Outer Billiards on Kites (AM-171)

Outer Billiards on Kites (AM-171)

by Richard Evan Schwartz
     
 

ISBN-10: 0691142483

ISBN-13: 9780691142487

Pub. Date: 10/05/2009

Publisher: Princeton University Press

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

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Overview

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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Product Details

ISBN-13:
9780691142487
Publisher:
Princeton University Press
Publication date:
10/05/2009
Series:
Annals of Mathematics Studies Series
Pages:
312
Product dimensions:
6.40(w) x 9.50(h) x 0.90(d)

Table of Contents

Contents

Preface....................xi
Chapter 1. Introduction....................1
PART 1. THE ERRATIC ORBITS THEOREM....................17
Chapter 2. The Arithmetic Graph....................19
Chapter 3. The Hexagrid Theorem....................33
Chapter 4. Period Copying....................41
Chapter 5. Proof of the Erratic Orbits Theorem....................45
PART 2. THE MASTER PICTURE THEOREM....................53
Chapter 6. The Master Picture Theorem....................55
Chapter 7. The Pinwheel Lemma....................69
Chapter 8. The Torus Lemma....................77
Chapter 9. The Strip Functions....................85
Chapter 10. Proof of the Master Picture Theorem....................93
PART 3. ARITHMETIC GRAPH STRUCTURE THEOREMS....................99
Chapter 11. Proof of the Embedding Theorem....................101
Chapter 12. Extension and Symmetry....................107
Chapter 13. Proof of Hexagrid Theorem I....................117
Chapter 14. The Barrier Theorem....................125
Chapter 15. Proof of Hexagrid Theorem II....................133
Chapter 16. Proof of the Intersection Lemma....................143
PART 4. PERIOD-COPYING THEOREMS....................151
Chapter 17. Diophantine Approximation....................153
Chapter 18. The Diophantine Lemma....................163
Chapter 19. The Decomposition Theorem....................171
Chapter 20. Existence of Strong Sequences....................181
PART 5. THE COMET THEOREM....................185
Chapter 21. Structure of the Inferior and Superior Sequences....................187
Chapter 22. The Fundamental Orbit....................193
Chapter 23. The Comet Theorem....................205
Chapter 24. Dynamical Consequences....................219
Chapter 25. Geometric Consequences....................227
PART 6. MORE STRUCTURE THEOREMS....................237
Chapter 26. Proof of the Copy Theorem....................239
Chapter 27. Pivot Arcs in the Even Case....................249
Chapter 28. Proof of the Pivot Theorem....................259
Chapter 29. Proof of the Period Theorem....................273
Chapter 30. Hovering Components....................279
Chapter 31. Proof of the Low Vertex Theorem....................287
Appendix....................295
Bibliography....................303
Index....................305

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