Cover

Title

Copyright

Contents

Preface

Chapter 1. Introduction

1.1 Definitions and History

1.2 The Erratic Orbits Theorem

1.3 Corollaries of the Comet Theorem

1.4 The Comet Theorem

1.5 Rational Kites

1.6 The Arithmetic Graph

1.7 The Master Picture Theorem

1.8 Remarks on Computation

1.9 Organization of the Book

PART 1. THE ERRATIC ORBITS THEOREM

Chapter 2. The Arithmetic Graph

2.1 Polygonal Outer Billiards

2.2 Special Orbits

2.3 The Return Lemma

2.4 The Return Map

2.5 The Arithmetic Graph

2.6 Low Vertices and Parity

2.7 Hausdorff Convergence

Chapter 3. The Hexagrid Theorem

3.1 The Arithmetic Kite

3.2 The Hexagrid Theorem

3.3 The Room Lemma

3.4 Orbit Excursions

Chapter 4. Period Copying

4.1 Inferior and Superior Sequences

4.2 Strong Sequences

Chapter 5. Proof of the Erratic Orbits Theorem

5.1 Proof of Statement 1

5.2 Proof of Statement 2

5.3 Proof of Statement 3

PART 2. THE MASTER PICTURE THEOREM

Chapter 6. The Master Picture Theorem

6.1 Coarse Formulation

6.2 The Walls of the Partitions

6.3 The Partitions

6.4 A Typical Example

6.5 A Singular Example

6.6 The Reduction Algorithm

6.7 The Integral Structure

6.8 Calculating with the Polytopes

6.9 Computing the Partition

Chapter 7. The Pinwheel Lemma

7.1 The Main Result

7.2 Discussion

7.3 Far from the Kite

7.4 No Sharps or Flats

7.5 Dealing with 4[sup(#)]

7.6 Dealing with 6[sup(b)]

7.7 The Last Cases

Chapter 8. The Torus Lemma

8.1 The Main Result

8.2 Input from the Torus Map

8.3 Pairs of Strips

8.4 Single-Parameter Proof

8.5 Proof in the General Case

Chapter 9. The Strip Functions

9.1 The Main Result

9.2 Continuous Extension

9.3 Local Affine Structure

9.4 Irrational Quintuples

9.5 Verification

9.6 An Example Calculation

Chapter 10. Proof of the Master Picture Theorem

10.1 The Main Argument

10.2 The First Four Singular Sets

10.3 Symmetry

10.4 The Remaining Pieces

10.5 Proof of the Second Statement

PART 3. ARITHMETIC GRAPH STRUCTURE THEOREMS

Chapter 11. Proof of the Embedding Theorem

11.1 No Valence 1 Vertices

11.2 No Crossings

Chapter 12. Extension and Symmetry

12.1 Translational Symmetry

12.2 A Converse Result

12.3 Rotational Symmetry

12.4 Near-Bilateral Symmetry

Chapter 13. Proof of Hexagrid Theorem I

13.1 The Key Result

13.2 A Special Case

13.3 Planes and Strips

13.4 The End of the Proof

13.5 A Visual Tour

Chapter 14. The Barrier Theorem

14.1 The Result

14.2 The Image of the Barrier Line

14.3 An Example

14.4 Bounding the New Crossings

14.5 The Other Case

Chapter 15. Proof of Hexagrid Theorem II

15.1 The Structure of the Doors

15.2 Ordinary Crossing Cells

15.3 New Maps

15.4 Intersection Results

15.5 The End of the Proof

15.6 The Pattern of Crossing Cells

Chapter 16. Proof of the Intersection Lemma

16.1 Discussion of the Proof

16.2 Covering Parallelograms

16.3 Proof of Statement 1

16.4 Proof of Statement 2

16.5 Proof of Statement 3

PART 4. PERIOD-COPYING THEOREMS

Chapter 17. Diophantine Approximation

17.1 Existence of the Inferior Sequence

17.2 Structure of the Inferior Sequence

17.3 Existence of the Superior Sequence

17.4 The Diophantine Constant

17.5 A Structural Result

Chapter 18. The Diophantine Lemma

18.1 Three Linear Functionals

18.2 The Main Result

18.3 A Quick Application

18.4 Proof of the Diophantine Lemma

18.5 Proof of the Agreement Lemma

18.6 Proof of the Good Integer Lemma

Chapter 19. The Decomposition Theorem

19.1 The Main Result

19.2 A Comparison

19.3 A Crossing Lemma

19.4 Most of the Parameters

19.5 The Exceptional Cases

Chapter 20. Existence of Strong Sequences

20.1 Step 1

20.2 Step 2

20.3 Step 3

PART 5. THE COMET THEOREM

Chapter 21. Structure of the Inferior and Superior Sequences

21.1 The Results

21.2 The Growth of Denominators

21.3 The Identities

Chapter 22. The Fundamental Orbit

22.1 Main Results

22.2 The Copy and Pivot Theorems

22.3 Half of the Result

22.4 The Inheritance of Low Vertices

22.5 The Other Half of the Result

22.6 The Combinatorial Model

22.7 The Even Case

Chapter 23. The Comet Theorem

23.1 Statement 1

23.2 The Cantor Set

23.3 A Precursor of the Comet Theorem

23.4 Convergence of the Fundamental Orbit

23.5 An Estimate for the Return Map

23.6 Proof of the Comet Precursor Theorem

23.7 The Double Identity

23.8 Statement 4

Chapter 24. Dynamical Consequences

24.1 Minimality

24.2 Tree Interpretation of the Dynamics

24.3 Proper Return Models and Cusped Solenoids

24.4 Some Other Equivalence Relations

Chapter 25. Geometric Consequences

25.1 Periodic Orbits

25.2 A Triangle Group

25.3 Modularity

25.4 Hausdorff Dimension

25.5 Quadratic Irrational Parameters

25.6 The Dimension Function

PART 6. MORE STRUCTURE THEOREMS

Chapter 26. Proof of the Copy Theorem

26.1 A Formula for the Pivot Points

26.2 A Detail from Part 5

26.3 Preliminaries

26.4 The Good Parameter Lemma

26.5 The End of the Proof

Chapter 27. Pivot Arcs in the Even Case

27.1 Main Results

27.2 Another Diophantine Lemma

27.3 Copying the Pivot Arc

27.4 Proof of the Structure Lemma

27.5 The Decrement of a Pivot Arc

27.6 An Even Version of the Copy Theorem

Chapter 28. Proof of the Pivot Theorem

28.1 An Exceptional Case

28.2 Discussion of the Proof

28.3 Confining the Bump

28.4 A Topological Property of Pivot Arcs

28.5 Corollaries of the Barrier Theorem

28.6 The Minor Components

28.7 The Middle Major Components

28.8 Even Implies Odd

28.9 Even Implies Even

Chapter 29. Proof of the Period Theorem

29.1 Inheritance of Pivot Arcs

29.2 Freezing Numbers

29.3 The End of the Proof

29.4 A Useful Result

Chapter 30. Hovering Components

30.1 The Main Result

30.2 Traps

30.3 Cases 1 and 2

30.4 Cases 3 and 4

Chapter 31. Proof of the Low Vertex Theorem

31.1 Overview

31.2 A Makeshift Result

31.3 Eliminating Minor Arcs

31.4 A Topological Lemma

31.5 The End of the Proof

Appendix

A.1 Structure of Periodic Points

A.2 Self-Similarity

A.3 General Orbits on Kites

A.4 General Quadrilaterals

Bibliography

Index

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