Richard Evan Schwartz is professor of mathematics at Brown University and the author of "Spherical CR Geometry and Dehn Surgery" (Princeton).

"This research is an exemplary sample of experimental mathematics."--Serge L. Tabachnikov, Mathematical Review

• 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

Richard Evan Schwartz is professor of mathematics at Brown University and the author of "Spherical CR Geometry and Dehn Surgery" (Princeton).

"This research is an exemplary sample of experimental mathematics."--Serge L. Tabachnikov, Mathematical Review

• 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

Richard Evan Schwartz is professor of mathematics at Brown University and the author of "Spherical CR Geometry and Dehn Surgery" (Princeton).

"This research is an exemplary sample of experimental mathematics."--Serge L. Tabachnikov, Mathematical Review

• 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