX1/2 < . 0 8 u∈O

150

7 Stability of Chafee-Infante Equations

Since iη op → 1 as η → 0, we have     iη uη − iη vη X1/2 − uη − vη X1/2  < ε/4 η

0

for uη , vη ∈ Aη . For any η ∈ [0, η2 ] and uη ∈ Aη , since Aη is invariant, there is vη ∈ Aη such that Tη (T )vη = uη . Then we have dist(iη uη , A0 ) = dist(iη Tη (T )vη , A0 ) ≤ iη Tη (T )vη − T0 (T )iη vη X1/2 + dist(T0 (T )iη vη , A0 ) < 0

ε , 4

1/2

where dist denotes the Hausdorff semi-distance in X0 . Consequently, we have iη Aη ⊂ B(A0 , ε/4)

for all η ∈ [0, η2 ].

Step 2. Next, we show that for any 0 < ε < ε0 , there is η3 > 0 such that jη A0 ⊂ B(Aη , ε/4) for all η ∈ [0, η3 ]. Suppose this is not the case Then for any k ∈ N, there are 0 < ηk < 1/k and uk ∈ A0 such that dist(jηk uk , Aηk ) ≥ ε/4.

(7.6)

Since A0 is compact, we can assume that uk → u0 as k → ∞ for some u0 ∈ A0 . Take u (u∗ , T (1)). By Lemma 7.9, there is a sequence of n ∈ N such that T0 (−n)u0 ∈ Wloc i,0 0 u ∗ ˜ v˜ηk ∈ Wloc (iηk ui,ηk , Tηk (1)) such that v˜ηk − T0 (−n)u0 X1/2 → 0 0

as k → ∞.

(7.7)

We have T˜ηk (n)v˜ηk − u0 X1/2 ≤ T˜ηk (n)v˜ηk − T0 (n)v˜ηk X1/2 0

0

+ T0 (n)v˜ηk − T0 (n)T0 (−n)u0 X1/2 . 0

Since T0 (n) − T˜ηk (n)O,∞ → 0 as k → ∞, by (7.7), we get T˜ηk (n)v˜ηk − u0 X1/2 → 0 0

Then we obtain

as k → ∞.

7.3 Continuity of Global Attractors

151

jηk uk − Tηk (n)jηk v˜ηk X1/2 ≤ 2uk − T˜ηk (n)v˜ηk X1/2 ηk 0   ≤ 2 uk − u0 X1/2 + u0 − T˜ηk (n)v˜ηk X1/2 0

0

→0 as k → ∞. Since jηk v˜ηk ∈ Aηk , we get a contradiction by (7.6). Step 3. Finally, we show that for any 0 < ε < ε0 , there is η0 > 0 such that if η ∈ [0, η0 ], then there are ε-isometries iˆη : Aη → A0 and jˆη : A0 → Aη . For each η ∈ [0, η2 ], we take a map iˆη : Aη → A0 satisfying iˆη uη ∈ A0 for uη ∈ Aη

iˆη uη − iη uη X1/2 < ε/4.

and

0

Then iˆη is an ε-isometry. In fact, for any uη , vη ∈ Aη , we have iˆη (uη ) − iˆη (vη )X1/2 − uη − vη X1/2 ≤ iˆη uη − iη uη X1/2 η

0

0

+ iη vη − iˆη vη X1/2 + iη uη 0

− iη vη X1/2 − uη − vη X1/2 η

0

< ε. Similarly, we can show that uη − vη X1/2 − iˆη uη − iˆη vη X1/2 < ε. η

0

By Step 2, for u0 ∈ A0 , we can choose uη ∈ Aη such that iη uη − u0 X1/2 < ε/2 for all 0 η ∈ [0, η2 ]. Since iˆη uη − u0 X1/2 ≤ iˆη uη − iη uη X1/2 + iη uη − u0 X1/2 < ε, 0

0

0

we get A0 ⊂ B(iˆη (Aη ), ε). Hence, iˆη is an ε-isometry. Take 0 < η4 < η3 such that if η ∈ [0, η4 ]. Then  jη u0 − jη v0 

1/2 Xη

 − u0 − v0 X1/2  < ε/4 0

for all u0 , v0 ∈ A0 . For each η ∈ [0, η4 ], take a map jˆη : A0 → Aη satisfying jˆη u0 ∈ Aη as u0 ∈ A0 and jˆη u0 − jη u0 X1/2 < ε/4. η

152

7 Stability of Chafee-Infante Equations

Then jˆη is an ε-isometry. In fact, for any u0 , v0 ∈ A0 , we have jˆη u0 − jˆη v0 X1/2 − u0 − v0 X1/2 ≤ jˆη u0 − jη u0 X1/2 η

η

0

+ jη v0 − jˆη v0 X1/2 + jη u0 − jη v0 X1/2 − u0 − v0 X1/2 η

η

0

< ε. Similarly, we can show that u0 − v0 X1/2 − jˆη u0 − jˆη v0 X1/2 < ε. η

0

By Step 1, for uη ∈ Aη , we can choose u0 ∈ A0 such that jη u0 − uη X1/2 < ε/2 for any η η ∈ [0, η4 ]. Since jˆη u0 − uη X1/2 ≤ jˆη u0 − jη u0 X1/2 + jη u0 − uη X1/2 < ε, η

η

η

we get Aη ⊂ B(jˆη (A0 ), ε). Hence, jˆη is an ε-isometry. Let η0 = min{η2 , η4 }. Then we see that for any η ∈ [0, η0 ], iˆη : Aη → A0 and ˆ jη : A0 → Aη are ε-isometries. Consequently, DGH0 (Aη , A0 ) < ε for all η ∈ [0, η0 ]. 

7.4

Geometric Stability

It is well known that every Morse-Smale dynamical system on a compact smooth manifold M is structurally stable under C r -perturbations (r ≥ 1), that is, there is a homeomorphism on M which sends the orbits in the perturbed system to the orbits in the original system (see, for example, [32, 33]). In what follows we study the stability of T0 on the global attractor A0 under Lipschitz perturbations of the domain and equation in (7.1). To this end, we introduce the notion of geometric stability of (7.1). Definition 7.3 We say that the system (7.1) is geometrically stable if for any ε > 0, there exists η0 > 0 such that for any η ∈ [0, η0 ], there exist an ε-isometry i˜η : Aη → A0 and α ∈ RepAη such that for any u ∈ Aη and t ∈ R, i˜η Tη (α(u, t))u = T0 (t)i˜η u. We can now state our main result of in this section.

7.4 Geometric Stability

153

Theorem 7.4 The dynamical system T0 on the global attractor A0 is geometrically stable. More precisely, for any ε > 0, there is η0 > 0 such that if η ∈ [0, η0 ], then there is an ε-isometry i˜η : Aη → A0 such that for any uη ∈ Aη and t ∈ R, i˜η Tη (t)uη = T0 (t)i˜η uη . Moreover, i˜η |Eη : Eη → E0 is a bijection such that s uη ∈ W u (u∗i,η , Tη (1)) ∩ Wloc (u∗j,η , Tη (1)) if and only if s ˜ ∗ i˜η uη ∈ W u (i˜η u∗i,η , T0 (1)) ∩ Wloc (iη uj,η , T0 (1)),

where 1 ≤ i, j ≤ p. Here, Eη := {u∗1,η , . . . , u∗p,η } is the set of all L-hyperbolic equilibria of Tη (1). 1/2

Remark 7.1 Theorem 7.2, implies that if there is a connection in Xη , then there is also a 1/2 connection in X0 , and vice versa. However, Theorem 7.4 gives us more information. In 1/2 1/2 fact, we see that i˜η maps a connection in Xη into a connection in X . 0

We observe that the system (7.4) induces a dynamical system Tη on its global attractor 1/2 Aη for sufficient small η. In fact, for a sufficiently small constant η0 > 0, let F˜η : Xη → Xη be a truncation of Fη such that ⎧ ⎨F (u) for u ∈ B(A , ε ), η η η F˜η (u) = ⎩0 for u ∈ / B(Aη , 2εη ), and F˜η is Lipschitz, where η ∈ [0, η0 ] and εη > 0. Since Gη is Lipschitz, the semidynamical system Sη induced by the equation du + Aη u = F˜η (u) + Gη (u) dt has an inertial manifold Mη that is invariant under Sη , that is, Sη (t)Mη = Mη for all t ∈ R (see, for example, [28, 53]). Since the global attractor Aη of Tη is contained in Mη and Sη |Aη = Tη |Aη , we see that Tη is a dynamical system on Aη for each η ∈ [0, η0 ]. To prove Theorem 7.4, we first study the stability in the Gromov-Hausdorff sense of the semidynamical system T0 induced by (7.1). For this, let us recall the concept of GromovHausdorff distance between two dynamical systems on compact metric spaces, introduced by Lee et al. in [49]. Let CDS be the collection of all dynamical systems on compact metric spaces up to isometry. For two systems (X, φ), (Y, ψ) ∈ CDS and T > 0, the GromovHausdorff distance D T 0 (φ, ψ) is defined as the infimum of ε for which that there exist GH

154

7 Stability of Chafee-Infante Equations

two ε-isometries i : X → Y and j : Y → X, and reparameterizations α ∈ RepX and β ∈ RepY such that for any x ∈ X, y ∈ Y and t ∈ [−T , T ], d(i(φ(x, α(x, t))), ψ(i(x), t)) < ε and d(j (ψ(y, β(y, t))), φ(j (y), t)) < ε; recall that RepX denotes the set of all continuous functions α : X × R → R such that for each fixed x ∈ X, α(x, ·) : R → R is an increasing homeomorphism with α(x, 0) = 0. Using the distance D T 0 , we introduce the Gromov-Hausdorff stability of the sysGH tem (7.1) as follows. Definition 7.4 We say that the system (7.1) is Gromov-Hausdorff stable if for any T > 0 and ε > 0, there exists η0 > 0 such that for any η ∈ [0, η0 ], there exist ε-isometries iˆη : Aη → A0 , jˆη : A0 → Aη , and reparameterizations α ∈ RepAη , β ∈ RepA0 such that iˆη Tη (α(u, t))u − T0 (t)iˆη uX1/2 < ε and jˆη T0 (β(v, t))v − Tη (t)jˆη vX1/2 < ε, η

0

for all u ∈ Aη , v ∈ A0 , and t ∈ [−T , T ]. To prove the Gromov-Hausdorff stability of the system (7.1), we need the following result. Lemma 7.10 For any T > 0 and ε > 0, there are η0 > 0 and δ > 0 such that for any η ∈ [0, η0 ], if uη − vη X1/2 < δ for uη , vη ∈ Aη and |t − s| < δ for s, t ∈ [−T , T ], then η Tη (t)uη − Tη (s)vη X1/2 < ε. η



Proof See Lemma 4.2 in [47].

Theorem 7.5 The dynamical system T0 on the global attractor A0 is Gromov-Hausdorff stable. Proof Step 1. We first show that for any T > 0 and ε > 0, there is η3 > 0 such that for any η ∈ [0, η3 ], there are ε/3-isometries iˆη : Aη → A0 and jˆη : A0 → Aη such that for any u0 ∈ A0 , uη ∈ Aη , and t ∈ [0, T ], iˆη Tη (t)uη − T0 (t)iˆη uη X1/2 < ε and jˆη T0 (t)u0 − Tη (t)jˆη u0 X1/2 < ε. 0

η

By Lemma 7.10, for any T > 0 and ε > 0, we can choose η1 > 0 and 0 < δ < ε/3 such that if η ∈ [0, η1 ] and uη −vη X1/2 +|t −s| < δ for uη , vη ∈ Aη and t, s ∈ [−T , T ], η then

7.4 Geometric Stability

155

Tη (t)uη − Tη (s)vη X1/2 < η

ε . 3

 1/2 Since the set 0≤η 0 is such that  if u, v ∈ 0≤η 0 such that iη Tη (t)jη u0 − T0 (t)u0 X1/2 < ε/9

(7.9)

0

for u0 ∈ A0 , t ∈ [0, T ] and η ∈ [0, η2 ]. By Step 3 in Theorem 7.3, there is 0 < η3 < η2 such that if η ∈ [0, η3 ], then there are δ/3-isometries iˆη : Aη → A0 and jˆη : A0 → Aη satisfying iˆη uη − iη uη X1/2 < 0

δ δ and jˆη u0 − jη u0 X1/2 < η 3 3

(7.10)

for uη ∈ Aη and u0 ∈ A0 . For any η ∈ [0, η3 ], u0 ∈ A0 , and t ∈ [0, T ], we have iˆη Tη (t)jˆη u0 − T0 (t)u0 X1/2 ≤ iˆη Tη (t)jˆη u0 − iη Tη (t)jˆη u0 X1/2 0

0

+ iη Tη (t)jˆη u0 − T0 (t)iη jˆη u0 X1/2 0

+ T0 (t)iη jˆη u0 − T0 (t)u0 X1/2 := I1 + I2 + I3 . 0

By (7.10), we get I1 < ε/9. By (7.9), we have I2 = iη Tη (t)jη iη jˆη u0 − T0 (t)iη jˆη u0 X1/2 < 0

ε . 9

Since iη jˆη u0 − u0 X1/2 ≤ 2jˆη u0 − jη u0 X1/2 ≤ 0

0

2δ , 3

by (7.8), we obtain I3 < ε/9. Consequently, we deduce iˆη Tη (t)jˆη u0 − T0 (t)u0 X1/2 < 0

ε . 3

(7.11)

156

7 Stability of Chafee-Infante Equations

For η ∈ [0, η3 ] and u0 ∈ A0 , we have iˆη ◦ jˆη u0 − u0 X1/2 ≤ iˆη ◦ jˆη u0 − iη ◦ jˆη u0 X1/2 + iη ◦ jˆη u0 − iη ◦ jη u0 X1/2 0

0

≤

0

δ + 2jˆη u − jη uX1/2 < δ. 0 3

(7.12)

For uη ∈ Aη , we see that jˆη ◦ iˆη uη − uη X1/2 ≤ jˆη ◦ iˆη uη − jη ◦ iˆη uη X1/2 0

0

+ jη ◦ iˆη uη − jη ◦ iη u0 X1/2 0

≤

δ + 2iˆη u − iη uX1/2 < δ. 0 3

(7.13)

For any uη ∈ Aη and t ∈ [0, T ], we have iˆη Tη (t)uη − T0 (t)iˆη uη X1/2 0

≤ iˆη Tη (t)uη − iˆη Tη (t)jˆη iˆη uη X1/2 + iˆη Tη (t)jˆη iˆη uη − T0 (t)iˆη uη X1/2 0

0

ε ≤ + Tη (t)uη − Tη (t)jˆη iˆη uη X1/2 + iˆη Tη (t)jˆη iˆη uη − T0 (t)iˆη uη X1/2 0 0 3 ε = + I I1 + I I2 . 3 By (7.13) and Lemma 7.10, I I1 < ε/3. By (7.11), I I2 < ε/3. Consequently, iˆη Tη (t)uη − T0 (t)iˆη uη X1/2 < ε. 0

For any u0 ∈ A0 and t ∈ [0, T ], we have jˆη T0 (t)u0 − Tη (t)jˆη u0 X1/2 ≤ iˆη ◦ jˆη T0 (t)u0 − iˆη Tη (t)jˆη u0 X1/2 + ε/3 0

0

≤ iˆη ◦ jˆη T0 (t)u0 − T0 (t)u0 X1/2 0

+ T0 (t)u0 − iˆη Tη (t)jˆη u0 X1/2 + ε/3 0

= I I I1 + I I I2 + ε/3. By (7.12), I I I1 < ε/3. By (7.11), I I I2 < ε/3. It follows that jˆη T0 (t)u0 − Tη (t)jˆη u0 X1/2 < ε. 0

7.4 Geometric Stability

157

Step 2. We show that the dynamical system T0 on A0 is Gromov-Hausdorff stable. For T > 0 and ε > 0, by Lemma 7.10, there are η4 > 0 and 0 < γ < ε/2 such that if η ∈ [0, η4 ] and uη − vη X1/2 + |t − s| < γ for uη , vη ∈ Aη and t, s ∈ [−2T , 2T ], then η

Tη (t)uη − Tη (s)vη X1/2 < η

ε . 2

(7.14)

By Step 1, we can take 0 < η0 < η4 such that for any η ∈ [0, η0 ], there exist γ -isometries iˆη : Aη → A0 and jˆη : A0 → Aη such that for any u0 ∈ A0 , uη ∈ Aη , and t ∈ [0, 2T ], iˆη Tη (t)uη − T0 (t)iˆη uη X1/2 < γ

and

0

jˆη T0 (t)u0 − Tη (t)jˆη u0 X1/2 < γ .

(7.15)

η

For any η ∈ [0, η0 ], u0 ∈ A0 , and t ∈ [−T , 0], we take w0 ∈ A0 such that u0 = T0 (−2t)w0 . Then we have jˆη T0 (t)u0 − Tη (t)jˆη u0 X1/2 = jˆη T0 (−t)w0 − Tη (t)jˆη T0 (−2t)w0 X1/2 η

η

≤ jˆη T0 (−t)w0 − Tη (−t)jˆη w0 X1/2 η

+ Tη (t)Tη (−2t)jˆη w0 − Tη (t)jˆη T0 (−2t)w0 X1/2 η

= I + I I. By (7.15), we have I < ε/2 and Tη (−2t)jˆη w0 − jˆη T0 (−2t)w0 X1/2 ≤ γ . η

Then, by (7.14), we get I I < ε/2. So, jˆη T0 (t)u0 − Tη (t)jˆη u0 X1/2 < ε. η For any η ∈ [0, η0 ], uη ∈ Aη , and t ∈ [−T , 0], we take wη ∈ Aη such that uη = Tη (−2t)wη . By (7.14) and (7.15), we have iˆη Tη (t)uη − T0 (t)iˆη uη X1/2 = iˆη Tη (−t)wη − T0 (t)iˆη Tη (−2t)wη X1/2 0

0

≤ iˆη Tη (−t)wη − T0 (−t)iˆη wη X1/2 0

+ T0 (t)T0 (−2t)iˆη wη −T0 (t)iˆη Tη (−2t)wη X1/2 0

≤ ε. This completes the proof of Theorem 7.5.



158

7 Stability of Chafee-Infante Equations

Now we are ready to prove the main theorem of in this section, which asserts that the system (7.1) is geometrically stable under Lipschitz perturbations of the domain and equation. To prove this, we use the shadowing property of (7.1) on the global attractor A0 , which was proved by S. Pilyugin. Let T be a homeomorphism on a compact metric space (X, d). A sequence ξ = {xn ∈ X : a < n < b} (−∞ ≤ a < b ≤ ∞) is called a δ-pseudo-orbit of T if d(T (xn ), xn+1 ) < δ for all a < n < b − 1. We say that a δ-pseudo-orbit ξ = {xn ∈ X : a < n < b} of T is ε-shadowed by a point x ∈ X if d(T n (x), xn ) < ε for all a < n < b; in this case, the point x is called a shadowing point of the pseudo-orbit ξ . By Theorem 3.4.1 in [62], we can derive that the time t-map T0 (t) (t > 0) induced by (7.1) has the forward shadowing property, that is, for any ε > 0, there is δ > 0 such that any δ-pseudo-orbit ξ = {xn }n∈N of T0 (t) in its global attractor A0 can be ε-shadowed by a point x ∈ M0 . Note that the shadowing point x does not necessarily belong to A0 . In the following lemma, we prove that the time t-map T0 (t) has the intrinsic shadowing property on A0 , that is, for any ε > 0, there is δ > 0 such that any δ-pseudo-orbit ξ = {xn }n∈Z in A0 can be ε-shadowed by a point x ∈ A0 . Lemma 7.11 (Intrinsic Shadowing Property) For any t > 0, the time t-map T0 (t) of T0 has the intrinsic shadowing property on the global attractor A0 . Proof For t > 0, we see that the time t-map T0 (t) satisfies the assumption of 1/2 Theorem 3.4.1 in [62]. Then there are d0 , L0 > 0 and a neighborhood W of A0 in X0 such that any d-pseudo-orbit ξ = {uk }k≥0 ⊂ W (d < d0 ) for which dist(u0 , M0 ) < d can be L0 d-shadowed by a point u ∈ M0 , that is, for any k ≥ 0, T0 (kt)u − uk X1/2 ≤ L0 d. 0

For ε > 0, we take δ = min{ d20 , Lε0 }. Let ξ = {uk }k∈Z be a δ-pseudo-orbit of T0 (t) in A0 . For any n ∈ N, the δ-pseudo orbit ξn = {uk }k≥−n can be ε-shadowed by vn ∈ M0 , that is, for any k ≥ 0, T0 (kt)vn − uk−n X1/2 ≤ ε. 0

Since {T0 (nt)vn }n∈N is bounded in the inertial manifold M0 , there is a subsequence {T0 (ni t)vni } such that T0 (ni t)vni → v for some v ∈ M0 . For any k ∈ Z, we have T0 (kt)v − uk X1/2 = lim T0 (kt)T0 (ni t)vni − uk X1/2 < ε. 0

i→∞

0

1/2

Then v is a shadowing point of ξ . Since ξ is bounded in X0 , {T0 (kt)v} is bounded in 1/2 X0 , and so v ∈ A0 . 

7.4 Geometric Stability

159

In view of Lemma 7.8, if every equilibrium of T0 (1) is hyperbolic, then there is η0 > 0 such that if η ∈ [0, η0 ], then T˜η (1) has the same number of equilibria that are L-hyperbolic. Then by Bortolan et al. [13, Proposition 3.9, Theorem 3.10], we can take a uniform r > 0 such that any equilibrium u∗i,η of Tη (1) is isolated in its r-neighborhood, that is, if {Tη (n)uη }n∈Z is a bounded global solution of Tη , then Tη (n)uη − u∗i,η X1/2 < r for n ∈ Z η

implies

uη = u∗i,η .

(7.16)

Proof (Proof of Theorem 7.4) Let r > 0 be a constant in (7.16). For any 0 < ε < r/2, take η0 > 0 and 0 < δ < 1 corresponding to ε/4 and T = 1 as in Lemma 7.10. Also, choose 0 < γ < ε/4 such that any γ -pseudo-orbit of T0 (δ) in A0 can be ε/4-shadowed by a point in A0 . By Theorem 7.5, we can choose a smaller η0 > 0 if necessary, such that for any η ∈ [0, η0 ], there is a γ -isometry iˆη : Aη → A0 such that for uη ∈ Aη and t ∈ [−δ, δ], iˆη Tη (t)uη − T0 (t)iˆη uη X1/2 < γ . 0

For η ∈ [0, η0 ], we take a subset Dη of Aη such that (i) for vη ∈ Aη , there is uη ∈ Dη such that vη = Tη (t)uη for some t ∈ R; (ii) if uη , vη ∈ Dη and vη = Tη (t)uη for some t ∈ R, then uη = vη . For uη ∈ Dη , consider a sequence {iˆη Tη (nδ)uη }n∈Z in A0 . For any n ∈ Z, we have T0 (δ)iˆη Tη (nδ)uη − iˆη Tη (δ)Tη (nδ)uη X1/2 < γ . 0

Then {iˆη Tη (nδ)uη }n∈Z is a γ -pseudo-orbit of T0 (δ). By Lemma 7.11, there is an element in A0 , denoted by i˜η (uη ), such that ε T0 (nδ)i˜η uη − iˆη Tη (nδ)uη X1/2 < . 0 4 For any t ∈ R, we can write t = nδ + s for some n ∈ Z and 0 ≤ s < δ. We have T0 (t)i˜η uη − iˆη Tη (t)uη X1/2 0

≤ T0 (s)T0 (nδ)i˜η uη − T0 (nδ)i˜η uη X1/2 + 0

T0 (nδ)i˜η uη − iˆη Tη (nδ)uη X1/2 0

< ε.

+ iˆη Tη (nδ)uη − iˆη Tη (s)Tη (nδ)uη X1/2 0

160

7 Stability of Chafee-Infante Equations

We define a map i˜η : Dη → A0 as follows: for any uη ∈ Dη , i˜η uη ∈ A0 is a shadowing point of the γ -pseudo orbit {iˆη Tη (nδ)uη }n∈Z . For any uη ∈ Aη , choose vη ∈ Dη such that uη = Tη (t)vη for some t ∈ R. We define i˜η (uη ) = T0 (t)i˜η vη . We observe that i˜η is an ε-isometry. In fact, for any uη , vη ∈ Aη , there are u˜ η , v˜η ∈ Dη such that uη = Tη (t)u˜ η and vη = Tη (s)v˜η for some t, s ∈ R. Then we have i˜η uη −i˜η vη X1/2 − uη − vη X1/2 η

0

≤ i˜η Tη (t)u˜ η − iˆη Tη (t)u˜ η X1/2 + iˆη Tη (t)u˜ η − iˆη Tη (s)v˜η X1/2 0

0

+ iˆη Tη (s)v˜η − i˜η Tη (s)v˜η X1/2 − uη − vη X1/2 η

0

:= I + I I + I I I − uη − vη X1/2 . η

By the shadowing property of T0 on A0 , I = T0 (t)i˜η u˜ η − iˆη Tη (t)u˜ η X1/2 < 0

ε 4

and ε I I I = T0 (s)i˜η v˜η − iˆη Tη (s)v˜η X1/2 < . 0 4 Moreover, we get ε I I = iˆη Tη (t)u˜ η − iˆη Tη (s)v˜η X1/2 ≤ + uη − vη X1/2 . η 0 4 Hence, i˜η uη − i˜η vη X1/2 − uη − vη X1/2 < ε. 0

η

Similarly, we have uη − vη X1/2 − i˜η uη − i˜η vη X1/2 < ε, η

0

and so    ˜  iη uη − i˜η vη X1/2 − uη − vη X1/2  < ε. 0

η

Moreover, for any u0 ∈ A0 , there is uη ∈ Aη such that u0 − iˆη uη X1/2 < ε/4. Then 0

7.4 Geometric Stability

161

i˜η uη − u0 X1/2 ≤ i˜η uη − iˆη uη X1/2 + iˆη uη − u0 X1/2 < ε. 0

0

0

Consequently, A0 ⊂ B(i˜η Aη , ε), and so i˜η is an ε-isometry. For any uη ∈ Aη and t ∈ R, there is vη ∈ Dη such that Tη (s)vη = uη for some s ∈ R. Then we have i˜η Tη (t)uη = i˜η Tη (t)Tη (s)vη = T0 (t)T0 (s)i˜η vη = T0 (t)i˜η Tη (s)vη = T0 (t)i˜η uη . Finally, we show that i˜η |Eη : Eη → E0 is a bijection and s uη ∈ W u (u∗i,η , Tη (1)) ∩ Wloc (u∗j,η , Tη (1))

if and only if s ˜ ∗ (iη uj,η , T0 (1)), i˜η uη ∈ W u (i˜η u∗i,η , T0 (1)) ∩ Wloc

where 1 ≤ i, j ≤ p. Note that Eη ⊂ Dη . Then we have ε T0 (n)i˜η u∗i,η − iˆη u∗i,η X1/2 < 0 4 for n ∈ Z. Hence, if η > 0 is sufficiently small, we get T0 (n)i˜η u∗i,η − u∗i,0 X1/2 ≤ T0 (n)i˜η u∗i,η − iˆη u∗i,η X1/2 0

0

+ iˆη u∗i,η

− iη u∗i,η X1/2 0

+ iη u∗i,η − u∗i,0 X1/2 < r. 0

Since any equilibrium of T0 (1) is isolated, we conclude that i˜η u∗i,η = u∗i,0 . s (u∗ , T (1)). Consider a bounded global solution Let uη ∈ W u (u∗i,η , Tη (1)) ∩ Wloc j,η η ξ : Z → Aη given by ξ(n) = Tη (n)uη for any n ∈ Z. Take N > 0 such that ξ(n) − u∗i,η X1/2 < r/2 for n ≤ −N η

and

ξ(n) − u∗j,η X1/2 < r/2 for n ≥ N. η

Then we have T0 (n)i˜η uη − u∗i,0 X1/2 = i˜η Tη (n)uη − i˜η u∗i,η X1/2 ≤ r. 0

0

In much the same way, T0 (n)i˜η uη − u∗j,0 X1/2 < r. 0

162

7 Stability of Chafee-Infante Equations

Since T0 (1) is dynamically gradient with respect to E0 , we have T0 (n)i˜η uη → u∗i,0 as n → −∞

and

T0 (n)i˜η uη → u∗j,0 as n → ∞.

s (i˜ u∗ , T (1)). This implies that i˜η uη ∈ W u (i˜η u∗i,η , T0 (1)) ∩ Wloc η j,η 0 Conversely, for i˜η uη ∈ W u (i˜η u∗ , T0 (1)) ∩ W s (i˜η u∗ , T0 (1)), we have loc

i,η

T0 (n)i˜η uη → i˜η u∗i,η as n → −∞

j,η

and

T0 (n)i˜η uη → i˜η u∗j,η as n → ∞.

and

T0 (n)i˜η uη − i˜η u∗j,η X1/2
0 such that T0 (n)i˜η uη − i˜η u∗i,η X1/2 < 0

r 2

0

r 2

for n ≤ −M and n ≥ M, respectively. Then for any n ≤ −M, Tη (n)uη − u∗i,η X1/2 ≤ ε + i˜η Tη (n)uη − i˜η u∗i,η X1/2 η

0

=ε

+ T0 (n)i˜η uη − u∗i,0 X1/2 0

< r, and for any n ≥ M, Tη (n)uη − u∗j,η X1/2 < r. η

Since Tη (1) is dynamically gradient with respect to Eη , Tη (n)uη → u∗i,η as n → −∞

and

Tη (n)uη → u∗j,η as n → ∞.

s (u∗ , T (1)). This completes the proof. Consequently, uη ∈ W u (u∗i,η , Tη (1)) ∩ Wloc j,η η



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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Lee, C. Morales, Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-12031-2

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