Dynamical Systems : An Introduction

214 Pages·2013·1.045 MB·English

by  Luis Barreira, Claudia Valls

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Luis Barreira (cid:2) Claudia Valls
Dynamical
Systems
An Introduction
LuisBarreira ClaudiaValls
DepartamentodeMatemática DepartamentodeMatemática
InstitutoSuperiorTécnico InstitutoSuperiorTécnico
Lisbon,Portugal Lisbon,Portugal
TranslationfromthePortugueselanguageedition:
SistemasDinâmicos:UmaIntroduçãobyLuísBarreiraandClaudiaValls
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ISBN978-1-4471-4834-0 ISBN978-1-4471-4835-7(eBook)
DOI10.1007/978-1-4471-4835-7
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Preface
This book is a self-contained introduction to the core of the theory of dynami-
cal systems, with emphasis on the study of maps. This includes topological, low-
dimensional,hyperbolicandsymbolicdynamics,aswellasabriefintroductionto
ergodic theory. It can be used primarily as a textbook for a one-semester or two-
semesterscourseondynamicalsystemsattheadvancedundergraduateorbeginning
graduatelevels.Itcanalsobeusedforindependentstudyandasarigorousstarting
pointforthestudyofmoreadvancedtopics.
Theexpositionisdirectandrigorous.Inparticular,alltheresultsformulatedin
thebookareproven.Wealsotriedtomakeeachproofassimpleaspossible.Some-
times,thisrequiredacarefulpreparationortherestrictiontoappropriateclassesof
dynamicalsystems,whichisfullyjustifiedinafirstintroduction.Thetextalsoin-
cludesmanyexamplesthatillustrateindetailthenewconceptsandresults,aswell
as140exercises,withdifferentlevelsofdifficulty.
Thetheoryofdynamicalsystemsisverybroadandisextremelyactiveinterms
ofresearch.Italsodependssubstantiallyonmostofthemainareasofmathematics.
So, in order to give a sufficiently broad view, but still self-contained and with a
controlledsize,itwasnecessarytomakeaselectionofthematerial.Inviewofthe
necessary details or the need for results from other areas, some topics have been
omitted, most notably Hamiltonian and holomorphic dynamics, although we have
indicated references for these and other topics. We have also provided references
for further reading on topics that are natural continuations of the material in the
book.Thesesuggestions,togetherwithashortdescriptionofthecontentsandofthe
interdependenceofthevariouschapters,aregroupedtogetherintheintroduction.
Lisbon,Portugal LuisBarreira
ClaudiaValls
v
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 ContentsandSuggestionsforFurtherReading . . . . . . . . . . . 1
1.1.1 BasicNotionsandExamples . . . . . . . . . . . . . . . . 1
1.1.2 TopologicalDynamics . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Low-DimensionalDynamics . . . . . . . . . . . . . . . . 2
1.1.4 HyperbolicDynamics . . . . . . . . . . . . . . . . . . . . 3
1.1.5 SymbolicDynamics . . . . . . . . . . . . . . . . . . . . . 4
1.1.6 ErgodicTheory . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 FurtherTopicsandSuggestedReferences . . . . . . . . . . . . . . 5
1.3 SuggestionsforCoursesBasedontheBook . . . . . . . . . . . . 5
2 BasicNotionsandExamples . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 TheNotionofaDynamicalSystem . . . . . . . . . . . . . . . . . 7
2.2 ExampleswithDiscreteTime . . . . . . . . . . . . . . . . . . . . 9
2.2.1 RotationsoftheCircle . . . . . . . . . . . . . . . . . . . . 9
2.2.2 ExpandingMapsoftheCircle . . . . . . . . . . . . . . . . 10
2.2.3 EndomorphismsoftheTorus . . . . . . . . . . . . . . . . 12
2.3 ExampleswithContinuousTime . . . . . . . . . . . . . . . . . . 14
2.3.1 AutonomousDifferentialEquations . . . . . . . . . . . . . 14
2.3.2 DiscreteTimeVersusContinuousTime . . . . . . . . . . . 18
2.3.3 DifferentialEquationsontheTorusT2 . . . . . . . . . . . 20
2.4 InvariantSets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 TopologicalDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 TopologicalDynamicalSystems. . . . . . . . . . . . . . . . . . . 27
3.2 LimitSetsandBasicProperties . . . . . . . . . . . . . . . . . . . 29
3.2.1 DiscreteTime . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 ContinuousTime. . . . . . . . . . . . . . . . . . . . . . . 34
3.3 TopologicalRecurrence . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 TopologicalTransitivity . . . . . . . . . . . . . . . . . . . 37
vii
viii Contents
3.3.2 TopologicalMixing . . . . . . . . . . . . . . . . . . . . . 39
3.4 TopologicalEntropy . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 BasicNotionsandExamples . . . . . . . . . . . . . . . . 41
3.4.2 TopologicalInvariance. . . . . . . . . . . . . . . . . . . . 43
3.4.3 AlternativeCharacterizations . . . . . . . . . . . . . . . . 45
3.4.4 ExpansiveMaps . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Low-DimensionalDynamics . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 HomeomorphismsoftheCircle . . . . . . . . . . . . . . . . . . . 57
4.1.1 Lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.2 RotationNumber . . . . . . . . . . . . . . . . . . . . . . 62
4.1.3 RationalRotationNumber . . . . . . . . . . . . . . . . . . 64
4.1.4 IrrationalRotationNumber . . . . . . . . . . . . . . . . . 67
4.2 DiffeomorphismsoftheCircle . . . . . . . . . . . . . . . . . . . 71
4.3 MapsoftheInterval . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1 ExistenceofPeriodicPoints . . . . . . . . . . . . . . . . . 74
4.3.2 Sharkovsky’sTheorem . . . . . . . . . . . . . . . . . . . 78
4.4 ThePoincaré–BendixsonTheorem . . . . . . . . . . . . . . . . . 82
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 HyperbolicDynamicsI . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1 SmoothManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 HyperbolicSets . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.2 SmaleHorseshoe . . . . . . . . . . . . . . . . . . . . . . 91
5.2.3 ContinuityoftheStableandUnstableSpaces . . . . . . . . 96
5.3 HyperbolicSetsandInvariantFamiliesofCones . . . . . . . . . . 100
5.3.1 FormulationoftheResult . . . . . . . . . . . . . . . . . . 100
5.3.2 ExistenceofInvariantFamiliesofCones . . . . . . . . . . 102
5.3.3 CriterionforHyperbolicity . . . . . . . . . . . . . . . . . 105
5.4 StabilityofHyperbolicSets . . . . . . . . . . . . . . . . . . . . . 107
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 HyperbolicDynamicsII . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1 BehaviorNearaHyperbolicFixedPoint . . . . . . . . . . . . . . 113
6.1.1 TheGrobman–HartmanTheorem . . . . . . . . . . . . . . 113
6.1.2 TheHadamard–PerronTheorem . . . . . . . . . . . . . . 120
6.2 StableandUnstableInvariantManifolds . . . . . . . . . . . . . . 131
6.2.1 ExistenceofInvariantManifolds . . . . . . . . . . . . . . 131
6.2.2 LocalProductStructure . . . . . . . . . . . . . . . . . . . 135
6.3 GeodesicFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.1 HyperbolicGeometry . . . . . . . . . . . . . . . . . . . . 137
6.3.2 QuotientsbyIsometries . . . . . . . . . . . . . . . . . . . 142
6.3.3 GeodesicFlow . . . . . . . . . . . . . . . . . . . . . . . . 144
6.3.4 HyperbolicFlows . . . . . . . . . . . . . . . . . . . . . . 147
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Contents ix
7 SymbolicDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.1.1 ShiftMap . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.1.2 TopologicalEntropy . . . . . . . . . . . . . . . . . . . . . 155
7.1.3 Two-SidedSequences . . . . . . . . . . . . . . . . . . . . 156
7.2 ExamplesofCodings . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2.1 ExpandingMaps . . . . . . . . . . . . . . . . . . . . . . . 157
7.2.2 QuadraticMaps . . . . . . . . . . . . . . . . . . . . . . . 161
7.2.3 TheSmaleHorseshoe . . . . . . . . . . . . . . . . . . . . 162
7.3 TopologicalMarkovChains . . . . . . . . . . . . . . . . . . . . . 164
7.3.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . 164
7.3.2 PeriodicPoints . . . . . . . . . . . . . . . . . . . . . . . . 165
7.3.3 TopologicalEntropy . . . . . . . . . . . . . . . . . . . . . 167
7.3.4 TopologicalRecurrence . . . . . . . . . . . . . . . . . . . 169
7.4 HorseshoesandTopologicalMarkovChains . . . . . . . . . . . . 172
7.5 ZetaFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8 ErgodicTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.1 NotionsfromMeasureTheory . . . . . . . . . . . . . . . . . . . . 181
8.2 InvariantMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.3 NontrivialRecurrence . . . . . . . . . . . . . . . . . . . . . . . . 188
8.4 TheErgodicTheorem . . . . . . . . . . . . . . . . . . . . . . . . 189
8.5 MetricEntropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.6 ProofoftheErgodicTheorem . . . . . . . . . . . . . . . . . . . . 194
8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Chapter 1
Introduction
Thisintroductorychapterisauser’sguideforthebook.Itincludesabriefdescrip-
tionofthecontentsofeachchapterandoftheinterdependenceofthevarioustopics.
Italsoincludessuggestionsforfurtherreadingandforspecificcoursesbasedonthe
book.
1.1 ContentsandSuggestionsfor FurtherReading
Wefirstsummarizethecontentsofeachchapter.Weemphasizethatwedonotstrive
forcompleteness.Instead,theideaistogiveabriefoverviewofthecontents,high-
lighting the main topics and pointing out those that can be considered more ad-
vanced. We also indicate references for further reading on topics that are natural
continuationsofthematerialinthebook.
1.1.1 BasicNotionsandExamples
Chapter2formsthebasisfortherestofthebook.Itisherethatweintroducetheno-
tionofadynamicalsystem,bothfordiscreteandcontinuoustime.Wealsodescribe
manyexamplesthat,togetherwithotherexamplesintroducedthroughoutthebook,
areusedtoillustratenewconceptsandresults.Theexamplesincluderotationsand
expandingmapsofthecircle,endomorphismsandautomorphismsofthetorus,and
autonomousdifferentialequationsandtheirflows.Inaddition,wedescribesomeba-
sicconstructionsthatdeterminenewdynamicalsystems.Thisincludesgoingback
andforthbetweendiscreteandcontinuoustime.
Theemphasisofthebookisprimarilyondynamicalsystemswithdiscretetime,
althoughwestilldeveloptoareasonableextentthecorrespondingtheoryforflows.
A natural playground for the study of flows is the theory of ordinary differential
equations. In particular, any (autonomous) differential equation x(cid:2) = F(x) with
L.Barreira,C.Valls,DynamicalSystems,Universitext,DOI10.1007/978-1-4471-4835-7_1, 1
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