Buch, Englisch, 312 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 491 g
Buch, Englisch, 312 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 491 g
Reihe: Annals of Mathematics Studies
ISBN: 978-0-691-14249-4
Verlag: Princeton University Press
Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950s, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids--connections that together allow for a fairly complete analysis of the dynamical system.
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Weitere Infos & Material
Preface xi
Chapter 1. Introduction 1
1.1 Definitions and History 1
1.2 The Erratic Orbits Theorem 3
1.3 Corollaries of the Comet Theorem 4
1.4 The Comet Theorem 7
1.5 Rational Kites 10
1.6 The Arithmetic Graph 12
1.7 The Master Picture Theorem 15
1.8 Remarks on Computation 16
1.9 Organization of the Book 16
PART 1. THE ERRATIC ORBITS THEOREM 17
Chapter 2. The Arithmetic Graph 19
2.1 Polygonal Outer Billiards 19
2.2 Special Orbits 20
2.3 The Return Lemma 21
2.4 The Return Map 25
2.5 The Arithmetic Graph 26
2.6 Low Vertices and Parity 28
2.7 Hausdorff Convergence 30
Chapter 3. The Hexagrid Theorem 33
3.1 The Arithmetic Kite 33
3.2 The Hexagrid Theorem 35
3.3 The Room Lemma 37
3.4 Orbit Excursions 38
Chapter 4. Period Copying 41
4.1 Inferior and Superior Sequences 41
4.2 Strong Sequences 43
Chapter 5. Proof of the Erratic Orbits Theorem 45
5.1 Proof of Statement 1 45
5.2 Proof of Statement 2 49
5.3 Proof of Statement 3 50
PART 2. THE MASTER PICTURE THEOREM 53
Chapter 6. The Master Picture Theorem 55
6.1 Coarse Formulation 55
6.2 The Walls of the Partitions 56
6.3 The Partitions 57
6.4 A Typical Example 59
6.5 A Singular Example 60
6.6 The Reduction Algorithm 62
6.7 The Integral Structure 63
6.8 Calculating with the Polytopes 65
6.9 Computing the Partition 66
Chapter 7. The Pinwheel Lemma 69
7.1 The Main Result 69
7.2 Discussion 71
7.3 Far from the Kite 72
7.4 No Sharps or Flats 73
7.5 Dealing with 4? 74
7.6 Dealing with 6? 75
7.7 The Last Cases 76
Chapter 8. The Torus Lemma 77
8.1 The Main Result 77
8.2 Input from the Torus Map 78
8.3 Pairs of Strips 79
8.4 Single-Parameter Proof 81
8.5 Proof in the General Case 83
Chapter 9. The Strip Functions 85
9.1 The Main Result 85
9.2 Continuous Extension 86
9.3 Local Affine Structure 87
9.4 Irrational Quintuples 89
9.5 Verification 90
9.6 An Example Calculation 91
Chapter 10. Proof of the Master Picture Theorem 93
10.1 The Main Argument 93
10.2 The First Four Singular Sets 94
10.3 Symmetry 95
10.4 The Remaining Pieces 96
10.5 Proof of the Second Statement 97
PART 3. ARITHMETIC GRAPH STRUCTURE THEOREMS 99
Chapter 11. Proof of the Embedding Theorem 101
11.1 No Valence 1 Vertices 101
11.2 No Crossings 104
Chapter 12. Extension and Symmetry 107
12.1 Translational Symmetry 107
12.2 A Converse Result 110
12.3 Rotational Symmetry 111
12.4 Near-Bilateral Symmetry 113
Chapter 13. Proof of Hexagrid Theorem I 117
13.1 The Key Result 117
13.2 A Special Case 118
13.3 Planes and Strips 119
13.4 The End of the Proof 120
13.5 A Visual Tour 121
Chapter 14. The Barrier Theorem 125
14.1 The Result 125
14.2 The Image of the Barrier Line 127
14.3 An Example 129
14.4 Bounding the New Crossings 130
14.5 The Other Case 132
Chapter 15. Proof of Hexagrid Theorem II 133
15.1 The Structure of the Doors 133
15.2 Ordinary Crossing Cells 135
15.3 New Maps 136
15.4 Intersection Results 138
15.5 The End of the Proof 141
15.6 The Pattern of Crossing Cells 142
Chapter 16. Proof of the Intersection Lemma 143
16.1 Discussion of the Proof 143
16.2 Covering Parallelograms 144
16.3 Proof of Statement 1 146
16.4 Proof of Statement 2 148
16.5 Proof of Statement 3 149
PART 4. PERIOD-COPYING THEOREMS 151
Chapter 17. Diophantine Approximation 153
17.1 Existence of the Inferior Sequence 153
17.2 Structure of the Inferior Sequence 155
17.3 Existence of the Superior Sequence 158
17.4 The Diophantine Constant 159
17.5 A Structural Result 161
Chapter 18. The Diophantine Lemma 163
18.1 Three Linear Functionals 163
18.2 The Main Result 164
18.3 A Quick Application 165
18.4 Proof of the Diophantine Lemma 166
18.5 Proof of the Agreement Lemma 167
18.6 Proof of the Good Integer Lemma 169
Chapter 19. The Decomposition Theorem 171
19.1 The Main Result 171
19.2 A Comparison 173
19.3 A Crossing Lemma
