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目次

Preface xi Chapter 1. Introduction 1 1.1 Definitions and History 11.2 The Erratic Orbits Theorem 3 1.3 Corollaries of the Comet Theorem 4 1.4The Comet Theorem 7 1.5 Rational Kites 10 1.6 The Arithmetic Graph 12 1.7The Master Picture Theorem 15 1.8 Remarks on Computation 16 1.9Organization of the Book 16 PART 1. THE ERRATIC ORBITS THEOREM 17Chapter 2. The Arithmetic Graph 19 2.1 Polygonal Outer Billiards 19 2.2Special Orbits 20 2.3 The Return Lemma 21 2.4 The Return Map 25 2.5 TheArithmetic Graph 26 2.6 Low Vertices and Parity 28 2.7 HausdorffConvergence 30 Chapter 3. The Hexagrid Theorem 33 3.1 The Arithmetic Kite33 3.2 The Hexagrid Theorem 35 3.3 The Room Lemma 37 3.4 Orbit Excursions38 Chapter 4. Period Copying 41 4.1 Inferior and Superior Sequences 41 4.2Strong Sequences 43 Chapter 5. Proof of the Erratic Orbits Theorem 45 5.1Proof of Statement 1 45 5.2 Proof of Statement 2 49 5.3 Proof of Statement3 50 PART 2. THE MASTER PICTURE THEOREM 53 Chapter 6. The MasterPicture Theorem 55 6.1 Coarse Formulation 55 6.2 The Walls of thePartitions 56 6.3 The Partitions 57 6.4 A Typical Example 59 6.5 ASingular Example 60 6.6 The Reduction Algorithm 62 6.7 The IntegralStructure 63 6.8 Calculating with the Polytopes 65 6.9 Computing thePartition 66 Chapter 7. The Pinwheel Lemma 69 7.1 The Main Result 69 7.2Discussion 71 7.3 Far from the Kite 72 7.4 No Sharps or Flats 73 7.5Dealing 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 788.3 Pairs of Strips 79 8.4 Single-Parameter Proof 81 8.5 Proof in theGeneral Case 83 Chapter 9. The Strip Functions 85 9.1 The Main Result 859.2 Continuous Extension 86 9.3 Local Affine Structure 87 9.4 IrrationalQuintuples 89 9.5 Verification 90 9.6 An Example Calculation 91 Chapter10. Proof of the Master Picture Theorem 93 10.1 The Main Argument 93 10.2The First Four Singular Sets 94 10.3 Symmetry 95 10.4 The Remaining Pieces96 10.5 Proof of the Second Statement 97 PART 3. ARITHMETIC GRAPHSTRUCTURE THEOREMS 99 Chapter 11. Proof of the Embedding Theorem 101 11.1No Valence 1 Vertices 101 11.2 No Crossings 104 Chapter 12. Extension andSymmetry 107 12.1 Translational Symmetry 107 12.2 A Converse Result 11012.3 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 Case118 13.3 Planes and Strips 119 13.4 The End of the Proof 120 13.5 A VisualTour 121 Chapter 14. The Barrier Theorem 125 14.1 The Result 125 14.2 TheImage of the Barrier Line 127 14.3 An Example 129 14.4 Bounding the NewCrossings 130 14.5 The Other Case 132 Chapter 15. Proof of Hexagrid TheoremII 133 15.1 The Structure of the Doors 133 15.2 Ordinary Crossing Cells 13515.3 New Maps 136 15.4 Intersection Results 138 15.5 The End of the Proof141 15.6 The Pattern of Crossing Cells 142 Chapter 16. Proof of theIntersection Lemma 143 16.1 Discussion of the Proof 143 16.2 CoveringParallelograms 144 16.3 Proof of Statement 1 146 16.4 Proof of Statement 2148 16.5 Proof of Statement 3 149 PART 4. PERIOD-COPYING THEOREMS 151Chapter 17. Diophantine Approximation 153 17.1 Existence of the InferiorSequence 153 17.2 Structure of the Inferior Sequence 155 17.3 Existence ofthe Superior Sequence 158 17.4 The Diophantine Constant 159 17.5 AStructural Result 161 Chapter 18. The Diophantine Lemma 163 18.1 ThreeLinear Functionals 163 18.2 The Main Result 164 18.3 A Quick Application165 18.4 Proof of the Diophantine Lemma 166 18.5 Proof of the AgreementLemma 167 18.6 Proof of the Good Integer Lemma 169 Chapter 19. TheDecomposition Theorem 171 19.1 The Main Result 171 19.2 A Comparison 17319.3 A Crossing Lemma 174 19.4 Most of the Parameters 175 19.5 TheExceptional Cases 178 Chapter 20. Existence of Strong Sequences 181 20.1Step 1 181 20.2 Step 2 182 20.3 Step 3 183 PART 5. THE COMET THEOREM185 Chapter 21. Structure of the Inferior and Superior Sequences 187 21.1The Results 187 21.2 The Growth of Denominators 188 21.3 The Identities 189Chapter 22. The Fundamental Orbit 193 22.1 Main Results 193 22.2 The Copyand Pivot Theorems 195 22.3 Half of the Result 197 22.4 The Inheritance ofLow Vertices 198 22.5 The Other Half of the Result 200 22.6 TheCombinatorial Model 201 22.7 The Even Case 203 Chapter 23. The CometTheorem 205 23.1 Statement 1 205 23.2 The Cantor Set 207 23.3 A Precursorof the Comet Theorem 208 23.4 Convergence of the Fundamental Orbit 209 23.5An Estimate for the Return Map 210 23.6 Proof of the Comet Precursor Theorem211 23.7 The Double Identity 213 23.8 Statement 4 216 Chapter 24.Dynamical Consequences 219 24.1 Minimality 219 24.2 Tree Interpretation ofthe Dynamics 220 24.3 Proper Return Models and Cusped Solenoids 221 24.4Some Other Equivalence Relations 225 Chapter 25. Geometric Consequences 22725.1 Periodic Orbits 227 25.2 A Triangle Group 228 25.3 Modularity 22925.4 Hausdorff Dimension 230 25.5 Quadratic Irrational Parameters 231 25.6The Dimension Function 234 PART 6. MORE STRUCTURE THEOREMS 237 Chapter26. Proof of the Copy Theorem 239 26.1 A Formula for the Pivot Points 23926.2 A Detail from Part 5 241 26.3 Preliminaries 242 26.4 The GoodParameter Lemma 243 26.5 The End of the Proof 247 Chapter 27. Pivot Arcs inthe Even Case 249 27.1 Main Results 249 27.2 Another Diophantine Lemma 25227.3 Copying the Pivot Arc 253 27.4 Proof of the Structure Lemma 254 27.5The Decrement of a Pivot Arc 257 27.6 An Even Version of the Copy Theorem257 Chapter 28. Proof of the Pivot Theorem 259 28.1 An Exceptional Case 25928.2 Discussion of the Proof 260 28.3 Confining the Bump 263 28.4 ATopological Property of Pivot Arcs 264 28.5 Corollaries of the BarrierTheorem 265 28.6 The Minor Components 266 28.7 The Middle Major Components268 28.8 Even Implies Odd 269 28.9 Even Implies Even 271 Chapter 29. Proofof the Period Theorem 273 29.1 Inheritance of Pivot Arcs 273 29.2 FreezingNumbers 275 29.3 The End of the Proof 276 29.4 A Useful Result 278 Chapter30. Hovering Components 279 30.1 The Main Result 279 30.2 Traps 280 30.3Cases 1 and 2 282 30.4 Cases 3 and 4 285 Chapter 31. Proof of the LowVertex Theorem 287 31.1 Overview 287 31.2 A Makeshift Result 288 31.3Eliminating Minor Arcs 290 31.4 A Topological Lemma 291 31.5 The End of theProof 292 Appendix 295 A.1 Structure of Periodic Points 295 A.2Self-Similarity 297 A.3 General Orbits on Kites 298 A.4 GeneralQuadrilaterals 300 Bibliography 303 Index 305