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Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950s, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids—connections that together allow for a fairly complete analysis of the dynamical system.

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Mathematical Review
This research is an exemplary sample of experimental mathematics.
— Serge L. Tabachnikov
Mathematical Review - Serge L. Tabachnikov
This research is an exemplary sample of experimental mathematics.
From the Publisher
"This research is an exemplary sample of experimental mathematics."—Serge L. Tabachnikov, Mathematical Review
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Product Details

  • ISBN-13: 9780691142494
  • Publisher: Princeton University Press
  • Publication date: 10/5/2009
  • Series: Annals of Mathematics Studies Series
  • Pages: 312
  • Product dimensions: 6.10 (w) x 9.20 (h) x 0.80 (d)

Meet the Author

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

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Outer Billiards on Kites

By Richard Evan Schwartz


Copyright © 2009 Princeton University Press
All right reserved.

ISBN: 978-0-691-14249-4

Chapter One



B. H. Neumann [N] introduced outer billiards in the late 1950s. In the 1970s, J. Moser [M1] popularized outer billiards as a toy model for celestial mechanics. See [T1], [T3], and [DT1] for expositions of outer billiards and many references on the subject.

Outer billiards is a dynamical system defined (typically) in the Euclidean plane. Unlike the more familiar variant, which is simply called billiards, outer billiards involves a discrete sequence of moves outside a convex shape rather than inside it. To define an outer billiards system, one starts with a bounded convex set K [subset] [R.sup.2] and considers a point [x.sub.0] [member of] [R.sup.2] - K. One defines [x.sub.1] to be the point such that the segment [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is tangent to K at its midpoint and K lies to the right of the ray [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The iteration [x.sub.0] [right arrow] [x.sub.1] [right arrow] [x.sub.2] [right arrow] ... is called the forward outer billiards orbit of [x.sub.0]. It is defined for almost every point of [R.sup.2] - K. The backward orbit is defined similarly.

One important feature of outer billiards is that it is an affinely invariant system. Since affine transformations carry lines to lines and respect the property of bisection, an affine transformation carrying one shape to another conjugates the one outer billiards system to the other.

It is worth recalling here a few basic definitions about orbits. An orbit is called periodic if it eventually repeats itself, and otherwise aperiodic. An orbit is called bounded if the whole orbit lies in a bounded portion of the plane. Otherwise, the orbit is called unbounded. Sometimes (un)bounded orbits are called (un)stable.

J. Moser [M2, p. 11] attributes the following question to Neumann ca. 1960, though it is sometimes called Moser's question. Is there an outer billiards system with an unbounded orbit? This is an idealized version of the question about the stability of the solar system. Here is a chronological list of much of the work related to this question.

J. Moser [M2] sketches a proof, inspired by KAM theory, that outer billiards on K has all bounded orbits provided that [partial derivative]K is at least [ITLITL.sup.6] smooth and positively curved. R. Douady gives a complete proof in his thesis [D].

In Vivaldi-Shaidenko [VS], Kolodziej [Ko], and Gutkin-Simanyi [GS], it is proved (each with different methods) that outer billiards on a quasirational polygon has all orbits bounded. This class of polygons includes rational polygons - i.e., polygons with rational-coordinate vertices - and also regular polygons. In the rational case, all defined orbits are periodic.

S. Tabachnikov [T3] analyzes the outer billiards system for a regular pentagon and shows that there are some nonperiodic (but bounded) orbits.

P. Boyland [B] gives examples of [ITLITL.sup.1] smooth convex domains for which an orbit can contain the domain boundary in its [omega]-limit set.

F. Dogru and S. Tabachnikov [DT2] show that, for a certain class of polygons in the hyperbolic plane, called large, all outer billiards orbits are unbounded. (One can define outer billiards in the hyperbolic plane, though the dynamics has a somewhat different feel to it.)

D. Genin [G] shows that all orbits are bounded for the outer billiards systems associated to trapezoids. See A.4. Genin also makes a brief numerical study of a particular irrational kite based on the square root of 2, observes possibly unbounded orbits, and indeed conjectures that this is the case.

In [S] we prove that outer billiards on the Penrose kite has unbounded orbits, thereby answering the Moser-Neumann question in the affirmative. The Penrose kite is the convex quadrilateral that arises in the Penrose tiling.

Recently, D. Dolgopyat and B. Fayad [DF] showed that outer billiards on a half-disk has some unbounded orbits. Their proof also works for regions obtained from a disk by nearly cutting it in half with a straight line. This is a second affirmative answer to the Moser-Neumann question.

The result in [S] naturally raises questions about generalizations. The purpose of this book is to develop the theory of outer billiards on kites and show that the phenomenon of unbounded orbits for polygonal outer billiards is (at least for kites) quite robust.


A kite is a convex quadrilateral K having a diagonal that is a line of symmetry. We say that K is (ir)rational if the other diagonal divides K into two triangles whose areas are (ir)rational multiples of each other. Equivalently, K is rational iff it is affinely equivalent to a quadrilateral with rational vertices. To avoid trivialities, we require that exactly one of the two diagonals of K is a line of symmetry. This means that a rhombus does not count as a kite.

Since outer billiards is an affinely natural system, we find it useful to normalize kites in a particular way. Any kite is affinely equivalent to the quadrilateral K(A) having vertices

(-1, 0), (0, 1), (0,-1), (A, 0), A [member of] (0, 1). (1.1)

Figure 1.1 shows an example. The omitted case A = 1 corresponds to rhombuses. Henceforth, when we say kite, we mean K(A) for some A. The kite K(A) is (ir)rational iff A is (ir)rational.

Let [Z.sub.odd] denote the set of odd integers. Reflection in each vertex of K(A) preserves R x [Z.sub.odd]. Hence outer billiards on K(A) preserves R x [Z.sub.odd]. We call an outer billiards orbit on K(A) special if (and only if) it is contained in R x [Z.sub.odd]. We discuss only special orbits in this book. The special orbits are hard enough for us already. In the appendix, we will say something about the general case. See A.3.

We call an orbit forward erratic if the forward orbit is unbounded and also returns to every neighborhood of a kite vertex. We state the same definition for the backward direction. We call an orbit erratic if it is both forward and backward erratic. In Parts 1-4 of the book we will prove the following result.

Theorem 1.1 (Erratic Orbits) The following hold for any irrational kite.

1. There are uncountably many erratic special orbits.

2. Every special orbit is either periodic or unbounded in both directions.

3. The set of periodic special orbits is open dense in R x [Z.sub.odd].

It follows from the work on quasirational polygons cited above that all orbits are periodic relative to a rational kite. (The analysis in this book gives another proof of this fact, at least for special orbits. See the remark at the end of 3.2.) Hence the Erratic Orbits Theorem has the following corollary.

Corollary 1.2 Outer billiards on a kite has an unbounded orbit if and only if the kite is irrational.

The Erratic Orbits Theorem is an intermediate result included so that the reader can learn a substantial theorem without having to read the whole book. We will describe our main result in the next two sections.


In Parts 5 and 6 of the book we will go deeper into the subject and establish our main result, the Comet Theorem. The Comet Theorem and its corollaries considerably sharpen the Erratic Orbits Theorem. We defer statement of the Comet Theorem until the next section. In this section, we describe some of its corollaries.

Given a Cantor set ITLITL contained in a line L, we let [ITLITL.sup.#] be the set obtained from ITLITL by deleting the endpoints of the components of L - C. We call [ITLITL.sup.#] a trimmed Cantor set. Note that ITLITL - [ITLITL.sup.#] is countable.

The interval

I = [0, 2] x {-1} (1.2)

turns out to be a very useful interval. Figure 1.2 shows I and its first 3 iterates under the outer billiards map.

Let [U.sub.A] denote the set of unbounded special orbits relative to A.

Theorem 1.3 Relative to any irrational A [member of] (0, 1), the following are true.

1. [U.sub.A] is minimal: Every orbit in [U.sub.A] is dense in [U.sub.A] and all but at most 2 orbits in [U.sub.A] are both forward dense and backward dense in [U.sub.A].

2. [U.sub.A] is locally homogeneous: Every two points in [U.sub.A] have arbitrarily small neighborhoods that are isometric to each other.

3. [U.sub.A] [intersection] I = [ITLITL.sup.#.sub.A] for some Cantor set [ITLITL.sub.A].


(i) One endpoint of [ITLITL.sub.A] is the kite vertex (0,-1). Hence Statement 1 implies that all but at most 2 unbounded special orbits are erratic. The remaining special orbits, if any, are each erratic in one direction.

(ii) Statements 2 and 3 combine to say that every point in [U.sub.A] lies in an interval that intersects [U.sub.A] in a trimmed Cantor set. This gives us a good local picture of [U.sub.A]. One thing we are missing is a good global picture of [U.sub.A].

(iii) The Comet Theorem describes [ITLITL.sub.A] explicitly.

Given Theorem 1.3, it makes good sense to speak of the first return map to any interval in R x [Z.sub.odd]. From the minimality result, the local nature of the return map is essentially the same around any point of [U.sub.A]. To give a crisp picture of this first return map, we consider the interval I discussed above.

For j = 1, 2, let [f.sub.j] : [X.sub.j] [right arrow] [X.sub.j] be a map such that [f.sub.j] and [f.sup.-1.sub.j] are defined on all but perhaps a finite subset of [X.sub.j]. We call [f.sub.1] and [f.sub.2] essentially conjugate if there are countable sets [ITLITL.sub.j] [subset] [X.sub.j], each one contained in a finite union of orbits, and a homeomorphism

h: [X.sub.1] - [ITLITL.sub.1] [right arrow] [X.sub.2] - [ITLITL.sub.2]

that conjugates [f.sub.1] to [f.sub.2].

An odometer is the map x [right arrow] x + 1 on the inverse limit of the system


The universal odometer is the map x [right arrow] x + 1 on the profinite completion of Z. This is the inverse limit taken over the system of all finite cyclic groups. For concreteness, Equation 1.3 defines the universal odometer when [D.sub.k] = k factorial. See [H] for a detailed discussion of the universal odometer.

Theorem 1.4 Let [[rho].sub.A] be the first return map to [U.sub.A] [intersection] I.

1. For any irrational A [member of] (0, 1), the map [[rho].sub.A] is defined on all but at most one point and is essentially conjugate to an odometer [Z.sub.A].

2. Any given odometer is essentially conjugate to [[rho].sub.A] for uncountably many difference choices of A.

3. [[rho].sub.A] is essentially conjugate to the universal odometer for almost all A.


(i) The Comet Theorem explicitly describes [Z.sub.A] in terms of a sequence we call the remormalization sequence. This sequence is related to the continued fraction expansion of A. We will give a description of this sequence in the next section. (ii) Theorem 1.4 is part of a larger result. There is a certain suspension flow over the odometer, which we call geodesic flow on the cusped solenoid. It turns out that the time-one map for this flow serves as a good model, in a certain sense, for the dynamics on [U.sub.A]. 24.3.

Our next result highlights an unexpected connection between outer billiards on kites and the modular group [SL.sub.2](Z). The group [SL.sub.2](Z) acts naturally on the upper half-plane model of the hyperbolic plane, [H.sup.2], by linear fractional transformations. Closely related to [SL.sub.2](Z) is the (2, [infinity], [infinity])-triangle group [Gamma] generated by reflections in the sides of the geodesic triangle with vertices (0, 1, i). The points 0 and 1 are the cusps, and the point i is the internal vertex corresponding to the right angle of the triangle. See 25.2 for more details. [Gamma] and [SL.sub.2](Z) are commensurable: Their intersection has finite index in both groups. In our next result, we interpret our kite parameter interval (0, 1) as the subset of the ideal boundary of [H.sup.2].

Theorem 1.5 Let S = [0, 1] - Q. Let u(A) be the Hausdorff dimension of [U.sub.A].

1. For all A [member of] S, the set [U.sub.A] has length 0. Hence almost all points in R x [Z.sub.odd] have periodic orbits relative to outer billiards on K(A).

2. If A, A' [member of] S are in the same [Gamma]-orbit, then [U.sub.A] and [U.sub.A'] are locally similar. In particular, u(A) = u(A').

3. If A [member of] S is quadratic irrational, then every point of [U.sub.A] lies in an interval that intersects [U.sub.A] in a self-similar trimmed Cantor set.

4. The function u is almost everywhere equal to some constant u0 and yet maps every open subset of S onto [0, 1].


(i)We do not know the value of [u.sub.0]. We guess that 0 < [u.sub.0] < 1. Theorem 25.9 gives a formula for u(A) in many cases.

(ii) The word similar in statement 2 means that the two sets have neighborhoods that are related by a similarity. In statement 3, a self-similar set is a disjoint finite union of similar copies of itself.

(iii)We will see that statement 2 essentially implies both statements 3 and 4. Statement 2 is the first hint that outer billiards on kites is connected to themodular group. The Comet Theorem says more about this.

(iv) Statement 3 of Theorem 1.4 combines with statement 4 of Theorem 1.5 to say that there is a "typical behavior" for outer billiards on kites, in a certain sense. For almost every parameter A, the dimension of [U.sub.A] is the (unknown) constant [u.sub.0] and the return map [[rho].sub.A] is essentially conjugate to the universal odometer.

We end this section by comparing our results here with the main theorems in [S] concerning the Penrose kite. The Penrose kite parameter is

A = [square root of 5] - 2 = [[phi].sup.-3],

where [phi] is the golden ratio. In [S], we prove that [ITLITL.sup.#.sub.A] [subset] [U.sub.A] and that the first return map to [ITLITL.sup.#.sub.A] is essentially conjugate to the 2-adic odometer. Theorems 1.3 and 1.4 subsume these results about the Penrose kite.


Excerpted from Outer Billiards on Kites by Richard Evan Schwartz Copyright © 2009 by Princeton University Press. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

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

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

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

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

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

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

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

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