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Hardcover ISBN:  9781470415228 
Product Code:  SURV/197 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470417185 
Product Code:  SURV/197.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470415228 
eBook ISBN:  9781470417185 
Product Code:  SURV/197.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 197; 2014; 212 ppMSC: Primary 37
A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is a translation wherever it is defined. The 1dimensional examples, interval exchange transformations, have been studied fruitfully for many years and have deep connections to other areas of mathematics, such as Teichmüller 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 1parameter family of polygon exchange transformations that turns out to be closely related to outer billiards on semiregular octagons. The 1parameter family admits a complete renormalization scheme, and this structure allows for a fairly complete analysis both of the system and of outer billiards on semiregular octagons. The material in this book was discovered through computer experimentation. On the other hand, the proofs are traditional, except for a few rigorous computerassisted calculations.
ReadershipGraduate students and research mathematicians interested in dynamical systems.

Table of Contents

Chapters

Chapter 1. Introduction

Part 1. Friends of the octagonal PETs

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

Part 2. Renormalization and symmetry

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

Part 3. Metric properties

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

Part 4. Topological properties

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

Part 5. Computational details

Chapter 26. Computational methods

Chapter 27. The calculations

Chapter 28. The raw data


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A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is a translation wherever it is defined. The 1dimensional examples, interval exchange transformations, have been studied fruitfully for many years and have deep connections to other areas of mathematics, such as Teichmüller 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 1parameter family of polygon exchange transformations that turns out to be closely related to outer billiards on semiregular octagons. The 1parameter family admits a complete renormalization scheme, and this structure allows for a fairly complete analysis both of the system and of outer billiards on semiregular octagons. The material in this book was discovered through computer experimentation. On the other hand, the proofs are traditional, except for a few rigorous computerassisted calculations.
Graduate students and research mathematicians interested in dynamical systems.

Chapters

Chapter 1. Introduction

Part 1. Friends of the octagonal PETs

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

Part 2. Renormalization and symmetry

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

Part 3. Metric properties

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

Part 4. Topological properties

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

Part 5. Computational details

Chapter 26. Computational methods

Chapter 27. The calculations

Chapter 28. The raw data