现代动力系统理论导论

thisbookprovidesthefirstself-containedcomprehensiveexpositionofthetheoryofdynamicalsystemsasacoremathematicaldisciplinecloselyintertwinedwithmostofthemainareasofmathematics.theauthorsintroduceandrigorouslydevelopthetheorywhileprovidingresearchersinterestedinapplicationswithfundamentaltoolsandparadigms.

thebookbeginswithadiscussionofseveralelementarybutfundamentalexamples.theseareusedtoformulateaprogramforthegeneralstudyofasymptoticpropertiesandtointroducetheprincipaltheoreticalconceptsandmethods.themainthemeofthesecondpartofthebookistheinterplaybetweenlocalanalysisnearindividualorbitsandtheglobalcomplexityoftheorbitstructure.thethirdandfourthpartsdevelopindepththetheoriesof！ow-dimensionaldynamicalsystemsandhyperbolicdynamicalsystems.

thebookisaimedatstudentsandresearchersinmathematicsatalllevelsfromad-vancedundergraduateup.scientistsandengineersworkinginapplieddynamics，non-linearscience，andchaoswillalsofindmanyfreshinsightsinthisconcreteandclearpresentation.itcontainsmorethanfourhundredsystematicexercises.

reface
0.introduction
1.principalbranchesofdynamics
2.flows，vectorfields，differentialequations
3.time-onemap，section，suspension
4.linearizationandlocalization

part1examplesandfundamentalconcepts
1.firstexamples
1.mapswithstableasymptoticbehaviorcontractingmaps；stabilityofcontractions；increasingintervalmaps
2.linearmaps
3.rotationsofthecircle
4.translationsonthetorus
5.linearflowonthetorusandcompletelyintegrablesystems
6.gradientflows
7.expandingmaps
8.hyperbolictoralautomorphisms
9.symbolicdynamicalsystemssequencespaces；theshifttransformation；topologicalmarkovchains；
theperron-frobeniusoperatorforpositivematrices

2.equivalence，classification，andinvariants
1.smoothconjugacyandmoduliformapsequivalenceandmoduli；localanalyticlinearization；varioustypesofmoduli
2.smoothconjugacyandtimechangeforflows
3.topologicalconjugacy，factors，andstructuralstability
4.topologicalclassificationofexpandingmapsonacircleexpandingmaps；conjugacyviacoding；thefixed-pointmethod
5.coding，horseshoes，andmarkovpartitionsmarkovpartitions；quadraticmaps；horseshoes；codingofthetoralautomor-phism
6.stabilityofhyperbolictotalautomorphisms
7.thefast-convergingiterationmethod（newtonmethod）fortheconjugacyproblemmethodsforfindingconjugacies；constructionoftheiterationprocess
8.thepoincare-siegeltheorem
9.cocyclesandcohomologicalequations

3.principalclassesofasymptotictopologicalinvariants
1.growthoforbits
periodicorbitsandthe-function；topologicalentropy；volumegrowth；topo-logicalcomplexity：growthinthefundamentalgroup；homologicalgrowth
2.examplesofcalculationoftopologicalentropy
isometries；gradientflows；expandingmaps；shiftsandtopologicalmarkovchains；thehyperbolictoralautomorphism；finitenessofentropyoflipschitzmaps；expansivemaps
3.recurrenceproperties

4.statisticalbehavioroforbitsandintroductiontoergodictheory
1.asymptoticdistributionandstatisticalbehavioroforbits
asymptoticdistribution，invariantmeasures；existenceofinvariantmeasures；thebirkhoffergodictheorem；existenceofsymptoticdistribution；ergod-icityanduniqueergodicity；statisticalbehaviorandrecurrence；measure-theoreticsomorphismandfactors
2.examplesofergodicity；mixing
rotations；extensionsofrotations；expandingmaps；mixing；hyperbolictotalautomorphisms；symbolicsystems
3.measure-theoreticentropy
entropyandconditionalentropyofpartitions；entropyofameasure-preservingtransformation；propertiesofentropy
4.examplesofcalculationofmeasure-theoreticentropy
rotationsandtranslations；expandingmaps；bernoulliandmarkovmeasures；hyperbolictotalautomorphisms
5.thevariationalprinciple

5.systemswithsmoothinvar1antmeasuresandmoreexamples
1.existenceofsmoothinvariantmeasures
thesmoothmeasureclass；theperron-frobeniusoperatoranddivergence；criteriaforexistenceofsmoothinvariantmeasures；absolutelycontinuousinvariantmeasuresforexpandingmaps；themosertheorem
2.examplesofnewtoniansystems
thenewtonequation；freeparticlemotiononthetorus；themathematicalpendulum；centralforces
3.lagrangianmechanics
uniquenessintheconfigurationspace；thelagrangeequation；lagrangiansystems；geodesicflows；thelegendretransform
4.examplesofgeodesicflowsmanifoldswithmanysymmetries；thesphereandthetoms；isometricsofthehyperbolicplane；geodesicsofthehyperbolicplane；compactfactors；thedynamicsofthegeodesicflowoncompacthyperbolicsurfaces
5.hamiltoniansystemssymplecticgeometry；cotangentbundles；hamiltonianvectorfieldsandflows；poissonbrackets；integrablesystems
6.contactsystemshamiltoniansystemspreservinga1-form；contactforms
7.algebraicdynamics：homogeneousandaflinesystemspart2localanalysisandorbitgrowth

6.localhyperbolictheoryanditsapplications
1.introduction
2.stableandunstablemanifolds
hyperbolicperiodicorbits；exponentialsplitting；thehadamard-perronthe-orem；proofofthehadamard-perrontheorem；theinclinationlemma
3.localstabilityofahyperbolicperiodicpoint
thehartman-grobmantheorem；localstructuralstability
4.hyperbolicsets
definitionandinvariantcones；stableandunstablemanifolds；closinglemmaandperiodicorbits；locallymaximalhyperbolicsets
5.homoclinicpointsandhorseshoes
generalhorseshoes；homoclinicpoints；horseshoesnearhomoclinicpoi
6.localsmoothlinearizationandnormalforms
jets，formalpowerseries，andsmoothequivalence；generalformalanalysis；thehyperbolicsmoothcase

7.transversalityandgenericity
1.genericpropertiesofdynamicalsystems
residualsetsandsetsoffirstcategory；hyperbolicityandgenericity
2.genericityofsystemswithhyperbolicperiodicpoints
transversefixedpoints；thekupka-smaletheorem
3.nontransversalityandbifurcations
structurallystablebifurcations；hopfbifurcations
4.thetheoremofartinandmazur

8.orbitgrowtharisingfromtopology
1.topologicalandfundamental-groupentropies
2.asurveyofdegreetheory
motivation；thedegreeofcirclemaps；twodefinitionsofdegreeforsmoothmaps；thetopologicaldefinitionofdegree
3.degreeandtopologicalentropy
4.indextheoryforanisolatedfixedpoint
5.theroleofsmoothness：theshub-sullivantheorem
6.thelefschetzfixed-pointformulaandapplications
7.nielsentheoryandperiodicpointsfortoralmaps

9.variationalaspectsofdynamics
1.criticalpointsoffunctions，morsetheory，anddynamics
2.thebilliardproblem
3.twistmaps
definitionandexamples；thegeneratingfunction；extensions；birkhoffperi-odicorbits；globalminimalityofbirkhoffperiodicorbits
4.variationaldescriptionoflagrangiansystems
5.localtheoryandtheexponentialmap
6.minimalgeodesics
7.minimalgeodesicsoncompactsurfaces
part3low-dimensionalphenomena

10.introduction：whatislow-dimensionaldynamics?
motivation；theintermediatevaluepropertyandconformality；vetlow-dimensionalandlow-dimensionalsystems；areasof！ow-dimensionaldynamics

11.homeomorphismsofthecircle
1.rotationnumber
2.thepoincareclassificationrationalrotationnumber；irrationalrotationnumber；orbittypesandmea-surableclassification

12.circlediffeomorphisms
1.thedenjoytheorem
2.thedenjoyexample
3.localanalyticconjugaciesfordiophantinerotationnumber
4.invariantmeasuresandregularityofconjugacies
5.anexamplewithsingularconjugacy
6.fast-approximationmethods
conjugaciesofintermediateregularity；smoothcocycleswithwildcobound-aries
7.ergodicitywithrespecttolebesguemeasure

13.twistmaps
1.theregularitylemma
2.existenceofaubry-mathersetsandhomoclinicorbits
aubry-mathersets；invariantcirclesandregionsofinstability
3.actionfunctionals，minimalandorderedorbits
minimalaction；minimalorbits；averageactionandminimalmeasures；stablesetsforaubry-mathersets
4.orbitshomoclinictoaubry-mathersets
5.nonexisienceofinvariantcirclesandlocalizationofaubry-mathersets

14.flowsonsurfacesandrelateddynamicalsystems
1.poincare-bendixsontheory
thepoincare-bendixsontheorem；existenceoftransversals
2.fixed-point-freeflowsonthetorus
globaltransversals；area-preservingflows
3.minimalsets
4.newphenomena
thecherryflow；linearflowontheoctagon
5.intervalexchangetransformations
definitionsandrigidintervals；coding；structureoforbitclosures；invariantmeasures；minimalnonuniquelyergodicintervalexchanges
6.applicationtoflowsandbilliards
classificationoforbits；parallelflowsandbilliardsinpolygons
7.generalizationsofrotationnumber
rotationvectorsforflowsonthetorus；asymptoticcycles；fundamentalclassandsmoothclassificationofarea-preservingflows

15.continuousmapsoftheinterval
1.markovcoversandpartitions
2.entropy，periodicorbits，andhorseshoes
3.thesharkovskytheorem
4.mapswithzerotopologicalentropy
5.thekneadingtheory
6.thetentmodel

16.smoothmapsoftheinterval
1.thestructureofhyperbolicrepellers
2.hyperbolicsetsforsmoothmaps
3.continuityofentropy
4.fullfamiliesofunimodalmapspart4hyperbolicdynamicalsystems

17.surveyofexamples
1.thesmaleattractor
2.theda（derivedfromanosov）mapandtheplykinattractor
thedamap；theplyklnattractor
3.expandingmapsandanosovautomorphismsofnilmanifolds
4.definitionsandbasicpropertiesofhyperbolicsetsforflows
5.geodesicflowsonsurfacesofconstantnegativecurvature
6.geodesicflowsoncompactriemannianmanifoldswithnegativesectionalcurvature
7.geodesicflowsonrank-onesymmetricspaces
8.hyperbolicjuliasetsinthecomplexplanerationalmapsoftheriemannsphere；holomorphicdynamics

18.topologicalpropertiesofhyperbolicsets
1.shadowingofpseudo-orbits
2.stabilityofhyperbolicsetsandmarkovapproximation
3.spectraldecompositionandspecification
spectraldecompositionformaps；spectraldecompositionforflows；specifica-tion
4.localproductstructure
5.densityandgrowthofperiodicorbits
6.globalclassificationofanosovdiffeomorphismsontori
7.markovpartitions

19.metricstructureofhyperbolicsets
1.holderstructures
theinvariantclassofhsider-continuonsfunctions；hsldercontinuityofconju-gacies；hsldercontinuityoforbitequivalenceforflows；hsldercontinuityanddifferentiabilityoftheunstabledistribution；hsldercontinuityofthejacobian
2.cohomologicalequationsoverhyperbolicdynamicalsystems
thelivschitztheorem；smoothinvariantmeasuresforanosovdiffeomor-phisms；timechangeandorbitequivalenceforhyperbolicflows；equivalenceoftorusextensions

20.equilibriumstatesandsmoothinvariantmeasures
1.bowenmeasure
2.pressureandthevariationalprinciple
3.uniquenessandclassificationofequilibriumstates
uniquenessofequilibriumstates；classificationofequilibriumstates
4.smoothinvariantmeasures
propertiesofsmoothinvariantmeasures；smoothclassificationofanosovdif-feomorphismsonthetorus；smoothclassificationofcontactanosovflowson3-manifolds
5.margulismeasure
6.multiplicativeasymptoticforgrowthofperiodicpoints
localproductflowboxes；themultiplicativeasymptoticoforbitgrowthsupplement
s.dynamicalsystemswithnonuniformlyhyperbolicbehaviorbyanatolekatokandleonardomendoza
1.introduction
2.lyapunovexponents
cocyclesoverdynamicalsystems；examplesofcocycles；themultiplicativeergodictheorem；osedelec-pesine-reductiontheorem；therue！！einequality
3.regularneighborhoods
existenceofregularneighborhoods；hyperbolicpoints，admissiblemanifolds，andthegraphtransform
4.hyperbolicmeasures
preliminaries；theclosinglemma；theshadowinglemma；pseudo-markovcovers；thelivschitztheorem
5.entropyanddynamicsofhyperbolicmeasures
hyperbolicmeasuresandhyperbolicperiodicpoints；continuousmeasuresandtransversehomoclinicpoints；thespectraldecompositiontheorem；entropy，horseshoes，andperiodicpointsforhyperbolicmeasures
appendix
a.backgroundmaterial
1.basictopology
topologicalspaces；homotopytheory；metricspaces
2.functionalanalysis
3.differentiablemanifolds
differentiablemanifolds；tensorbundles；exteriorcalculus；transversality
4.differentialgeometry
5.topologyandgeometryofsurfaces
6.measuretheory
basicnotions；measureandtopology
7.homologytheory
8.locallycompactgroupsandliegroups
notes
hintsandanswerstotheexercises
references
index