INTRODUCTION TO THE MODERN THEORY OF DYNAMICAL SYSTEMS 现代动力系统理论导论2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

INTRODUCTION TO THE MODERN THEORY OF DYNAMICAL SYSTEMS 现代动力系统理论导论PDF格式电子书版下载

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0．INTRODUCTION1

1．Principal branches of dynamics1

2．Flows,vector fields,differential equations6

3．Time-one map,section,suspension8

4．Linearization and localization10

Part 1 Examples and fundamental concepts15

1．FIRST EXAMPLES15

1．Maps with stable asymptotic behavior15

2．Linear maps19

3．Rotations of the circle26

4．Translations on the torus28

5．Linear flow on the torus and completely integrable systems32

6．Gradient flows35

7．Expanding maps39

8．Hyperbolic toral automorphisms42

9．Symbolic dynamical systems47

2．EQUIVALENCE,CLASSIFICATION,AND INVARIANTS57

1．Smooth conjugacy and moduli for maps57

2．Smooth conjugacy and time change for flows64

3．Topological conjugacy,factors,and structural stability68

4．Topological classification of expanding maps on a circle71

5．Coding,horseshoes,and Markov partitions79

6．Stability of hyperbolic toral automorphisms87

7．The fast-converging iteration method(Newton method)for the conjugacy problem90

8．The Poincaré-Siegel Theorem94

9．Cocycles and cohomological equations100

3．PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS105

1．Growth of orbits105

2．Examples of calculation of topological entropy119

3．Recurrence properties128

4．STATISTICAL BEHAVIOR OF ORBITS AND INTRODUCTION TO ERGODIC THEORY133

1．Asymptotic distribution and statistical behavior of orbits133

2．Examples of ergodicity;mixing146

3．Measure-theoretic entropy161

4．Examples of calculation of measure-theoretic entropy173

5．The Variational Principle179

5．SYSTEMS WITH SMOOTH INVARIANT MEASURES AND MORE EXAMPLES183

1．Existence of smooth invariant measures183

2．Examples of Newtonian systems196

3．Lagrangian mechanics200

4．Examples of geodesic flows205

5．Hamiltonian systems219

6．Contact systems229

7．Algebraic dynamics:Homogeneous and affine systems233

Part 2 Local analysis and orbit growth237

6．LOCAL HYPERBOLIC THEORY AND ITS APPLICATIONS237

1．Introduction237

2．Stable and unstable manifolds239

3．Local stability of a hyperbolic periodic point260

4．Hyperbolic sets263

5．Homoclinic points and horseshoes273

6．Local smooth linearization and normal forms278

7．TRANSVERSALITY AND GENERICITY287

1．Generic properties of dynamical systems287

2．Genericity of systems with hyperbolic periodic points290

3．Nontransversality and bifurcations298

4．The theorem of Artin and Mazur304

8．ORBIT GROWTH ARISING FROM TOPOLOGY307

1．Topological and fundamental-group entropies308

2．A survey of degree theory310

3．Degree and topological entropy316

4．Index theory for an isolated fixed point318

5．The role of smoothness:The Shub-Sullivan Theorem323

6．The Lefschetz Fixed-Point Formula and applications326

7．Nielsen theory and periodic points for toral maps330

9．VARIATIONAL ASPECTS OF DYNAMICS335

1．Critical points of functions,Morse theory,and dynamics336

2．The billiard problem339

3．Twist maps349

4．Variational description of Lagrangian systems365

5．Local theory and the exponential map367

6．Minimal geodesics372

7．Minimal geodesics on compact surfaces376

Part 3 Low-dimensional phenomena381

10．INTRODUCTION:WHAT IS LOW-DIMENSIONAL DYNAMICS?381

11．HOMEOMORPHISMS OF THE CIRCLE387

1．Rotation number387

2．The Poincaré classification393

12．CIRCLE DIFFEOMORPHISMS401

1．The Denjoy Theorem401

2．The Denjoy example403

3．Local analytic conjugacies for Diophantine rotation number405

4．Invariant measures and regularity of conjugacies410

5．An example with singular conjugacy412

6．Fast-approximation methods415

7．Ergodicity with respect to Lebesgue measure419

13．TWIST MAPS423

1．The Regularity Lemma424

2．Existence of Aubry-Mather sets and homoclinic orbits425

3．Action functionals,minimal and ordered orbits434

4．Orbits homoclinic to Aubry-Mather sets441

5．Nonexistence of invariant circles and localization of Aubry-Mather sets447

14．FLOWS ON SURFACES AND RELATED DYNAMICAL SYSTEMS451

1．Poincaré-Bendixson theory452

2．Fixed-point-free flows on the torus457

3．Minimal sets460

4．New phenomena464

5．Interval exchange transformations470

6．Application to flows and billiards479

7．Generalizations of rotation number483

15．CONTINUOUS MAPS OF THE INTERVAL489

1．Markov covers and partitions489

2．Entropy,periodic orbits,and horseshoes493

3．The Sharkovsky Theorem500

4．Maps with zero topological entropy505

5．The kneading theory511

6．The tent model514

16．SMOOTH MAPS OF THE INTERVAL519

1．The structure of hyperbolic repellers519

2．Hyperbolic sets for smooth maps520

3．Continuity of entropy525

4．Full families of unimodal maps526

Part 4 Hyperbolic dynamical systems531

17．SURVEY OF EXAMPLES531

1．The Smale attractor532

2．The DA(derived from Anosov)map and the Plykin attractor537

3．Expanding maps and Anosov automorphisms of nilmanifolds541

4．Definitions and basic properties of hyperbolic sets for flows544

5．Geodesic flows on surfaces of constant negative curvature549

6．Geodesic flows on compact Riemannian manifolds with negative sectional curvature551

7．Geodesic flows on rank-one symmetric spaces555

8．Hyperbolic Julia sets in the complex plane559

18．TOPOLOGICAL PROPERTIES OF HYPERBOLIC SETS565

1．Shadowing of pseudo-orbits565

2．Stability of hyperbolic sets and Markov approximation571

3．Spectral decomposition and specification574

4．Local product structure581

5．Density and growth of periodic orbits583

6．Global classification of Anosov diffeomorphisms on tori587

7．Markov partitions591

19．METRIC STRUCTURE OF HYPERBOLIC SETS597

1．H？lder structures597

2．Cohomological equations over hyperbolic dynamical systems608

20．EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES615

1．Bowen measure615

2．Pressure and the variational principle623

3．Uniqueness and classification of equilibrium states628

4．Smooth invariant measures637

5．Margulis measure643

6．Multiplicative asymptotic for growth of periodic points651

Supplement659

S．DYNAMICAL SYSTEMS WITH NONUNIFORMLY HYPERBOLIC BEHAVIOR BY ANATOLE KATOK AND LEONARDO MENDOZA659

1．Introduction659

2．Lyapunov exponents660

3．Regular neighborhoods672

4．Hyperbolic measures678

5．Entropy and dynamics of hyperbolic measures693

Appendix703

A．BACKGROUND MATERIAL703

1．Basic topology703

2．Functional analysis711

3．Differentiable manifolds715

4．Differential geometry727

5．Topology and geometry of surfaces730

6．Measure theory731

7．Homology theory735

8．Locally compact groups and Lie groups738

NOTES741

HINTS AND ANSWERS TO THE EXERCISES765

REFERENCES781

INDEX793