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Outer billiards on kites
Author
Title
Outer billiards on kites / Richard Evan Schwartz.
ISBN
9780691142487 (hardcover : alk. paper)
0691142483 (hardcover : alk. paper)
9780691142494 (pbk. : alk. paper)
0691142491 (pbk. : alk. paper)
Published
Princeton : Princeton University Press, 2009.
Physical Description
xii, 306 p. : ill. ; 24 cm.
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 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."--Publisher website.
Format
Books
Language
English
Added to Catalog
November 16, 2009
Series
Annals of mathematics studies ; no. 171
Bibliography
Includes bibliographical references and index.
Contents
Introduction
The arithmetic graph
The hexagrid theorem
Period copying
Proof of the erratic orbits theorem
The master picture theorem
The pinwheel lemma
The torus lemma
The strip functions
Proof of the master picture theorem - Proof of the embedding theorem
Extension and symmetry
Proof of hexagrid theorem I
The barrier theorem
Proof of hexagrid theorem II
Proof of the intersection lemma
Diophantine approximation
The diophantine lemma
The decomposition theorem
Existence of strong sequences
Structure of the inferior and superior sequences
The comet theorem
Dynamical consequences
Geometric consequences
Proof of the copy theorem
Pivot arcs in the even case
Proof of the pivot theorem
Proof of the period theorem
Hovering components
Proof of the low vertex theorem
Structure of periodic points
Self-similarity
General orbits on kites
General quadrilaterals.
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