Preface ......................................................... v
Notation ....................................................... ix
0 Introduction ................................................. 1
0.1 Definition and a (very brief) historical overview ....... 1
0.2 Continuous vs. discrete time ............................ 3
0.3 The dynamical systems point of view ..................... 7
0.4 Examples ................................................ 9
1 Basic notions ............................................... 17
1.1 Invariant and periodic points .......................... 17
1.2 Invariant sets ......................................... 23
1.3 Transitivity ........................................... 28
1.4 Limit sets ............................................. 33
1.5 Topological conjugacy and factor mappings .............. 35
1.6 Equicontinuity and weak mixing ......................... 44
1.7 Miscellaneous examples ................................. 57
2 Dynamical systems on the real line .......................... 73
2.1 Graphical iteration .................................... 73
2.2 Existence of periodic orbits ........................... 80
2.3 The truncated tent map ................................. 84
2.4 The double of a mapping ................................ 87
2.5 The Markov graph of a periodic orbit in an interval .... 91
2.6 Transitivity of mappings of an interval ............... 101
3 Limit behaviour ............................................ 117
3.1 Limit sets and attraction ............................. 117
3.2 Stability ............................................. 126
3.3 Stability and attraction for periodic orbits .......... 132
3.4 Asymptotic stability in locally compact spaces ........ 143
3.5 The structure of (asymptotically) stable sets ......... 153
4 Recurrent behaviour ........................................ 165
4.1 Recurrent points ...................................... 165
4.2 Almost periodic points and minimal orbit closures ..... 169
4.3 Non-wandering points .................................. 175
4.4 Chain-recurrence ...................................... 182
4.5 Asymptotic stability and basic sets ................... 197
5 Shift systems .............................................. 218
5.1 Notation and terminology .............................. 218
5.2 The shift mapping ..................................... 223
5.3 Shift spaces .......................................... 226
5.4 Factor maps ........................................... 236
5.5 Subshifts and graphs .................................. 244
5.6 Recurrence, almost periodicity and mixing ............. 253
6 Symbolic representations ................................... 282
6.1 Topological partitions ................................ 282
6.2 Expansive systems ..................................... 293
6.3 Applications .......................................... 302
7 Erratic behaviour .......................................... 325
7.1 Stability revisited ................................... 325
7.2 Chaos(1): sensitive systems ........................... 336
7.3 Chaos(2): scrambled sets .............................. 342
7.4 Horseshoes for interval maps .......................... 355
7.5 Existence of a horseshoe .............................. 365
8 Topological entropy ........................................ 378
8.1 The definition ........................................ 378
8.2 Independence of the metric; factor maps ............... 387
8.3 Maps on intervals and circles ......................... 391
8.4 The definition with covers ............................ 394
8.5 Miscellaneous results ................................. 402
8.6 Positive entropy and horseshoes for interval maps ..... 406
A Topology ................................................... 423
A.l Elementary notions .................................... 423
A.2 Compactness ........................................... 426
A.3 Continuous mappings ................................... 428
A.4 Convergence ........................................... 430
A.5 Subspaces, products and quotients ..................... 432
A.6 Connectedness ......................................... 434
A.7 Metric spaces ......................................... 437
A.8 Baire category ........................................ 444
A.9 Irreducible mappings .................................. 446
A.10 Miscellaneous results ................................. 449
В The Cantor set ............................................. 453
B.l The construction ...................................... 453
B.2 Proof of Brouwer's Theorem ............................ 456
B.3 Cantor spaces ......................................... 461
С Hints to th e Exercises .................................... 465
Literature .................................................... 481
Index ......................................................... 485
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