Outer Billiards on Kites 1st Edition by Richard Evan Schwartz 1400831970 9781400831975

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ISBN 10: 1400831970 

ISBN 13: 9781400831975

Author: Richard Evan Schwartz

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.

Table of contents:

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

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Tags: Richard Evan Schwartz, Outer, Billiards, Kites