Fractal dimensions for Poincare recurrences

259 Pages·2006·1.245 MB·English

by  Valentin Afraimovich, Edgardo Ugalde, Jesus Urias

Fractal Dimensions for Poincaré
Recurrences
MONOGRAPH SERIES ON NONLINEAR
SCIENCE AND COMPLEXITY
SERIESEDITORS
AlbertC.J.Luo
SouthernIllinoisUniversity,Edwardsville,USA
GeorgeZaslavsky
NewYorkUniversity,NewYork,USA
ADVISORYBOARD
ValentinAfraimovich, SanLuisPotosiUniversity,SanLuisPotosi,Mexico
MauriceCourbage, UniversitéParis7,Paris,France
Ben-JacobEshel, SchoolofPhysicsandAstronomy,TelAvivUniversity,
TelAviv,Israel
BernoldFiedler, FreieUniversitätBerlin,Berlin,Germany
JamesA.Glazier, IndianaUniversity,Bloomington,USA
NailIbragimov, IHN,BlekingeInstituteofTechnology,Karlskrona,Sweden
AnatolyNeishtadt, SpaceResearchInstituteRussianAcademyofSciences,
Moscow,Russia
LeonidShilnikov, ResearchInstituteforAppliedMathematics&Cybernetics,
NizhnyNovgorod,Russia
MichaelShlesinger, OfficeofNavalResearch,Arlington,USA
DietrichStauffer, UniversityofCologne,Köln,Germany
JianQiaoSun, UniversityofDelaware,Newark,USA
DimitryTreschev, MoscowStateUniversity,Moscow,Russia
VladimirV.Uchaikin, UlyanovskStateUniversity,Ulyanovsk,Russia
AngeloVulpiani, UniversityLaSapienza,Roma,Italy
PeiYu, TheUniversityofWesternOntario,London,OntarioN6A5B7,Canada
Fractal Dimensions for Poincaré
Recurrences
V.AFRAIMOVICH
SanLuisPotosiUniversity,Mexico
E.UGALDE
SanLuisPotosiUniversity,Mexico
J.URÍAS
SanLuisPotosiUniversity,Mexico
AMSTERDAM BOSTON HEIDELBERG LONDON NEWYORK OXFORD
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PARIS SANDIEGO SANFRANCISCO SINGAPORE SYDNEY TOKYO
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Preface
Researchbyseveralprominentresearchgroups(includingPennStateUniversity,
CNRSatLuminyandEcolePolithecnique,UniversitySimónBolivar,ourgroup
inSanLuisPotosí,etc.)hasshownthatthedimensiontheoryofdynamicalsys-
tems is a powerful tool to analyze the (multi)fractal behavior which appears in
realsystemsandtheirmathematicalmodels.Thisbookisdevotedtoanimportant
branch of this theory: the study of the fine (fractal) structure of Poincaré recur-
rences–instantsoftimewhenthesystemalmostrepeatsitsinitialstate.Because
ofthisrestrictionwewereabletowriteanentirelyself-containedtextincluding
manyinsightsandexamples,aswellasprovidingcompletedetailsofproofs.The
onlyprerequisitesareabasicknowledgeofanalysisandtopology.Thusthisbook
canserveasagraduatetextorself-studyguideforcoursesinappliedmathematics
ornonlineardynamics(inthenaturalsciences).
To motivate our study of Poincare recurrences, imagine that the phase space
(or invariant subset) is partitioned into different colors according to their “tem-
perature”.Theorbitforinitialconditionschoseninhotareasreturnsnearbyfaster
thanforinitialconditionschosenfromcoldareas.Moreprecisely,thereturntime
foranεballcenteredonahotinitialconditionismuchlessthanforacoldinitial
condition.Itistrueevenforuniformlyhyperbolicsystems,providedthattheyare
“sufficientlynonlinear”,i.e.,theirlinearizationdependsonthepoint.
Onecanthenfixalargecollectionofεballsandcalculatean“average”return
time, and study this average as ε goes to 0. For many hyperbolic systems the
average behaves as γ logε and for non-chaotic systems as ε γ, where γ is a
−
−
keydimension-likecharacteristicobtainedfromthefractaldimensionmachinery:
thedimensionforPoincarérecurrences.Itdependsonthesetofinitialpointswe
dealwith,i.e.,itisafunctionofaset.Ifwechooseasetsituatedaroundhotspots,
evidentlywewillobtainadimensionthatisdifferentfromthataroundcoldspots.
Thishasprofoundpracticalapplications,foritprovidesauseful“measure”of
howchaoticisadynamicalsystemwithintheclassofchaoticdynamicalsystems
while for nonchaotic dynamical systems it is a new and useful measure of the
complexity of the orbit structure. Furthermore, given an invariant measure, it is
naturaltointroduceadimensionofmeasurewhichequals,asusual,thedimension
ofthesmallestsetofthefullmeasure.So,ourdimensioncandistinguishdiffer-
ent measures according to the behavior of Poincaré recurrences. If the measure
is ergodic then the behavior of Poincaré recurrences is asymptotically the same
for any ball centered at a typical point. Thus, one can obtain the dimension of
v
vi Preface
measureforPoincarérecurrences,whichisaglobalquantity,byknowingalocal
dimension.
Webelievethatthedimensioncouldbequiteusefulformanyappliedproblems.
Letusemphasizethatthedimensioniscomputableandcanbefoundnumerically
for specific systems. Let us mention now a problem related to synchronization
phenomena. If two (or more) coupled subsystems are synchronized then their
behaviorintimehastobesimilarandsincethedimensiondefinitelyreflectssuch
abehavior,thenthedimensionforPoincarérecurrencesinasynchronizedregime
hastobethesameforallindividualsubsystems.Thus,itcanserveasanindicator
ofsynchronization.
Thesecondproblemwewanttomentionistheproblemoffractalandmultifrac-
tal features of Poincaré recurrences. In situations where a system is nonergodic
andcontainsboth,chaoticinvariantsubsetsandsubsetswithzerotopologicalen-
tropy (such as in standard map) a normalized distribution of return times to a
regionbehavesasfollows
P(τ) τ γ, τ
−
∼ →∞
(whereP(τ)dτ isinfacttheprobabilitytoreturntotheregionduringtheinter-
val of time (τ,(τ dτ))). We explained in Chapter 15 that this exponent γ is
+
directlyrelatedtothedimensionforPoincarérecurrences.So,ourquantityhasan
importantphysicalmeaning.
The book includes figures already published in our papers. Kind permissions
werereceivedfromthepublishersWorldScientific,DiscreteandContinuousDy-
namicalSystemsandTheAmericanPhysicalSocietyforthereproductionofthe
following figures of this book: Figures 3.4 and 3.5 from reference [52], Fig-
ure15.3fromreference[12]andFigures16.3–16.6fromreference[13].
Contents
Preface v
Chapter1. Introduction 1
PARTI. FUNDAMENTALS 7
Chapter2. SymbolicSystems 9
2.1. Specifiedsubshifts 9
2.1.1 Ultrametricspace 11
2.2. OrderedtopologicalMarkovchains 12
2.3. Multipermutativesystems 17
2.3.1 Polysymbolicgeneralization 19
2.3.2 Topologicalconjugationofpolysymbolicminimalsystems 20
2.3.3 Nonminimalmultipermutativesystems 23
2.4. Topologicalpressure 28
2.4.1 Dimension-likedefinitionoftopologicalpressure 32
Chapter3. GeometricConstructions 35
3.1. Moranconstructions 35
3.1.1 GeneralizedMoranconstructions 37
3.1.2 InvariantsubsetsofMarkovmaps 40
3.2. TopologicalpressureandHausdorffdimension 43
3.2.1 Hausdorffandboxdimensions 43
3.2.2 Bowen’sequation 45
3.2.3 Morancovers 45
3.3. StrongMoranconstruction 48
3.4. Controlledpackingofcylinders 48
3.5. Stickysets 49
3.5.1 Geometricconstructionsofstickysets 51
vii
viii Contents
Chapter4. TheSpectrumofDimensionsforPoincaréRecurrences 53
4.1. GeneralizedCarathéodoryconstruction 53
4.1.1 Examples 54
4.2. Thespectrumofdimensionsforrecurrences 57
4.3. Dimensionandcapacities 58
4.4. Theappropriategaugefunctions 59
4.5. Generalpropertiesofthedimensionforrecurrences 63
4.6. Dimensionforminimalsets 65
4.6.1 Thegaugefunctionξ(t) 1/t 66
=
4.6.2 Rotationsofthecircle 66
4.6.3 Denjoyexample 69
4.6.4 Multidimensionalrotation 72
PARTII. ZERO-DIMENSIONALINVARIANTSETS 75
Chapter5. UniformlyHyperbolicRepellers 77
5.1. SpectrumofLyapunovexponents 78
5.2. Thecontrolled-packingcondition 79
5.2.1 ProofofLemma5.1 80
5.2.2 ProofofLemma5.2 82
5.3. Spectraunderthegapcondition 83
Chapter6. Non-UniformlyHyperbolicRepellers 87
6.1. Noorbitsinthecriticalset 88
6.2. Thecriticalsetcontainsanorbit 90
Chapter7. TheSpectrumforaStickySet 95
7.1. ThespectrumforPoincarérecurrences 95
Chapter8. RhythmicalDynamics 99
8.1. Set-up 99
8.2. DimensionsforPoincarérecurrences 100
8.2.1 Thecaseofanautonomousrhythmfunctionφ 100
8.2.2 Thecaseofnon-autonomousrhythmfunctionφ 101
8.3. Thespectrumofdimensions 102
8.3.1 Autonomousφ 102
8.3.2 Non-autonomousφ 103
Contents ix
PARTIII. ONE-DIMENSIONALSYSTEMS 107
Chapter9. MarkovMapsoftheInterval 109
9.1. Thespectrumofdimensions 110
Chapter10. SuspendedFlows 117
10.1. Suspendedflowsoverspecifiedsubshifts 117
10.1.1 Poincarérecurrences 118
10.1.2 Suspendedflow 118
10.2. Bowen–Walters’distance 118
10.3. Spectrumofdimensions 119
10.3.1 ThePoincarérecurrence 119
10.3.2 Thespectrum 120
10.3.3 Mainresults 120
10.3.4 ProofofClaim10.1 127
10.3.5 ProofofClaim10.2 129
PARTIV. MEASURETHEORETICALRESULTS 133
Chapter11. InvariantMeasuresandPoincaréRecurrences 135
11.1. Pointwisedimensionandlocalrates 135
11.2. TheSMBtheorem 137
11.3. KolmogorovcomplexityandBrudno’stheorem 137
11.4. Thelocalrateofreturntimes 138
11.4.1 ProofofTheorem11.3basedontheSMBTheorem 138
11.4.2 ProofofTheorem11.3basedonBrudno’sTheorem 140
11.4.3 Rotationsofthecircle 141
11.5. Remarksonlocalrates 143
11.6. Theq-pointwisedimension 145
Chapter12. DimensionsforMeasuresandq-PointwiseDimension 149
12.1. Preliminariesandmotivation 149
12.2. Aformulaformeasures 151
12.3. Theq-pointwisedimension 153
12.4. Stickysets 156
12.5. Remarksontheq-pointwisedimension 161