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
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 6b ........................................... 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 ......................................... 174
19.4 Most of the Parameters ................................... 175
19.5 The Exceptional Cases .................................... 178
Chapter 20. Existence of Strong Sequences ..................... 181
20.1 Step 1 ................................................... 181
20.2 Step 2 ................................................... 182
20.3 Step 3 ................................................... 183
PART 5. THE COMET THEOREM ..................................... 185
Chapter 21. Structure of the Inferior and Superior
Sequences ......................................... 187
21.1 The Results .............................................. 187
21.2 The Growth of Denominators ............................... 188
21.3 The Identities ........................................... 189
Chapter 22. The Fundamental Orbit ............................. 193
22.1 Main Results ............................................. 193
22.2 The Copy and Pivot Theorems .............................. 195
22.3 Half of the Result ....................................... 197
22.4 The Inheritance of Low Vertices .......................... 198
22.5 The Other Half of the Result ............................. 200
22.6 The Combinatorial Model .................................. 201
22.7 The Even Case ............................................ 203
Chapter 23. The Comet Theorem ................................. 205
23.1 Statement 1 .............................................. 205
23.2 The Cantor Set ........................................... 207
23.3 A Precursor of the Comet Theorem ......................... 208
23.4 Convergence of the Fundamental Orbit ..................... 209
23.5 An Estimate for the Return Map ........................... 210
23.6 Proof of the Comet Precursor Theorem ..................... 211
23.7 The Double Identity ...................................... 213
23.8 Statement 4 .............................................. 216
Chapter 24. Dynamical Consequences ............................ 219
24.1 Minimality ............................................... 219
24.2 Tree Interpretation of the Dynamics ...................... 220
24.3 Proper Return Models and Cusped Solenoids ................ 221
24.4 Some Other Equivalence Relations ......................... 225
Chapter 25. Geometric Consequences ............................ 227
25.1 Periodic Orbits .......................................... 227
25.2 A Triangle Group ......................................... 228
25.3 Modularity ............................................... 229
25.4 Hausdorff Dimension ...................................... 230
25.5 Quadratic Irrational Parameters .......................... 231
25.6 The Dimension Function ................................... 234
PART 6. MORE STRUCTURE THEOREMS .............................. 237
Chapter 26. Proof of the Copy Theorem ......................... 239
26.1 A Formula for the Pivot Points ........................... 239
26.2 A Detail from Part 5 ..................................... 241
26.3 Preliminaries ............................................ 242
26.4 The Good Parameter Lemma ................................. 243
26.5 The End of the Proof ..................................... 247
Chapter 27. Pivot Arcs in the Even Case ....................... 249
27.1 Main Results ............................................. 249
27.2 Another Diophantine Lemma ................................ 252
27.3 Copying the Pivot Arc .................................... 253
27.4 Proof of the Structure Lemma ............................. 254
27.5 The Decrement of a Pivot Arc ............................. 257
27.6 An Even Version of the Copy Theorem ...................... 257
Chapter 28. Proof of the Pivot Theorem ........................ 259
28.1 An Exceptional Case ...................................... 259
28.2 Discussion of the Proof .................................. 260
28.3 Confining the Bump ....................................... 263
28.4 A Topological Property of Pivot Arcs ..................... 264
28.5 Corollaries of the Barrier Theorem ....................... 265
28.6 The Minor Components ..................................... 266
28.7 The Middle Major Components .............................. 268
28.8 Even Implies Odd ......................................... 269
28.9 Even Implies Even ........................................ 271
Chapter 29. Proof of the Period Theorem ....................... 273
29.1 Inheritance of Pivot Arcs ................................ 273
29.2 Freezing Numbers ......................................... 275
29.3 The End of the Proof ..................................... 276
29.4 A Useful Result .......................................... 278
Chapter 30. Hovering Components ............................... 279
30.1 The Main Result .......................................... 279
30.2 Traps .................................................... 280
30.3 Cases 1 and 2 ............................................ 282
30.4 Cases 3 and 4 ............................................ 285
Chapter 31. Proof of the Low Vertex Theorem ................... 287
31.1 Overview ................................................. 287
31.2 A Makeshift Result ....................................... 288
31.3 Eliminating Minor Arcs ................................... 290
31.4 A Topological Lemma ...................................... 291
31.5 The End of the Proof ..................................... 292
Appendix ...................................................... 295
A.1 Structure of Periodic Points ............................. 295
A.2 Self-Similarity .......................................... 297
A.3 General Orbits on Kites .................................. 298
A.4 General Quadrilaterals ................................... 300
Bibliography .................................................. 303
Index ......................................................... 305
| 
