Normally Hyperbolic Invariant Manifolds: The Noncompact Case

197 Pages·2013·3.588 MB·English

by  Jaap Eldering

Atlantis Series in Dynamical Systems
Series Editors: Henk Broer · Boris Hasselblatt
Jaap Eldering
Normally Hyperbolic
Invariant Manifolds
The Noncompact Case
Atlantis Series in Dynamical Systems
Volume 2
For furthervolumes:
http://www.springer.com/series/11155
Jaap Eldering
Normally Hyperbolic
Invariant Manifolds
The Noncompact Case
JaapEldering
Department of Mathematics
Utrecht University
Utrecht
The Netherlands
ISSN 2213-3526
ISBN 978-94-6239-002-7 ISBN 978-94-6239-003-4 (eBook)
DOI 10.2991/978-94-6239-003-4
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(cid:2)AtlantisPressandtheauthor2013
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Preface
Inthiswork,weprovethepersistenceofnormallyhyperbolicinvariantmanifolds.
This result iswell knownwhenthe invariant manifoldiscompact; weextendthis
toasettingwheretheinvariantmanifoldaswellastheambientspaceareallowed
to be noncompact manifolds. The ambient space is assumed to be a Riemannian
manifold of bounded geometry.
Normally hyperbolic invariant manifolds (NHIMs) are a generalization of
hyperbolic fixed points. Many of the concepts, results, and proofs for hyperbolic
fixed points carry over to NHIMs. Two important properties that generalize to
NHIMs are persistence of the invariant manifold and existence of stable and
unstable manifolds.
We shall focus on the first property. Persistence of a hyperbolic fixed point
follows as a straightforward application of the implicit function theorem. For a
NHIM the situation is significantly more subtle, although the basic idea is the
same. In the case of a hyperbolic fixed point, we only have stable and unstable
directions. When we consider a NHIM, there is a third direction, tangent to the
manifold itself. The dynamics in the tangential directions is assumed to be dom-
inated by the stable and unstable directions in terms of the respective Lyapunov
exponents. Thus the dynamics on the invariant manifold is approximately neutral
andthedynamicsinthenormaldirectionsishyperbolic;hencethenamenormally
hyperbolic. The system is called r-normally hyperbolic, if the spectral gap con-
dition holds that the tangential dynamics is dominated by a factor r C 1. An r-
NHIMpersistsunderC1smallperturbationsofthesystem.Thepersistentmanifold
willbeCrifthesystemis,butitmaynotbemoresmooth,evenifthesystemisC?
or analytic. This can also be formulated as follows: r-normal hyperbolicity is an
‘openproperty’inthespaceofCrsystemsundertheC1topology.Thedescription
above shows that the spectral properties of NHIMs and center manifolds are
similar. The difference is that NHIMs are globally uniquely defined, while center
manifolds are not.
There are two basic methods of proof for hyperbolic fixed points and center
manifolds: Hadamard’s graph transform and Perron’s variation of constants inte-
gral method. Both can be extended to prove persistence of NHIMs, as well as
existence of its stable and unstable manifolds. We employ the Perron method.
vii
viii Preface
Both methods of proof construct a contraction scheme to find the persistent
NHIM(andasimilarcontractionschemecanbeusedtofinditsstableandunstable
manifolds). Heuristically, we can construct the implicit function F(M, v) = Ut
(M) - M = 0, where M is the NHIM and Ut is the flow of the vector field v after
some fixed time t. Normal hyperbolicity of M implies that D F is invertible.
1
Hence, there is a function M~ ¼Gð~vÞ that maps perturbed vector fields ~v to per-
sistent manifolds M~, at least in a neighborhood of v. This idea does not work
directly for higher derivatives. An inductive scheme can be set up that typically
uses some form of the fiber contraction theorem. This scheme will break down
afterriterations,hencethelimitedsmoothness.Example1.1showsthatthisisan
intrinsic problem.
To tackle the noncompact case, we replace compactness by uniformity condi-
tions.Theseincludeuniformcontinuityandglobalboundednessofthevectorfield
and the invariant manifold and their derivatives up to order r. We require addi-
tional uniformity conditions on the ambient manifold, namely ‘bounded geome-
try’.ThismeansthattheRiemanniancurvatureisgloballybounded,andasaresult
wehaveauniformatlaswhichallowsustoretainuniformestimatesthroughoutall
constructions in the proof.
Organization of the Book
Thisbookisorganizedasfollows.Intheintroduction,wegiveabroadoverviewof
thetheoryofNHIMswithreferencestomoredetailsinthelaterchapters.Westart
by describing how NHIMs are related to hyperbolic fixed points and center
manifolds. Then we give some basic examples and motivation for studying the
noncompact case. We give a brief overview of the history and literature and
comparethetwomethodsofproofinthebasicsettingofahyperbolicfixedpoint.
Then we continue to introduce the concept of bounded geometry and a precise
statement of the main result of this work and discuss its relation to the literature.
We describe a few extensions and details of the results and conclude the chapter
with notation used throughout this book.
Chapter 2 treats Riemannian manifolds of bounded geometry. We first intro-
duce the definition of bounded geometry and some basic implications. We
explicitlyworkouttherelationbetweencurvatureandholonomyinSect.2.2.This
we use in Sect. 3.7 to prove the smoothness of the persistent manifold. In the
subsequent sections, we develop the theory required to prove persistence of
noncompact NHIMs in general ambient manifolds of bounded geometry. We
extend results for submanifolds to uniform versions in bounded geometry, to
finally show how to reduce the main theorem to a setting in a trivial bundle. A
number of these results are new and may be of independent interest, namely the
uniform tubular neighborhood theorem, the uniform smooth approximation of a
submanifold, and a uniform embedding into a trivial bundle.
Preface ix
InChap.3,wefinallyprovethemainresultinthetrivialbundlesetting.Wefirst
stateboththisandthegeneralversionofthemaintheoremanddiscusstheseinfull
detail. We include aprecise comparison with resultsintheliterature,followed by
an outline of the proof. Section 3.3 contains a discussion of the differences to the
compact case and presentsdetailed examplestoillustratethese.Then we start the
actual proof. We first prepare the system: we put it in a suitable form and obtain
estimates for the perturbed system. Then we prove that there exists a unique
persistent invariant manifold and that it is Lipschitz. Second, we set up an elab-
orate scheme in Sect. 3.7 to prove that this manifold is Cr smooth by induction
over the smoothness degree.
In Chap. 4, we discuss how the main result can be extended in a number of
differentwaysthatmayspecificallybeusefulforapplications.Weshowhowtime
and parameter dependence can be added and we present a slightly more general
definition of overflow invariance that might be applicable to systems that are not
overflowing invariant under the standard definition.
Finally, the appendices contain technical and reference material. These are
referencedfromthemaintextwhereappropriate.AppendixAshowsanimportant
idea that permeates this work: the implicit function theorem allows for explicit
estimatesintermsoftheinput,henceit‘preservesuniformityestimates’.Thiscan
then directly be applied to dependence of a flow on the vector field. In Appendix
B,theNemytskiioperatorisintroducedasatechniquetoprovecontinuityofpost-
composition with a function. This is an essential basic part in the smoothness
proof, together with the results on the exponential growth behavior of higher
derivatives offlows in Appendix C. Here, we also develop a framework to work
withhigherderivativesonRiemannianmanifolds.Thelastappendicesincludethe
fibercontractiontheoremofHirschandPughthatisusedinthesmoothnessproof,
Alekseev’s nonlinear variation of constants integral defined on manifolds, and a
brief overview of those parts of Riemannian geometry that we use.
What is (Not) New
Normal hyperbolicity can nowadays be called a classical subject; it was first
formulatedandstudiedinthelate1960sand1970s,seethehistoricaloverviewin
Sect. 1.3. Although initial results were formulated for compact NHIMs, more
recentworkbySakamotoandespeciallyBates,Lu,andZenghavebroughtthisto
the noncompact setting, and even to semi-flows in Banach spaces.
Thespecificaspectthatisnewinthisworkisthedifferentialgeometriccontext
in which our results on noncompact NHIMs are formulated. Uniformity of the
ambient space seems not to have been addressed before in the literature, and our
useofboundedgeometryallowsustoextendpersistenceofNHIMstothissetting,
seeSect. 1.5andChap. 2.Additionally,someofourresultsonboundedgeometry
appear to be new, including the uniform tubular neighborhood theorem and the
theorem on uniform smooth approximation of a submanifold.
x Preface
The ‘core’ persistence proof itself is based on the Perron method; it can
probably be replaced by the proof of Bates, Lu, and Zeng, that is based on the
graph transform, when taking into account the necessary bounded geometry
technical details from Chap. 2. Our proof uses ideas of Henry and Vander-
bauwhede and Van Gils to extend the Perron method to NHIMs and higher
smoothness.Anovelaspectisthatwedeveloptheseideasonatrivialbundlewith
a bounded geometrymanifoldas base. Thisrequires a whole framework tobe set
up, including representations of higher jets of a flow in Appendix C and formal
tangent bundles of spaces of curves (see Sect. 3.7.4) to study derivatives of the
Perron contraction operator. These ideas might be of interest in other contexts of
dynamics in noncompact differential geometry.
Finally, our Definition 4.4 of a priori overflowing invariance might be new
(althoughprobablynotsurprisingtoexpertsinthefield)andcouldproveusefulfor
certain applications where the original definition of over- or inflowing invariance
does not hold.
Acknowledgments
ThisbookisbasedonmyPh.D.thesisworkcompletedatUtrechtUniversity.Iam
grateful to having had Hans Duistermaat as my supervisor while working on this
project. His inspiring enthusiasm and deep insights have been greatly beneficial,
andIfeelhonoredtohavingbeenhisPh.D.student.AfterHans’untimelydeathin
2010, the guidance of Erik van den Ban and Heinz Hanßmann has been instru-
mental in completing this project. I am indebted to both, and it was a pleasure
working together.
Themembersofmythesiscommitteehaveprovidedvaluablecomments,andI
would especially like to thank Charles Pugh for stimulating discussions and
finding an error in a lemma.
Finally, I would like tothank the editors Henk Broer and Boris Hasselblatt for
their support in preparing this book.
March 2013 Jaap Eldering
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Normally Hyperbolic Invariant Manifolds. . . . . . . . . . . . . . . . . 1
1.1.1 Persistence and (Un)stable Manifolds . . . . . . . . . . . . . . 3
1.1.2 The Relation to Center Manifolds. . . . . . . . . . . . . . . . . 5
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The Spectral Gap Condition. . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Motivation for Noncompact NHIMs . . . . . . . . . . . . . . . 10
1.3 Historical Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Comparison of Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Hadamard’s Graph Transform. . . . . . . . . . . . . . . . . . . . 16
1.4.2 Perron’s Variation of Constants Method . . . . . . . . . . . . 16
1.4.3 Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Bounded Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Problem Statement and Results. . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.1 Non-Autonomous Systems. . . . . . . . . . . . . . . . . . . . . . 24
1.6.2 Immersed Submanifolds. . . . . . . . . . . . . . . . . . . . . . . . 25
1.6.3 Overflowing Invariant Manifolds . . . . . . . . . . . . . . . . . 29
1.7 Induced Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.8 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Manifolds of Bounded Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1 Bounded Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Curvature and Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3 Submanifolds and Tubular Neighborhoods . . . . . . . . . . . . . . . . 51
2.4 Smoothing of Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.5 Embedding into a Trivial Bundle. . . . . . . . . . . . . . . . . . . . . . . 68
2.6 Reduction of a NHIM to a Trivial Bundle . . . . . . . . . . . . . . . . 73
3 Persistence of Noncompact NHIMs. . . . . . . . . . . . . . . . . . . . . . . . 75
3.1 Statement of the Main Theorems. . . . . . . . . . . . . . . . . . . . . . . 76
3.2 Outline of the Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
xi