Outer billiards on kites / Richard Evan Schwartz.

Format:

Book

Author/Creator:

Schwartz, Richard Evan.

Series:

Annals of mathematics studies ; 171.

Annals of mathematics studies ; 171

Language:

English

Subjects (All):

Hyperbolic spaces.

Singularities (Mathematics).

Transformations (Mathematics).

Geometry, Plane.

Physical Description:

1 online resource (321 p.)

Edition:

Course Book

Place of Publication:

Princeton, NJ : Princeton University Press, c2009.

Language Note:

English

Summary:

Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950's, 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.

Contents:

Front matter

Contents

Preface

Chapter 1. Introduction

Part 1. The Erratic Orbits Theorem

Chapter 2. The Arithmetic Graph

Chapter 3. The Hexagrid Theorem

Chapter 4. Period Copying

Chapter 5. Proof of the Erratic Orbits Theorem

Part 2. The Master Picture Theorem

Chapter 6. The Master Picture Theorem

Chapter 7. The Pinwheel Lemma

Chapter 8. The Torus Lemma

Chapter 9. The Strip Functions

Chapter 10. Proof of the Master Picture Theorem

Part 3. Arithmetic Graph Structure Theorems

Chapter 11. Proof of the Embedding Theorem

Chapter 12. Extension and Symmetry

Chapter 13. Proof of Hexagrid Theorem I

Chapter 14. The Barrier Theorem

Chapter 15. Proof of Hexagrid Theorem II

Chapter 16. Proof of the Intersection Lemma

Part 4. Period-Copying Theorems

Chapter 17. Diophantine Approximation

Chapter 18. The Diophantine Lemma

Chapter 19. The Decomposition Theorem

Chapter 20. Existence of Strong Sequences

Part 5. The Comet Theorem

Chapter 21. Structure of the Inferior and Superior Sequences

Chapter 22. The Fundamental Orbit

Chapter 23. The Comet Theorem

Chapter 24. Dynamical Consequences

Chapter 25. Geometric Consequences

Part 6. More Structure Theorems

Chapter 26. Proof of the Copy Theorem

Chapter 27. Pivot Arcs in the Even Case

Chapter 28. Proof of the Pivot Theorem

Chapter 29. Proof of the Period Theorem

Chapter 30. Hovering Components

Chapter 31. Proof of the Low Vertex Theorem

Appendix

Bibliography

Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786612458583

9781282458581

1282458582

9781400831975

1400831970

OCLC:

592756158