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 |  موضوع: كتاب A First Course in Dynamics - with a Panorama of Recent Developments الخميس 10 أكتوبر 2024, 6:26 pm | |
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أخواني في الله
أحضرت لكم كتاب
A First Course in Dynamics - with a Panorama of Recent Developments
Boris Hasselblatt
Tufts University
Anatole Katok
The Pennsylvania State University
و المحتوى كما يلي :
Contents
Preface page ix
1 Introduction 1
1.1 Dynamics 1
1.2 Dynamics in Nature 4
1.3 Dynamics in Mathematics 19
PART 1. A COURSE IN DYNAMICS: FROM SIMPLE TO
COMPLICATED BEHAVIOR 29
2 Systems with Stable Asymptotic Behavior 31
2.1 Linear Maps and Linearization 31
2.2 Contractions in Euclidean Space 32
2.3 Nondecreasing Maps of an Interval and Bifurcations 45
2.4 Differential Equations 49
2.5 Quadratic Maps 57
2.6 Metric Spaces 61
2.7 Fractals 69
3 Linear Maps and Linear Differential Equations 73
3.1 Linear Maps in the Plane 73
3.2 Linear Differential Equations in the Plane 86
3.3 Linear Maps and Differential Equations in Higher Dimension 90
4 Recurrence and Equidistribution on the Circle 96
4.1 Rotations of the Circle 96
4.2 Some Applications of Density and Uniform Distribution 109
4.3 Invertible Circle Maps 123
4.4 Cantor Phenomena 135
5 Recurrence and Equidistribution in Higher Dimension 143
5.1 Translations and Linear Flows on the Torus 143
5.2 Applications of Translations and Linear Flows 152
vvi Contents
6 Conservative Systems 155
6.1 Preservation of Phase Volume and Recurrence 155
6.2 Newtonian Systems of Classical Mechanics 162
6.3 Billiards: Definition and Examples 177
6.4 Convex Billiards 186
7 Simple Systems with Complicated Orbit Structure 196
7.1 Growth of Periodic Points 196
7.2 Topological Transitivity and Chaos 205
7.3 Coding 211
7.4 More Examples of Coding 221
7.5 Uniform Distribution 229
7.6 Independence, Entropy, Mixing 235
8 Entropy and Chaos 242
8.1 Dimension of a Compact Space 242
8.2 Topological Entropy 245
8.3 Applications and Extensions 251
PART 2. PANORAMA OF DYNAMICAL SYSTEMS 257
9 Simple Dynamics as a Tool 259
9.1 Introduction 259
9.2 Implicit- and Inverse-Function Theorems in Euclidean Space 260
9.3 Persistence of Transverse Fixed Points 265
9.4 Solutions of Differential Equations 267
9.5 Hyperbolicity 273
10 Hyperbolic Dynamics 279
10.1 Hyperbolic Sets 279
10.2 Orbit Structure and Orbit Growth 284
10.3 Coding and Mixing 291
10.4 Statistical Properties 294
10.5 Nonuniformly Hyperbolic Dynamical Systems 298
11 Quadratic Maps 299
11.1 Preliminaries 299
11.2 Development of Simple Behavior Beyond the First Bifurcation 303
11.3 Onset of Complexity 307
11.4 Hyperbolic and Stochastic Behavior 314
12 Homoclinic Tangles 318
12.1 Nonlinear Horseshoes 318
12.2 Homoclinic Points 320
12.3 The Appearance of Horseshoes 322
12.4 The Importance of Horseshoes 324
12.5 Detecting Homoclinic Tangles: The Poincare´–Melnikov
Method 327
12.6 Homoclinic Tangencies 328Contents vii
13 Strange Attractors 331
13.1 Familiar Attractors 331
13.2 The Solenoid 333
13.3 The Lorenz Attractor 335
14 Variational Methods, Twist Maps, and Closed Geodesics 342
14.1 The Variational Method and Birkhoff Periodic Orbits
for Billiards 342
14.2 Birkhoff Periodic Orbits and Aubry–Mather Theory
for Twist Maps 346
14.3 Invariant Circles and Regions of Instability 357
14.4 Periodic Points for Maps of the Cylinder 360
14.5 Geodesics on the Sphere 362
15 Dynamics, Number Theory, and Diophantine Approximation 365
15.1 Uniform Distribution of the Fractional Parts of Polynomials 365
15.2 Continued Fractions and Rational Approximation 369
15.3 The Gauß Map 374
15.4 Homogeneous Dynamics, Geometry, and Number Theory 377
15.5 Quadratic Forms in Three Variables 383
Reading 386
APPENDIX 389
A.1 Metric Spaces 389
A.2 Differentiability 400
A.3 Riemann Integration in Metric Spaces 401
Hints and Answers 408
Solutions 414
Index 419
Index
absolute continuity, 295–296
accumulation point, 390
action functional, 175, 176, 343
adapted
inner product, 95
metric, 67
norm, 95
adding machine, 109, 312
admissible, 217
Alaoglu, Leonidas, 296
Alekseev, Vladimir Mihkailovich, 326
almost everywhere, 233
alphabet, 214, 216
Angenent, Sigurd B., 358
angular momentum, 174
annulus, 38
Anosov Closing Lemma, 285
Arnold tongues, 142
astroid, 195
asymptotic distribution, 109–110
Atela, Pau, 8
attracting, 300
attracting fixed point, 41, 331
attractor, 296, 331, 332
Aubry, Serge J., 342
Aubry–Mather set, 352–354, 357, 358
autonomous differential equation, 271
Baire Category, 392
ball, 62
Bangert, Victor, 364
Barton, Reid, 310
basic set, 280
basin of attraction, 301, 305
Bernoulli measure, 238, 239
Bernoulli scheme, 236
bifurcation, 48, 305
period doubling, 305
saddle-node, 48
bijection, 62
billiard flow, 116, 179
billiard map, 179
binary search, 20
Birkhoff average, 103, 104, 230, 233
Birkhoff Ergodic Theorem, 234, 295
Birkhoff periodic orbit, 343, 344, 349–351, 358,
359
Birkhoff, George David, 103, 178, 195, 322, 324,
343, 360, 364
Birkhoff–Smale Theorem, 322
Borel Density Theorem, 383
boundary, 390
bounded, 389
bounded variation, 138
Bowen, Rufus, 297
box dimension, 243, 244
Burns, Keith, 310
butterfly, 12, 51, 209
C 1-topology, 400
Cr -topology, 400
Cantor function, 136, 142
Cantor set, 69–70, 132, 135, 138, 213–214, 216,
225, 243, 312, 342, 390, 398, 399
capacity, 242, 246
capture, 326
Cartwright, Mary Lucy, 324
Cassels, J. W. S., 383
Cauchy sequence, 35, 62, 391
caustic, 181, 183, 190, 192
cellular automata, 26
center, 88
central force, 171, 174
chain recurrence, 285
chaos, 205, 310, 342
chaotic, 205, 208, 209, 217, 219
characteristic function, 103
characters, 150
419420 Index
Chebyshev, Pafnuty Lvovich, 316
Chuba, Sharon, 299
circle, 62, 96, 123
circle map, 123
classification, 223
climate, 316
closed, 390
closed geodesics, 23, 326, 363
closure, 390
cobweb picture, 46
coding, 211, 212, 214, 216, 223–224, 226, 252,
288, 291, 293
coin tossing, 235
Collet, 317
compact, 393, 399
complete integrability, 122
completeness, 62, 391
cone, 281
configuration space, 163
confocal ellipse, 183
confocal hyperbola, 184
conjugacy, 134, 135, 216–217, 223, 246
connected, 390
conservative, 165
constant of motion, 171
continued fraction, 100, 371, 373, 376, 378
continuity, 62, 392
continuous functions, space of, 391, 396
contracting subspace, 94
contraction, 33–40, 62, 392
Contraction Principle, 35–36, 66, 259–263, 265,
268, 276, 284
convergence, 62, 390
convergents, 370
convex, 38, 45, 186, 189
caustic, 192–195
strictly differentiably, 186
Coxeter, Harold Scott Macdonald, 7
Cremona map, 321
curvature, 191
cylinder, 65, 167, 179, 180, 215
Dani, Shrikrishna Gopatrao, 384
decreasing, 45
degree, 124, 200
Denjoy example, 137
Denjoy Theorem, 354
dense, 390
derivative, 400
determinant, 157
Devaney, Robert, 205, 386
devil’s staircase, 71, 135, 140
diagonalization, 75
diameter, 182, 189
diffeomorphism, 401
differential, 37
digits of polynomials, 25–26
Diophantine, 374
Dirichlet, Johann Peter Gustav Lejeune, 365
distance function, 61, 389; see also metric
distribution function, 110, 238, 403
divergence, 158
Dragt, Alex J., 321, 325
Drake, Sir Francis, 17
dyadic integers, 109, 399
Eckmann, Jean-Pierre, 317
eigenfunction, 150
eigenspace, 74
generalized, 92
eigenvalue, 74, 150
eigenvector, 74
ellipsoid, 326
elliptic
integrals, 168
islands, 359, 360
linear map, 78
embedding, 401
entropy, 237, 246, 299, 314, 318, 326, 327, 358
envelope, 190–191
equidistribution, 102, 296
of squares, 368
see also uniform distribution
equilibrium, 49
equivalent metrics, 393
ergodicity, 234, 295
Euler–Lagrange equation, 175
eventually contracting, 67, 79, 93–94
expanding map, 196–198, 200–203, 207–208,
211–213, 221, 223, 230, 232, 234, 251–252,
291, 294
expanding subspace, 94
expansivity, 283–284, 286, 288
exponential convergence, 37
factor, 134, 216, 221
map, 216
Feigenbaum, Mitchell J., 307–308, 311–312
Fermat’s principle, 182
Fibonacci, 6, 8, 35, 44, 84, 85, 204
Finn, John M., 325
fireflies, 17, 141
first integral, 171
fixed point, 34
attracting, 41
repelling, 48
flow, 53–54, 272
box, 170
equivalence, 254
flux, 195
focus, 81, 88
focussing, 181
forced oscillator, 348
fractals, 244
Franks, John, 361, 362, 364
free particle motion, 153, 165
frequency, 100, 103, 112, 145
locking, 141Index 421
Fuchsian group, 379
fundamental domain, 118, 121, 143
Furstenberg, Hillel, 366, 384
Galilei, Galileo, 1
game of life, 26
Gauß map, 375, 382
Gelfreich, Vasily G., 328
general relativity, 173
generalized eigenspace, 92–93
generating function, 182, 186–189, 347,
348
geodesic flow, 153, 165, 177, 326, 378, 382
geodesics, 23
geometric optics, 191
Gibbs, Josiah Willard, 296
Gibbs measure, 296
Gole, Christophe, 8 ´
Graczyk, Jacek, 315
graphical computing, 46
group, 107, 143
Haar measure, 379
Hakluyt, Richard, 17
Handel, Michael, 362
harmonic oscillator, 115, 141, 164, 168, 174
Hartman–Grobman Theorem, 277–278
Hausdorff metric, 394–395
Heine–Borel Theorem, 393, 399
Herman, Michael , 354
Heron of Alexandria, 19
heteroclinic, 47, 211, 321, 325, 359
Hilbert cube, 397
Hofmeister rules, 8
homeomorphism, 62, 392
homoclinic, 47, 159, 166–167, 169, 320, 322, 325,
358
tangency, 328
tangles, 326
homogeneous action, 378
homogeneous space, 377
homothety, 76
horocycle, 380
flow, 380
horseshoe, 213, 224, 244, 280, 290, 318, 322,
324–327, 329, 334, 358, 359
Hotton, Scott, 8
hyperbolic
attractor, 335
dynamical systems, 289
fixed point, 273
linear map, 76
metric, 378
quadratic map, 302, 315
repeller, 280
set, 279–280, 283–288, 290–294
locally maximal, 280
Hyperbolic Fixed-Point Theorem, 277, 284
hyperbolicity, 380
Implicit-Function Theorem, 264
incompressibility, 155, 157
increasing, 45
induced metric, 389
integrable, 402
twist, 153
integral, 171
interior, 389
Intermediate-Value Theorem, 45, 47
invariant
circle, 185, 192, 357, 358
density, 295
measure, 295–296, 300
invariants, 246
inverse limit, 203–204, 335
Inverse-Function Theorem, 263
irrational rotation, 99
isometry, 62, 155, 207, 392
itinerary, 211
Jacobi, Carl Gustav Jacob, 365
Jacobian, 157
Jakobson, Michael V., 303, 317
Jordan Curve Theorem, 60
Jordan normal form, 92
Kampfer, Engelbert, 17 ¨
Kepler problem, 121
Kepler’s Second Law, 173
Kepler, Johannes, 6–7, 163
Khinchine, Alexander Ya., 374
kinetic energy, 164
Klingenberg, Wilhelm, 364
kneading theory, 300, 307
Knieper, Gerhard, 326
Koch snowflake, 71, 244
Kronecker, Leopold, 365
Kronecker–Weyl method, 107, 150, 366
Lagrange, 175
Lagrange equation, 175
Lanford, Oscar III, 308
Laplace, Pierre Simon de, 1, 2
lattice, 381, 384
law of large numbers, 234, 316
Lazutkin parameter, 194
Lazutkin, Vladimir F., 194
Le Calvez, Patrice, 362
lemmings, 44
Levinson, Norman , 324
Li, Tien-Yien, 310
lift, 124, 346
limit cycle, 11, 55, 332
linear
approximation, 32
flow, 145
on the torus, 112, 121
twist, 156
linearization, 114, 168422 Index
Liouville measure, 379
Liouvillian phenomena, 373
Lipschitz, 33, 50, 62, 138, 268, 271, 353, 357, 392,
398
Lissajous figures, 115
Littlewood, John Edensor, 324
lobsters, 7
local entropy, 255
localization, 276
locally compact, 395
logistic differential equation, 51–53
logistic equation, 14–17, 57–60, 86; see also
quadratic family
Lorenz attractor, 280, 338–341
Lorenz, Edward Norton, 12, 209, 316, 331, 335,
338
Lotka–Volterra equation, 10
Lyapunov
function, 336
metric, 67
norm, 94
Lyusternik, Lazar A., 363, 364
Lyusternik–Shnirelman category, 363
Mahler criterion, 381–382
Margulis, Gregory A., 383–385
Markov
graph, 217–218, 309
partition, 291, 293, 300
mathematical pendulum, 122, 159, 169
Mather, John, 342, 355
matrix exponential, 89
May, Robert M., 13, 310
Man˜e, Ricardo, 290 ´
Mean Value Theorem, 400
measurable, 404
partition, 405
measure, 404
Melnikov, V. K., 327, 328
Menger curve, 71
Mercury, 173
metric, 61, 97, 214, 389
adapted, 67
Lyapunov, 67
metric space, 389
minimal, 99, 108, 113
minimality, 99, 108, 113–114, 144, 146, 148, 153
minimax orbit, 344, 351, 359
mirror equation, 191
Misiurewicz, Michal, 312, 317, 327
mixing, 238–240
modular surface, 379, 382
momentum, 164
mountain pass, 344
Myrberg, Pekka Juhana, 300
Mobius transformation, 378 ¨
neighborhood, 389
Newhouse Phenomenon, 329
Newhouse, Sheldon, 327
Newton method, 21, 42, 261
Newton’s Law, 163
Newton, Sir Isaac, 1, 163
Nguyen, An, 307
node, 80, 87
degenerate, 81, 87
nondecreasing, 45
nondimensionalizing, 166
nonincreasing, 45
nonwandering, 294
norm, 78, 92, 395
of a matrix, 37, 45, 91
normal form, 341
nowhere dense, 390
null set, 233, 296, 302, 402
open, 389
open map, 392
Oppenheim conjecture, 365
orbit, 34
ordered states, 349
ordering, 123, 128, 138
orientation-preserving, 125
outer billiard, 348
parabolic linear map, 77
parameter exclusion method, 317
Peano curve, 72
perfect, 132, 390, 399
perihelion angle, 173
period, 34
period doubling, 299, 304–305
periodic, 97, 116, 144
coefficients, 89
orbit, 196–198
orbits, 217, 314
point, 34, 300
attracting, 300
points, 196–201, 205, 217, 219, 226, 234,
254, 284–285, 288, 310–311, 314, 360,
361
phase
portrait, 51
space, 163, 180
volume, 155, 158, 164
phyllotaxis, 7–9
Picard, Charles Emile, 268
Picard iteration, 267
piecewise monotone, 110
Poincare Classi ´ fication, 133
Poincare´’s Last Geometric Theorem, 360, 364
Poincare, Jules Henri, 2, 324, 341, 360 ´
Poincare´–Bendixson Theorem, 60
Poincare´–Melnikov method, 327
Poincare Recurrence Theorem, 159 ´
potential energy, 164, 174
powers of 2, 23, 111
precession, 173Index 423
prime period, 34
product metric, 396–397
pseudo-orbit, 285
pseudosphere, 379
pullback, 391
quadratic family, 57–60, 198–200, 223, 299–317;
see also logistic equation
quadratic form, 381, 383–384
quasiperiodic, 96
rabbits, and Leonardo of Pisa, 5, 35
rabbits, and predators, 10
rabbits, antipodal, 4
radioactive decay, 9
Raghunathan, Madabusi S., 383–384
randomness, 235
rational independence, 144, 151
Ratner, Marina, 384–385
rectangle, 291–292
recurrence, 96, 137, 155
recurrent, 159
recursion, 84, 85
reduction to first order, 84, 164, 166
region of instability, 358
renormalization, 305, 376–377
repelling fixed point, 48
rescaling property, 70
resonance, 80
return map, 55, 113, 179
Riemann integral, 407
Riemann sum, 405
Riesz integral, 404
Robbin, Joel, 290
Robinson, R. Clark, 290
Rom-Kedar, Vered, 326
root space, 92–93
rotation number, 125, 128–129, 133, 138–139,
246, 352–354, 356, 361
rotation of a circle, 96–109, 161
rotation set, 135
Ruelle, David, 297, 316
saddle, 82, 88
Sanders, Jan A., 328
Scherer, Andrew, 299
Schwarzian derivative, 306
section map, 179
self-similarity, 70, 136
semiconjugacy, 216, 228
semistable, 49, 130
sensitive dependence, 209, 210, 217, 219, 255,
283
separable, 395
sequence space, 214, 398
shadowing, 286, 296
Shadowing Lemma, 286
Shadowing Theorem, 289
Sharkovsky, Olexandr Mikolaiovich, 300
Sharkovsky Theorem, 311
shear, 77
shift, 27, 215, 217, 240
Shnirelman, Lev Genrikhovich, 363, 364
Sierpinski carpet, 71, 243
Sierpinski sponge, 71
Sinai, Yakov, 297
Sinai–Ruelle–Bowen Measure, 297
sliding block codes, 27
Smale attractor, 333–335
Smale, Steven, 224, 307, 322, 324, 333
small oscillation, 168
smooth dependence on initial conditions, 270
sofic system, 288
solenoid, 203, 296, 333
space average, 105
space-filling curve, 72
specification, 287
property, 287, 288
Specification Theorem, 287, 296
spectral
decomposition, 294
radius, 90, 218
spectrum, 90
sphere, 165
spherical pendulum, 114, 174
square root, 19
Stability Conjecture, 290
Stable Manifold Theorem, 275, 282, 320
state space, 163
statistical properties, 294
steady state, 32
Stirling formula, 231
stochasticity, 316
string construction, 194
structural stability, 277, 289–290, 300
Størmer problem, 325
subadditivity, 126, 147
subshift of finite type, 216
superattracting, 43
Swinnerton-Dyer, Sir Peter, 383
´ Swi
atek, Grzegorz, 303, 315 
sword, 334
symbolic dynamics, 27
synchronization, 18, 141
syndetic, 160
taffy, 334
tangles, 321, 324, 325, 359
tent map, 315–316
Thurston, William, 362
tiling, 118
time average, 105
time change, 272
topological
conjugacy, 128
entropy, 246–249, 251, 254, 288
group, 108
invariants, 217424 Index
topological (cont.)
Markov chain, 216–217, 228, 252, 288, 291,
293, 302
mixing, 207, 210, 217, 226, 239, 287,
293–294
transitivity, 99, 108, 133, 205, 212, 217,
285
torus, 65, 112, 118, 121, 143, 152, 165, 226
automorphism, 201–203, 208, 226, 240, 252,
280, 283, 291
total energy, 164, 166
totally bounded, 393
totally disconnected, 390, 399
transitive matrix, 219
transverse periodic point, 266
triangle inequality, 62, 389, 396
trigonometric polynomial, 107, 150
Tucker, Warwick, 339, 340
twist, 179
interval, 347, 360
map, 325, 346
Tychonoff, Andrey Nikolayevich, 397
Ulam, 316
unfolding, 117
uniform distribution, 26, 102, 107, 112, 116, 120,
145–147, 229, 234, 295, 316
of polynomials, 368
of squares, 368
see also equidistribution
uniform recurrence, 160–161
uniformly equivalent, 62
unimodal, 199
unimodular, 381, 384
unipotence, 384
unique ergodicity, 106, 230, 234, 295, 313, 365,
367, 380, 384
universal repeller, 302
van der Mark, J., 11, 12
van der Pol, Balthasar, 11, 12, 324
von Neumann, John, 316
wandering, 137
Weierstraß, Karl Theodor Wilhelm, 107, 151, 366
Weiss, Howard, 326
Weyl, Hermann, 365, 366
width, 182, 189
Wiener, Norbert, 310
Wintner, Aurel, 310
Yomdin, Yosef, 327
Yorke, James A., 310
Zariski, Oscar, 383
ω-limit set, 132, 159
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