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.
Author(s): Richard Evan Schwartz
Series: Ann.Math.Stud.171
Publisher: PUP
Year: 2009
Language: English
Pages: 321
Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 12
1.1 Definitions and History......Page 16
1.2 The Erratic Orbits Theorem......Page 18
1.3 Corollaries of the Comet Theorem......Page 19
1.4 The Comet Theorem......Page 22
1.5 Rational Kites......Page 25
1.6 The Arithmetic Graph......Page 27
1.7 The Master Picture Theorem......Page 30
1.9 Organization of the Book......Page 31
PART 1. THE ERRATIC ORBITS THEOREM......Page 32
2.1 Polygonal Outer Billiards......Page 34
2.2 Special Orbits......Page 35
2.3 The Return Lemma......Page 36
2.4 The Return Map......Page 40
2.5 The Arithmetic Graph......Page 41
2.6 Low Vertices and Parity......Page 43
2.7 Hausdorff Convergence......Page 45
3.1 The Arithmetic Kite......Page 48
3.2 The Hexagrid Theorem......Page 50
3.3 The Room Lemma......Page 52
3.4 Orbit Excursions......Page 53
4.1 Inferior and Superior Sequences......Page 56
4.2 Strong Sequences......Page 58
5.1 Proof of Statement 1......Page 60
5.2 Proof of Statement 2......Page 64
5.3 Proof of Statement 3......Page 65
PART 2. THE MASTER PICTURE THEOREM......Page 68
6.1 Coarse Formulation......Page 70
6.2 The Walls of the Partitions......Page 71
6.3 The Partitions......Page 72
6.4 A Typical Example......Page 74
6.5 A Singular Example......Page 75
6.6 The Reduction Algorithm......Page 77
6.7 The Integral Structure......Page 78
6.8 Calculating with the Polytopes......Page 80
6.9 Computing the Partition......Page 81
7.1 The Main Result......Page 84
7.2 Discussion......Page 86
7.3 Far from the Kite......Page 87
7.4 No Sharps or Flats......Page 88
7.5 Dealing with 4[sup(#)]......Page 89
7.6 Dealing with 6[sup(b)]......Page 90
7.7 The Last Cases......Page 91
8.1 The Main Result......Page 92
8.2 Input from the Torus Map......Page 93
8.3 Pairs of Strips......Page 94
8.4 Single-Parameter Proof......Page 96
8.5 Proof in the General Case......Page 98
9.1 The Main Result......Page 100
9.2 Continuous Extension......Page 101
9.3 Local Affine Structure......Page 102
9.4 Irrational Quintuples......Page 104
9.5 Verification......Page 105
9.6 An Example Calculation......Page 106
10.1 The Main Argument......Page 108
10.2 The First Four Singular Sets......Page 109
10.3 Symmetry......Page 110
10.4 The Remaining Pieces......Page 111
10.5 Proof of the Second Statement......Page 112
PART 3. ARITHMETIC GRAPH STRUCTURE THEOREMS......Page 114
11.1 No Valence 1 Vertices......Page 116
11.2 No Crossings......Page 119
12.1 Translational Symmetry......Page 122
12.2 A Converse Result......Page 125
12.3 Rotational Symmetry......Page 126
12.4 Near-Bilateral Symmetry......Page 128
13.1 The Key Result......Page 132
13.2 A Special Case......Page 133
13.3 Planes and Strips......Page 134
13.4 The End of the Proof......Page 135
13.5 A Visual Tour......Page 136
14.1 The Result......Page 140
14.2 The Image of the Barrier Line......Page 142
14.3 An Example......Page 144
14.4 Bounding the New Crossings......Page 145
14.5 The Other Case......Page 147
15.1 The Structure of the Doors......Page 148
15.2 Ordinary Crossing Cells......Page 150
15.3 New Maps......Page 151
15.4 Intersection Results......Page 153
15.5 The End of the Proof......Page 156
15.6 The Pattern of Crossing Cells......Page 157
16.1 Discussion of the Proof......Page 158
16.2 Covering Parallelograms......Page 159
16.3 Proof of Statement 1......Page 161
16.4 Proof of Statement 2......Page 163
16.5 Proof of Statement 3......Page 164
PART 4. PERIOD-COPYING THEOREMS......Page 166
17.1 Existence of the Inferior Sequence......Page 168
17.2 Structure of the Inferior Sequence......Page 170
17.3 Existence of the Superior Sequence......Page 173
17.4 The Diophantine Constant......Page 174
17.5 A Structural Result......Page 176
18.1 Three Linear Functionals......Page 178
18.2 The Main Result......Page 179
18.3 A Quick Application......Page 180
18.4 Proof of the Diophantine Lemma......Page 181
18.5 Proof of the Agreement Lemma......Page 182
18.6 Proof of the Good Integer Lemma......Page 184
19.1 The Main Result......Page 186
19.2 A Comparison......Page 188
19.3 A Crossing Lemma......Page 189
19.4 Most of the Parameters......Page 190
19.5 The Exceptional Cases......Page 193
20.1 Step 1......Page 196
20.2 Step 2......Page 197
20.3 Step 3......Page 198
PART 5. THE COMET THEOREM......Page 200
21.1 The Results......Page 202
21.2 The Growth of Denominators......Page 203
21.3 The Identities......Page 204
22.1 Main Results......Page 208
22.2 The Copy and Pivot Theorems......Page 210
22.3 Half of the Result......Page 212
22.4 The Inheritance of Low Vertices......Page 213
22.5 The Other Half of the Result......Page 215
22.6 The Combinatorial Model......Page 216
22.7 The Even Case......Page 218
23.1 Statement 1......Page 220
23.2 The Cantor Set......Page 222
23.3 A Precursor of the Comet Theorem......Page 223
23.4 Convergence of the Fundamental Orbit......Page 224
23.5 An Estimate for the Return Map......Page 225
23.6 Proof of the Comet Precursor Theorem......Page 226
23.7 The Double Identity......Page 228
23.8 Statement 4......Page 231
24.1 Minimality......Page 234
24.2 Tree Interpretation of the Dynamics......Page 235
24.3 Proper Return Models and Cusped Solenoids......Page 236
24.4 Some Other Equivalence Relations......Page 240
25.1 Periodic Orbits......Page 242
25.2 A Triangle Group......Page 243
25.3 Modularity......Page 244
25.4 Hausdorff Dimension......Page 245
25.5 Quadratic Irrational Parameters......Page 246
25.6 The Dimension Function......Page 249
PART 6. MORE STRUCTURE THEOREMS......Page 252
26.1 A Formula for the Pivot Points......Page 254
26.2 A Detail from Part 5......Page 256
26.3 Preliminaries......Page 257
26.4 The Good Parameter Lemma......Page 258
26.5 The End of the Proof......Page 262
27.1 Main Results......Page 264
27.2 Another Diophantine Lemma......Page 267
27.3 Copying the Pivot Arc......Page 268
27.4 Proof of the Structure Lemma......Page 269
27.6 An Even Version of the Copy Theorem......Page 272
28.1 An Exceptional Case......Page 274
28.2 Discussion of the Proof......Page 275
28.3 Confining the Bump......Page 278
28.4 A Topological Property of Pivot Arcs......Page 279
28.5 Corollaries of the Barrier Theorem......Page 280
28.6 The Minor Components......Page 281
28.7 The Middle Major Components......Page 283
28.8 Even Implies Odd......Page 284
28.9 Even Implies Even......Page 286
29.1 Inheritance of Pivot Arcs......Page 288
29.2 Freezing Numbers......Page 290
29.3 The End of the Proof......Page 291
29.4 A Useful Result......Page 293
30.1 The Main Result......Page 294
30.2 Traps......Page 295
30.3 Cases 1 and 2......Page 297
30.4 Cases 3 and 4......Page 300
31.1 Overview......Page 302
31.2 A Makeshift Result......Page 303
31.3 Eliminating Minor Arcs......Page 305
31.4 A Topological Lemma......Page 306
31.5 The End of the Proof......Page 307
A.1 Structure of Periodic Points......Page 310
A.2 Self-Similarity......Page 312
A.3 General Orbits on Kites......Page 313
A.4 General Quadrilaterals......Page 315
Bibliography......Page 318
P......Page 320
W......Page 321
