现代动力系统理论导论
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四川成都
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作者（美）卡托克 著
出版社世界图书出版公司
ISBN9787510032929
出版时间2011-04
装帧平装
开本16开
纸张胶版纸
定价99元
货号22511149
上书时间2024-07-10
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【目录】:
preface
0. introduction
1. principal branches of dynamics
2. flows, vector fields, differential equations
3. time-one map, section, suspension
4. linearization and localization
part 1examples and fundamental concepts
1. firstexamples
1. maps with stable asymptotic behavior
contracting maps; stability of contractions; increasing intervalmaps
2. linear maps
3. rotations of the circle
4. translations on the torus
5. linear flow on the torus and completely integrable systems
6. gradient flows
7. expanding maps
8. hyperbolic toral automorphisms
9. symbolic dynamical systems
sequence spaces; the shift transformation; topological markovchains; the
.perron-frobenius operator for positive matrices
2. equivalence, classification, andinvariants
1. smooth conjugacy and moduli for maps equivalence and moduli;local analytic linearization; various types of moduli
2. smooth conjugacy and time change for flows
3. topological conjugacy, factors, and structural stability
4. topological classification of expanding maps on a circleexpanding maps; conjugacy via coding; the fixed-point method
5. coding, horseshoes, and markov partitions
markov partitions; quadratic maps; horseshoes; coding of the toralautomor- phism
6. stability of hyperbolic total automorphisms
7. the fast-converging iteration method (newton method) forthe
conjugacy problem
methods for finding conjugacies; construction of the iterationprocess
8. the poincare-siegel theorem
9. cocycles and cohomological equations
3. principalclassesofasymptotictopologicalinvariants
1. growth of orbits
periodic orbits and the-function; topological entropy; volumegrowth; topo-logical complexity: growth in the fundamental group;homological growth
2. examples of calculation of topological entropy
isometries; gradient flows; expanding maps; shifts and topologicalmarkov chains; the hyperbolic toral automorphism; finiteness ofentropy of lipschitz maps; expansive maps
3. recurrence properties
4.statistical behavior of orbits and introduction to ergodictheory
1. asymptotic distribution and statistical behavior of orbits
asymptotic distribution, invariant measures; existence of invariantmeasures;the birkhoff ergodic theorem; existence of symptoticdistribution; ergod-icity and unique ergodicity; statisticalbehavior and recurrence; measure-theoretic somorphism andfactors
2. examples of ergodicity; mixing
rotations; extensions of rotations; expanding maps; mixing;hyperbolic total automorphisms; symbolic systems
3. measure-theoretic entropy
entropy and conditional entropy of partitions; entropy of ameasure-preserving transformation; properties of entropy
4. examples of calculation of measure-theoretic entropy
rotations and translations; expanding maps; bernoulli and markovmeasures;hyperbolic total automorphisms
5. the variational principle
5.systems with smooth invar1ant measures and more examples
1. existence of smooth invariant measures
the smooth measure class; the perron-frobenius operator anddivergence;criteria for existence of smooth invariant measures;absolutely continuous invariant measures for expanding maps; themoser theorem
2. examples of newtonian systems
the newton equation; free particle motion on the torus; themathematical pendulum; central forces
3. lagrangian mechanics
uniqueness in the configuration space; the lagrange equation;lagrangian systems; geodesic flows; the legendre transform
4. examples of geodesic flows
manifolds with many symmetries; the sphere and the toms; isometricsof the hyperbolic plane; geodesics of the hyperbolic plane; compactfactors; the dynamics of the geodesic flow on compact hyperbolicsurfaces
5. hamiltonian systems
symplectic geometry; cotangent bundles; hamiltonian vector fieldsand flows;poisson brackets; integrable systems
6. contact systems
hamiltonian systems preserving a 1-form; contact forms
7. algebraic dynamics: homogeneous and afline systems
part 2local analysis and orbit growth
6.local hyperbolic theory and its applications
1. introduction
2. stable and unstable manifolds
hyperbolic periodic orbits; exponential splitting; thehadamard-perron the-orem; proof of the hadamard-perron theorem; theinclination lemma
3. local stability of a hyperbolic periodic point
the hartman-grobman theorem; local structural stability
4. hyperbolic sets
definition and invariant cones; stable and unstable manifolds;closing lemma and periodic orbits; locally maximal hyperbolicsets
5. homoclinic points and horseshoes
general horseshoes; homoclinic points; horseshoes near homoclinicpoi
6. local smooth linearization and normal forms
jets, formal power series, and smooth equivalence; general formalanalysis; the hyperbolic smooth case
7.transversality and genericity
1. generic properties of dynamical systems
residual sets and sets of first category; hyperbolicity andgenericity
2. genericity of systems with hyperbolic periodic points
transverse fixed points; the kupka-smale theorem
3. nontransversality and bifurcations
structurally stable bifurcations; hopf bifurcations
4. the theorem of artin and mazur
8.orbitgrowtharisingfromtopology
1. topological and fundamental-group entropies
2. a survey of degree theory
motivation; the degree of circle maps; two definitions of degreefor smooth maps; the topological definition of degree
3. degree and topological entropy
4. index theory for an isolated fixed point
5. the role of smoothness: the shub-sullivan theorem
6. the lefschetz fixed-point formula and applications
7. nielsen theory and periodic points for toral maps
9.variational aspects of dynamics
1. critical points of functions, morse theory, and dynamics
2. the billiard problem
3. twist maps
definition and examples; the generating function; extensions;birkhoff peri-odic orbits; global minimality of birkhoff periodicorbits
4. variational description of lagrangian systems
5. local theory and the exponential map
6. minimal geodesics
7. minimal geodesics on compact surfaces
part 3low-dimensional phenomena
10. introduction: what is low-dimensional dynamics?
motivation; the intermediate value property and conformality; vetlow-dimensional and low-dimensional systems; areas of!ow-dimensional dynamics
11.homeomorphismsofthecircle
1. rotation number
2. the poincare classification
rational rotation number; irrational rotation number; orbit typesand mea-surable classification
12. circle diffeomorphisms
1. the denjoy theorem
2. the denjoy example
3. local analytic conjugacies for diophantine rotation number
4. invariant measures and regularity of conjugacies
5. an example with singular conjugacy
6. fast-approximation methods
conjugacies of intermediate regularity; smooth cocycles with wildcobound-aries
7. ergodicity with respect to lebesgue measure
13. twist maps
1. the regularity lemma
2. existence of aubry-mather sets and homoclinic orbits
aubry-mather sets; invariant circles and regions ofinstability
3. action functionals, minimal and ordered orbits
minimal action; minimal orbits; average action and minimalmeasures; stable sets for aubry-mather sets
4. orbits homoclinic to aubry-mather sets
5. nonexisience of invariant circles and localization ofaubry-mather sets
14.flowsonsurfacesandrelateddynamicalsystems
1. poincare-bendixson theory
the poincare-bendixson theorem; existence of transversals
2. fixed-point-free flows on the torus
global transversals; area-preserving flows
3. minimal sets
4. new phenomena
the cherry flow; linear flow on the octagon
5. interval exchange transformations
definitions and rigid intervals; coding; structure of orbitclosures; invariant measures; minimal nonuniquely ergodic intervalexchanges
6. application to flows and billiards
classification of orbits; parallel flows and billiards inpolygons
7. generalizations of rotation number
rotation vectors for flows on the torus; asymptotic cycles;fundamental class and smooth classification of area-preservingflows
15.continuousmapsoftheinterval
1. markov covers and partitions
2. entropy, periodic orbits, and horseshoes
3. the sharkovsky theorem
4. maps with zero topological entropy
5. the kneading theory
6. the tent model
16.smoothmapsoftheinterval
1. the structure of hyperbolic repellers
2. hyperbolic sets for smooth maps
3. continuity of entropy
4. full families of unimodal maps
part 4hyperbolic dynamical systems
17.surveyofexamples
1. the smale attractor
2. the da (derived from anosov) map and the plykin attractor
the da map; the plykln attractor
3. expanding maps and anosov automorphisms of nilmanifolds
4. definitions and basic properties of hyperbolic sets forflows
5. geodesic flows on surfaces of constant negative curvature
6. geodesic flows on compact riemannian manifolds with negativesectional curvature
7. geodesic flows on rank-one symmetric spaces
8. hyperbolic julia sets in the complex plane
rational maps of the riemann sphere; holomorphic dynamics
18.topologicalpropertiesofhyperbolicsets
1. shadowing of pseudo-orbits
2. stability of hyperbolic sets and markov approximation
3. spectral decomposition and specification
spectral decomposition for maps; spectral decomposition for flows;specifica- tion
4. local product structure
5. density and growth of periodic orbits
6. global classification of anosov diffeomorphisms on tori
7. markov partitions
19. metric structure of hyperbolic sets
1. holder structures
the invariant class of hsider-continuons functions; hsldercontinuity of conju-gacies; hslder continuity of orbit equivalencefor flows; hslder continuity and differe
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