The octogonal PETs / Richard Evan Schwartz.

Format:

Book

Author/Creator:

Schwartz, Richard Evan, author.

Series:

Mathematical surveys and monographs ; Volume 197.

Mathematical surveys and monographs ; Volume 197

Language:

English

Subjects (All):

Geometry.

Polytopes.

Physical Description:

1 online resource (x, 212 pages) : illustrations.

Edition:

1st ed.

Other Title:

Octogonal polytope exchange transformations

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2014]

Language Note:

English

Summary:

A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is a translation wherever it is defined. The 1-dimensional examples, interval exchange transformations, have been studied fruitfully for many years and have deep connections to other areas of mathematics, such as Teichm�ller theory. This book introduces a general method for constructing polytope exchange transformations in higher dimensions and then studies the simplest example of the construction in detail. The simplest case is a 1-parameter family of polygon exchange transformations that turns out to be closely related to outer billiards on semi-regular octagons. The 1-parameter family admits a complete renormalization scheme, and this structure allows for a fairly complete analysis both of the system and of outer billiards on semi-regular octagons. The material in this book was discovered through computer experimentation. On the other hand, the proofs are traditional, except for a few rigorous computer-assisted calculations.

Contents:

Cover

Title page

Contents

Preface

Chapter 1. Introduction

Chapter 2. Background

Chapter 3. Multigraph PETs

Chapter 4. The alternating grid system

Chapter 5. Outer billiards on semiregular octagons

Chapter 6. Quarter turn compositions

Chapter 7. Elementary properties

Chapter 8. Orbit stability and combinatorics

Chapter 9. Bilateral symmetry

Chapter 10. Proof of the main theorem

Chapter 11. The renormalization map

Chapter 12. Properties of the tiling

Chapter 13. The filling lemma

Chapter 14. The covering lemma

Chapter 15. Further geometric results

Chapter 16. Properties of the limit set

Chapter 17. Hausdorff convergence

Chapter 18. Recurrence relations

Chapter 19. Hausdorff dimension bounds

Chapter 20. Controlling the limit set

Chapter 21. The arc case

Chapter 22. Further symmetries of the tiling

Chapter 23. The forest case

Chapter 24. The Cantor set case

Chapter 25. Dynamics in the arc case

Chapter 26. Computational methods

Chapter 27. The calculations

Chapter 28. The raw data

Bibliography

Back Cover.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-1718-9

OCLC:

907370751