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Outer Billiards on Kites

R. E. Schwartz
Publisher: 
Princeton University Press
Publication Date: 
2009
Number of Pages: 
306
Format: 
Paperback
Series: 
Annals of Mathematics Studies 171
Price: 
45.00
ISBN: 
9780691142494
Category: 
Monograph
[Reviewed by
Fernando Q. Gouvêa
, on
05/19/2010
]

Mathematical billiards are dynamical systems in which a point bounces around the inside of a convex shape on the plane (or, more generally, on some surface). Outer billiards is the dual situation. Choose a convex shape K on the plane (say, a convex polygon). Choose a starting point x0 outside the shape. Draw a ray from x0 to x1 such that

  • the line segment x0x1 is tangent to K and the tangency point is the midpoint of the segment, and
  • moving from x0 to x1 the shape K is on the right.

Iterating this process gives a (affinely invariant) dynamical system. The basic questions are the expected ones: one wants to know whether orbits are periodic, whether they are bounded, etc.

The idea of outer billiards seems to be due to B. H. Neumann. They were popularized in the 1970s by J. Moser, who related them to celestial mechanics. This research monograph deals with outer billiards in which the shape K is a kite, i.e., a convex quadrilateral having a diagonal which is a line of symmetry. The main result is that if the ratio of the diagonals in the kite is irrational, then there are uncountably many orbits that are unbounded (both forward and backward).

The author clearly has a knack for inventing names (kites, erratic orbits, the comet theorem, the hexagrid theorem, the room lemma…), which helps make the book a pleasant read. There are lots of pictures. The exposition is a bit breathless, at times seeming to rush forward in its enthusiasm without filling in all the details. That is to be expected, I guess, from a book in the Annals of Mathematics Studies series.

Most of the results, the author tells us, were first discovered experimentally, using a program called Billiard King, available from the author's web site. Also on the site is an interactive guide to the book that uses Java applets to explain what is going on. The online materials make the book much more accessible than it might otherwise have been.

Not many books combine serious mathematics, neat software, and an element of play. This one does. It would be a great source for independent projects with students.


Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.

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

Dummy View - NOT TO BE DELETED