一致双曲性之外的动力学
作者:
[法] 博纳蒂
(Bonatti C) 著
出版时间:
2007-01
版次:
1
ISBN:
9787030182906
定价:
68.00
装帧:
精装
开本:
16开
纸张:
胶版纸
页数:
384页
字数:
473千字
正文语种:
简体中文
-
广义而言,动力学的目的是描述由“极少的”演化规律所决定的系统(如微分方程或映射)的长期动态。
20世纪60年代早期,SteveSmale引入一臻双曲性概念,统一了动力系统理论的重要结果,导致了关于一大类系统的一个非常成功的理论:一致双曲系统理论。一致双曲系统的动态非常复杂,然而,无论是从几何角度还是统计层面,它们都已得到很好的理解。
在过去的20年中,动力系统理论发生了另一个巨大变化:研究人员试图建立一个统一理论,适合“大多数”动力系统;在该理论下,一致双曲情形的尽可能多的结论依然成立。
《一致双曲线之外的动力学:一种整体的几何学的与概率论的观点》尝试由最新进展出发,统一地展望动力系统理论,提出一些公共开问题,指出未来的可能发展方向。
《一致双曲线之外的动力学:一种整体的几何学的与概率论的观点》面向希望快速而广泛地了解动力学这一方面发展的初学者及研究人员,深度不等地讨论了主要的思想、方法以及结果,给出了相关参考文献,读者可以从文献中获知详细细节和补充信息。
《一致双曲线之外的动力学:一种整体的几何学的与概率论的观点》共12章,各章保持相当的独立性,以方便读者阅读特定主题。
书后五个附录涵盖了一些重要的补充材料。 1HyperbolicityandBeyond
1.1Spectraldecomposition
1.2Structuralstability
1.3Sinai-Ruelle-Bowentheory
1.4Heterodimensionalcycles
1.5Homoclinictangencies
1.6Attractorsandphysicalmeasures
1.7Aconjectureonfinitudeofattractors
2One-DimensionalDynamics
2.1Hyperbolicity
2.2Non-criticalbehavior
2.3Densityofhyperbolicity
2.4Chaoticbehavior
2.5Therenormalizationtheorem
2.6Statisticalpropertiesofunimodalmaps
3HomoclinicTangencies
3.1HomoclinictangenciesandCantorsets
3.2Persistenttangencies,coexistenceofattractors
3.3Hyperbolicityandfractaldimensions
3.4StableintersectionsofregularCantorsets
3.5Homoclinictangenciesinhigherdimensions
3.6Ontheboundaryofhyperbolicsystems
4HenonlikeDynamics
4.1Henon-likefamilies
4.2Abundanceofstrangeattractors
4.3Sinai-Ruelle-Bowenmeasures
4.4Decayofcorrelationsandcentrallimittheorem
4.5Stochasticstability
4.6Chaoticdynamicsnearhomoclinictangencies
5Non-CriticalDynamicsandHyperbolicity
5.1Non-criticalsurfacedynamics
5.2Dominationimpliesalmosthyperbolicity
5.3Homoclinictangenciesvs.AxiomA
5.4Entropyandhomoclinicpointsonsurfaces
5.5Non-criticalbehaviorinhigherdimensions
6HeterodimensionalCyclesandBlenders
6.1Heterodimensionalcycles
6.2Blenders
6.3Partiallyhyperboliccycles
7RobustTransitivity
7.1Examplesofrobusttransitivity
7.2Consequencesofrobusttransitivity
7.3Invariantfoliation
8StableErgodieity
8.1Examplesofstablyergodicsystems
8.2Accessibilityandergodicity
8.3ThetheoremofPugh-Shub
8.4Stableergodicityoftorusautomorphisms
8.5Stableergodicityandrobusttransitivity
8.6Lyapunovexponentsandstableergodicity
9RobustSingularDynamics
9.1Singularinvariantsets
9.2Singularcycles
9.3Robusttransitivityandsingularhyperbolicity
9.4Consequencesofsingularhyperbolicity
9.5SingularAxiomAflows
9.6Persistentsingularattractors
10GenericDiffeomorphisms
10.1Aquickoverview
10.2Notionsofrecurrence
10.3Decomposingthedynamicstoelementarypieces
10.4Homoclinicclassesandelementarypieces
10.5Wildbehaviorvs.tamebehavior
10.6Asampleofwilddynamics
11SRBMeasuresandGibbsStates
11.1SRBmeasuresforcertainnon-hyperbolicmaps
11.2Gibbsu-statesforEuEcssystems
11.3SRBmeasuresfordominateddynamics
11.4GenericexistenceofSRBmeasures
11.5Extensionsandrelatedresults
12LyapunovExponents
12.1ContinuityofLyapunovexponents
12.2Adichotomyforconservativesystems
12.3Deterministicproductsofmatrices
12.4Abundanceofnon-zeroexponents
12.5Lookingfornon-zeroLyapunovexponents
12.6Hyperbolicmeasuresareexactdimensiona
APerturbationLemmas
A.1Closinglemmas
A.2Ergodicclosinglemma
A.3Connectinglemmas
A.4Someideasoftheproofs
A.5Aconnectinglemmaforpseudo-orbits
A.6Realizingperturbationsofthederivative
BNormalHyperbolicityandFoliations
B.1Dominatedsplittings
B.2Invariantfoliations
B.3LinearPoincareflows
CNon-UniformlyHyperbolicTheory
C.1Thelineartheory
C.2Stablemanifoldtheorem
C.3Absolutecontinuityoffoliations
C.4Conditionalmeasuresalonginvariantfoliations
C.5Localproductstructure
C.6Thedisintegrationtheorem
DRandomPerturbations
D.1Markovchainmodel
D.2Iterationsofrandommaps
D.3Stochasticstability
D.4RealizingMarkovchainsbyrandommaps
D.5Shadowingversusstochasticstability
D.6Randomperturbationsofflows
EDecayofCorrelations
E.1Transferoperators:spectralgapproperty
E.2Expandingandpiecewiseexpandingmaps
E.3Invariantconesandprojectivemetrics
E.4Uniformlyhyperbolicdiffeomorphisms
E.5Uniformlyhyperbolicflows
E.6Non-uniformlyhyperbolicsystems
E.7Non-exponentialconvergence
E.8Mapswithneutralfixedpoints
E.9Centrallimittheorem
Conclusion
References
Index
-
内容简介:
广义而言,动力学的目的是描述由“极少的”演化规律所决定的系统(如微分方程或映射)的长期动态。
20世纪60年代早期,SteveSmale引入一臻双曲性概念,统一了动力系统理论的重要结果,导致了关于一大类系统的一个非常成功的理论:一致双曲系统理论。一致双曲系统的动态非常复杂,然而,无论是从几何角度还是统计层面,它们都已得到很好的理解。
在过去的20年中,动力系统理论发生了另一个巨大变化:研究人员试图建立一个统一理论,适合“大多数”动力系统;在该理论下,一致双曲情形的尽可能多的结论依然成立。
《一致双曲线之外的动力学:一种整体的几何学的与概率论的观点》尝试由最新进展出发,统一地展望动力系统理论,提出一些公共开问题,指出未来的可能发展方向。
《一致双曲线之外的动力学:一种整体的几何学的与概率论的观点》面向希望快速而广泛地了解动力学这一方面发展的初学者及研究人员,深度不等地讨论了主要的思想、方法以及结果,给出了相关参考文献,读者可以从文献中获知详细细节和补充信息。
《一致双曲线之外的动力学:一种整体的几何学的与概率论的观点》共12章,各章保持相当的独立性,以方便读者阅读特定主题。
书后五个附录涵盖了一些重要的补充材料。
-
目录:
1HyperbolicityandBeyond
1.1Spectraldecomposition
1.2Structuralstability
1.3Sinai-Ruelle-Bowentheory
1.4Heterodimensionalcycles
1.5Homoclinictangencies
1.6Attractorsandphysicalmeasures
1.7Aconjectureonfinitudeofattractors
2One-DimensionalDynamics
2.1Hyperbolicity
2.2Non-criticalbehavior
2.3Densityofhyperbolicity
2.4Chaoticbehavior
2.5Therenormalizationtheorem
2.6Statisticalpropertiesofunimodalmaps
3HomoclinicTangencies
3.1HomoclinictangenciesandCantorsets
3.2Persistenttangencies,coexistenceofattractors
3.3Hyperbolicityandfractaldimensions
3.4StableintersectionsofregularCantorsets
3.5Homoclinictangenciesinhigherdimensions
3.6Ontheboundaryofhyperbolicsystems
4HenonlikeDynamics
4.1Henon-likefamilies
4.2Abundanceofstrangeattractors
4.3Sinai-Ruelle-Bowenmeasures
4.4Decayofcorrelationsandcentrallimittheorem
4.5Stochasticstability
4.6Chaoticdynamicsnearhomoclinictangencies
5Non-CriticalDynamicsandHyperbolicity
5.1Non-criticalsurfacedynamics
5.2Dominationimpliesalmosthyperbolicity
5.3Homoclinictangenciesvs.AxiomA
5.4Entropyandhomoclinicpointsonsurfaces
5.5Non-criticalbehaviorinhigherdimensions
6HeterodimensionalCyclesandBlenders
6.1Heterodimensionalcycles
6.2Blenders
6.3Partiallyhyperboliccycles
7RobustTransitivity
7.1Examplesofrobusttransitivity
7.2Consequencesofrobusttransitivity
7.3Invariantfoliation
8StableErgodieity
8.1Examplesofstablyergodicsystems
8.2Accessibilityandergodicity
8.3ThetheoremofPugh-Shub
8.4Stableergodicityoftorusautomorphisms
8.5Stableergodicityandrobusttransitivity
8.6Lyapunovexponentsandstableergodicity
9RobustSingularDynamics
9.1Singularinvariantsets
9.2Singularcycles
9.3Robusttransitivityandsingularhyperbolicity
9.4Consequencesofsingularhyperbolicity
9.5SingularAxiomAflows
9.6Persistentsingularattractors
10GenericDiffeomorphisms
10.1Aquickoverview
10.2Notionsofrecurrence
10.3Decomposingthedynamicstoelementarypieces
10.4Homoclinicclassesandelementarypieces
10.5Wildbehaviorvs.tamebehavior
10.6Asampleofwilddynamics
11SRBMeasuresandGibbsStates
11.1SRBmeasuresforcertainnon-hyperbolicmaps
11.2Gibbsu-statesforEuEcssystems
11.3SRBmeasuresfordominateddynamics
11.4GenericexistenceofSRBmeasures
11.5Extensionsandrelatedresults
12LyapunovExponents
12.1ContinuityofLyapunovexponents
12.2Adichotomyforconservativesystems
12.3Deterministicproductsofmatrices
12.4Abundanceofnon-zeroexponents
12.5Lookingfornon-zeroLyapunovexponents
12.6Hyperbolicmeasuresareexactdimensiona
APerturbationLemmas
A.1Closinglemmas
A.2Ergodicclosinglemma
A.3Connectinglemmas
A.4Someideasoftheproofs
A.5Aconnectinglemmaforpseudo-orbits
A.6Realizingperturbationsofthederivative
BNormalHyperbolicityandFoliations
B.1Dominatedsplittings
B.2Invariantfoliations
B.3LinearPoincareflows
CNon-UniformlyHyperbolicTheory
C.1Thelineartheory
C.2Stablemanifoldtheorem
C.3Absolutecontinuityoffoliations
C.4Conditionalmeasuresalonginvariantfoliations
C.5Localproductstructure
C.6Thedisintegrationtheorem
DRandomPerturbations
D.1Markovchainmodel
D.2Iterationsofrandommaps
D.3Stochasticstability
D.4RealizingMarkovchainsbyrandommaps
D.5Shadowingversusstochasticstability
D.6Randomperturbationsofflows
EDecayofCorrelations
E.1Transferoperators:spectralgapproperty
E.2Expandingandpiecewiseexpandingmaps
E.3Invariantconesandprojectivemetrics
E.4Uniformlyhyperbolicdiffeomorphisms
E.5Uniformlyhyperbolicflows
E.6Non-uniformlyhyperbolicsystems
E.7Non-exponentialconvergence
E.8Mapswithneutralfixedpoints
E.9Centrallimittheorem
Conclusion
References
Index
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