Outer Billiards on Kites

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 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