Cvitanovic et al.
Table of contents :
Contributors……Page 2
Overture……Page 12
Why this book?……Page 13
Chaos ahead……Page 14
A game of pinball……Page 15
Periodic orbit theory……Page 24
Evolution operators……Page 29
From chaos to statistical mechanics……Page 33
Semiclassical quantization……Page 34
Guide to literature……Page 36
Guide to exercises……Page 38
Resumé……Page 39
Exercises……Page 43
Dynamical systems……Page 44
Flows……Page 48
Changing coordinates……Page 52
Computing trajectories……Page 55
Infinite-dimensional flows……Page 56
Resumé……Page 61
Exercises……Page 63
Poincaré sections……Page 68
Constructing a Poincaré section……Page 71
Hénon map……Page 73
Billiards……Page 75
Exercises……Page 80
Flows transport neighborhoods……Page 84
Linear flows……Page 86
Nonlinear flows……Page 91
Hamiltonian flows……Page 93
Billiards……Page 94
Maps……Page 97
Cycle stabilities are metric invariants……Page 98
Going global: Stable/unstable manifolds……Page 102
Resumé……Page 103
Exercises……Page 105
Measures……Page 108
Density evolution……Page 110
Invariant measures……Page 113
Koopman, Perron-Frobenius operators……Page 116
Resumé……Page 121
Exercises……Page 123
Dynamical averaging……Page 128
Evolution operators……Page 135
Lyapunov exponents……Page 137
Resumé……Page 142
Exercises……Page 143
Trace of an evolution operator……Page 146
An asymptotic trace formula……Page 153
Resumé……Page 156
Exercises……Page 157
Spectral determinants……Page 158
Spectral determinants for maps……Page 159
Spectral determinant for flows……Page 160
Dynamical zeta functions……Page 162
More examples of spectral determinants……Page 166
All too many eigenvalues?……Page 169
Resumé……Page 172
Exercises……Page 174
Why does it work?……Page 180
The simplest of spectral determinants: A single fixed point……Page 181
Analyticity of spectral determinants……Page 184
Hyperbolic maps……Page 192
Physics of eigenvalues and eigenfunctions……Page 196
Why not just run it on a computer?……Page 199
Resumé……Page 203
Exercises……Page 205
Qualitative dynamics……Page 208
Temporal ordering: Itineraries……Page 209
Symbolic dynamics, basic notions……Page 211
3-disk symbolic dynamics……Page 215
Spatial ordering of “stretch & fold” flows……Page 217
Unimodal map symbolic dynamics……Page 221
Spatial ordering: Symbol square……Page 226
Pruning……Page 231
Topological dynamics……Page 233
Resumé……Page 241
Exercises……Page 244
Counting itineraries……Page 250
Topological trace formula……Page 252
Determinant of a graph……Page 254
Topological zeta function……Page 258
Counting cycles……Page 260
Infinite partitions……Page 263
Shadowing……Page 266
Resumé……Page 268
Exercises……Page 271
Fixed points, and how to get them……Page 280
One-dimensional mappings……Page 281
d-dimensional mappings……Page 285
Flows……Page 286
Periodic orbits as extremal orbits……Page 290
Stability of cycles for maps……Page 294
Exercises……Page 299
Pseudocycles and shadowing……Page 304
Cycle formulas for dynamical averages……Page 312
Cycle expansions for finite alphabets……Page 315
Stability ordering of cycle expansions……Page 316
Dirichlet series……Page 319
Resumé……Page 322
Exercises……Page 325
Escape rates……Page 330
Flow conservation sum rules……Page 334
Correlation functions……Page 336
Trace formulas vs. level sums……Page 337
Resumé……Page 340
Exercises……Page 342
Rényi entropies……Page 344
Fractal dimensions……Page 349
Resumé……Page 353
Exercises……Page 354
Intermittency……Page 358
Intermittency everywhere……Page 359
Intermittency for beginners……Page 363
General intermittent maps……Page 376
Probabilistic or BER zeta functions……Page 382
Resumé……Page 387
Exercises……Page 389
Discrete symmetries……Page 392
Preview……Page 393
Discrete symmetries……Page 397
Dynamics in the fundamental domain……Page 400
Factorizations of dynamical zeta functions……Page 404
C2 factorization……Page 406
C3v factorization: 3-disk game of pinball……Page 408
Resumé……Page 411
Exercises……Page 414
Deterministic diffusion……Page 418
Diffusion in periodic arrays……Page 419
Diffusion induced by chains of 1-d maps……Page 423
Resumé……Page 432
Exercises……Page 435
Irrationally winding……Page 436
Mode locking……Page 437
Local theory: “Golden mean” renormalization……Page 444
Global theory: Thermodynamic averaging……Page 446
Hausdorff dimension of irrational windings……Page 447
Thermodynamics of Farey tree: Farey model……Page 449
Resumé……Page 455
Exercises……Page 458
The thermodynamic limit……Page 460
Ising models……Page 463
Fisher droplet model……Page 466
Scaling functions……Page 472
Geometrization……Page 476
Resumé……Page 484
Exercises……Page 486
Semiclassical evolution……Page 490
Quantum mechanics: A brief review……Page 491
Semiclassical evolution……Page 495
Semiclassical propagator……Page 504
Semiclassical Green’s function……Page 508
Resumé……Page 516
Exercises……Page 518
Trace formula……Page 524
Semiclassical spectral determinant……Page 529
One-dimensional systems……Page 531
Resumé……Page 533
Exercises……Page 538
Helium atom……Page 540
Classical dynamics of collinear helium……Page 541
Semiclassical quantization of collinear helium……Page 554
Resumé……Page 564
Exercises……Page 566
Quantum eavesdropping……Page 568
An application……Page 575
Resumé……Page 582
Exercises……Page 584
Cycles as the skeleton of chaos……Page 586
Index……Page 590
II Material available on www.nbi.dk/ChaosBook/……Page 606
Professor Gatto Nero……Page 608
Chaos is born……Page 610
Chaos grows up……Page 614
Chaos with us……Page 615
Death of the Old Quantum Theory……Page 619
Symplectic invariance……Page 622
Monodromy matrix for Hamiltonian flows……Page 624
Material invariants……Page 628
Implementing evolution……Page 629
Exercises……Page 634
Topological zeta functions for infinite subshifts……Page 636
Prime factorization for dynamical itineraries……Page 645
Counting curvatures……Page 650
Exercises……Page 652
Evolution operator for Lyapunov exponents……Page 654
Advection of vector fields by chaotic flows……Page 659
Exercises……Page 666
Preliminaries and Definitions……Page 668
C4v factorization……Page 673
C2v factorization……Page 678
Symmetries of the symbol square……Page 681
Curvature expansions: geometric picture……Page 682
Ma-the-matical caveats……Page 686
Estimate of the nth cumulant……Page 688
Matrix-valued functions……Page 690
Trace class and Hilbert-Schmidt class……Page 692
Determinants of trace class operators……Page 694
Von Koch matrices……Page 698
Regularization……Page 700
Solutions……Page 704
Projects……Page 734
Deterministic diffusion, zig-zag map……Page 736
Deterministic diffusion, sawtooth map……Page 743
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