0．INTRODUCTION 1
1．Principal branches of dynamics 1
2．Flows,vector fields,differential equations 6
3．Time-one map,section,suspension 8
4．Linearization and localization 10
Part 1 Examples and fundamental concepts 15
1．FIRST EXAMPLES 15
1．Maps with stable asymptotic behavior 15
2．Linear maps 19
3．Rotations of the circle 26
4．Translations on the torus 28
5．Linear flow on the torus and completely integrable systems 32
6．Gradient flows 35
7．Expanding maps 39
8．Hyperbolic toral automorphisms 42
9．Symbolic dynamical systems 47
2．EQUIVALENCE,CLASSIFICATION,AND INVARIANTS 57
1．Smooth conjugacy and moduli for maps 57
2．Smooth conjugacy and time change for flows 64
3．Topological conjugacy,factors,and structural stability 68
4．Topological classification of expanding maps on a circle 71
5．Coding,horseshoes,and Markov partitions 79
6．Stability of hyperbolic toral automorphisms 87
7．The fast-converging iteration method(Newton method)for the conjugacy problem 90
8．The Poincaré-Siegel Theorem 94
9．Cocycles and cohomological equations 100
3．PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS 105
1．Growth of orbits 105
2．Examples of calculation of topological entropy 119
3．Recurrence properties 128
4．STATISTICAL BEHAVIOR OF ORBITS AND INTRODUCTION TO ERGODIC THEORY 133
1．Asymptotic distribution and statistical behavior of orbits 133
2．Examples of ergodicity;mixing 146
3．Measure-theoretic entropy 161
4．Examples of calculation of measure-theoretic entropy 173
5．The Variational Principle 179
5．SYSTEMS WITH SMOOTH INVARIANT MEASURES AND MORE EXAMPLES 183
1．Existence of smooth invariant measures 183
2．Examples of Newtonian systems 196
3．Lagrangian mechanics 200
4．Examples of geodesic flows 205
5．Hamiltonian systems 219
6．Contact systems 229
7．Algebraic dynamics:Homogeneous and affine systems 233
Part 2 Local analysis and orbit growth 237
6．LOCAL HYPERBOLIC THEORY AND ITS APPLICATIONS 237
1．Introduction 237
2．Stable and unstable manifolds 239
3．Local stability of a hyperbolic periodic point 260
4．Hyperbolic sets 263
5．Homoclinic points and horseshoes 273
6．Local smooth linearization and normal forms 278
7．TRANSVERSALITY AND GENERICITY 287
1．Generic properties of dynamical systems 287
2．Genericity of systems with hyperbolic periodic points 290
3．Nontransversality and bifurcations 298
4．The theorem of Artin and Mazur 304
8．ORBIT GROWTH ARISING FROM TOPOLOGY 307
1．Topological and fundamental-group entropies 308
2．A survey of degree theory 310
3．Degree and topological entropy 316
4．Index theory for an isolated fixed point 318
5．The role of smoothness:The Shub-Sullivan Theorem 323
6．The Lefschetz Fixed-Point Formula and applications 326
7．Nielsen theory and periodic points for toral maps 330
9．VARIATIONAL ASPECTS OF DYNAMICS 335
1．Critical points of functions,Morse theory,and dynamics 336
2．The billiard problem 339
3．Twist maps 349
4．Variational description of Lagrangian systems 365
5．Local theory and the exponential map 367
6．Minimal geodesics 372
7．Minimal geodesics on compact surfaces 376
Part 3 Low-dimensional phenomena 381
10．INTRODUCTION:WHAT IS LOW-DIMENSIONAL DYNAMICS? 381
11．HOMEOMORPHISMS OF THE CIRCLE 387
1．Rotation number 387
2．The Poincaré classification 393
12．CIRCLE DIFFEOMORPHISMS 401
1．The Denjoy Theorem 401
2．The Denjoy example 403
3．Local analytic conjugacies for Diophantine rotation number 405
4．Invariant measures and regularity of conjugacies 410
5．An example with singular conjugacy 412
6．Fast-approximation methods 415
7．Ergodicity with respect to Lebesgue measure 419
13．TWIST MAPS 423
1．The Regularity Lemma 424
2．Existence of Aubry-Mather sets and homoclinic orbits 425
3．Action functionals,minimal and ordered orbits 434
4．Orbits homoclinic to Aubry-Mather sets 441
5．Nonexistence of invariant circles and localization of Aubry-Mather sets 447
14．FLOWS ON SURFACES AND RELATED DYNAMICAL SYSTEMS 451
1．Poincaré-Bendixson theory 452
2．Fixed-point-free flows on the torus 457
3．Minimal sets 460
4．New phenomena 464
5．Interval exchange transformations 470
6．Application to flows and billiards 479
7．Generalizations of rotation number 483
15．CONTINUOUS MAPS OF THE INTERVAL 489
1．Markov covers and partitions 489
2．Entropy,periodic orbits,and horseshoes 493
3．The Sharkovsky Theorem 500
4．Maps with zero topological entropy 505
5．The kneading theory 511
6．The tent model 514
16．SMOOTH MAPS OF THE INTERVAL 519
1．The structure of hyperbolic repellers 519
2．Hyperbolic sets for smooth maps 520
3．Continuity of entropy 525
4．Full families of unimodal maps 526
Part 4 Hyperbolic dynamical systems 531
17．SURVEY OF EXAMPLES 531
1．The Smale attractor 532
2．The DA(derived from Anosov)map and the Plykin attractor 537
3．Expanding maps and Anosov automorphisms of nilmanifolds 541
4．Definitions and basic properties of hyperbolic sets for flows 544
5．Geodesic flows on surfaces of constant negative curvature 549
6．Geodesic flows on compact Riemannian manifolds with negative sectional curvature 551
7．Geodesic flows on rank-one symmetric spaces 555
8．Hyperbolic Julia sets in the complex plane 559
18．TOPOLOGICAL PROPERTIES OF HYPERBOLIC SETS 565
1．Shadowing of pseudo-orbits 565
2．Stability of hyperbolic sets and Markov approximation 571
3．Spectral decomposition and specification 574
4．Local product structure 581
5．Density and growth of periodic orbits 583
6．Global classification of Anosov diffeomorphisms on tori 587
7．Markov partitions 591
19．METRIC STRUCTURE OF HYPERBOLIC SETS 597
1．H？lder structures 597
2．Cohomological equations over hyperbolic dynamical systems 608
20．EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES 615
1．Bowen measure 615
2．Pressure and the variational principle 623
3．Uniqueness and classification of equilibrium states 628
4．Smooth invariant measures 637
5．Margulis measure 643
6．Multiplicative asymptotic for growth of periodic points 651
Supplement 659
S．DYNAMICAL SYSTEMS WITH NONUNIFORMLY HYPERBOLIC BEHAVIOR BY ANATOLE KATOK AND LEONARDO MENDOZA 659
1．Introduction 659
2．Lyapunov exponents 660
3．Regular neighborhoods 672
4．Hyperbolic measures 678
5．Entropy and dynamics of hyperbolic measures 693
Appendix 703
A．BACKGROUND MATERIAL 703
1．Basic topology 703
2．Functional analysis 711
3．Differentiable manifolds 715
4．Differential geometry 727
5．Topology and geometry of surfaces 730
6．Measure theory 731
7．Homology theory 735
8．Locally compact groups and Lie groups 738
NOTES 741
HINTS AND ANSWERS TO THE EXERCISES 765
REFERENCES 781
INDEX 793