**Mathematical Surveys and Monographs**

Volume: 197;
2014;
212 pp;
Hardcover

MSC: Primary 37;

Print ISBN: 978-1-4704-1522-8

Product Code: SURV/197

List Price: $90.00

Individual Member Price: $72.00

**Electronic ISBN: 978-1-4704-1718-5
Product Code: SURV/197.E**

List Price: $90.00

Individual Member Price: $72.00

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#### Supplemental Materials

# The Octagonal PETs

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*Richard Evan Schwartz*

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 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 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.

#### Table of Contents

# Table of Contents

## The Octagonal PETs

- Cover Cover11 free
- Title page i2 free
- Contents iii4 free
- Preface ix10 free
- Chapter 1. Introduction 112 free
- Chapter 2. Background 1728 free
- Chapter 3. Multigraph PETs 2738
- Chapter 4. The alternating grid system 3344
- Chapter 5. Outer billiards on semiregular octagons 4354
- Chapter 6. Quarter turn compositions 5566
- Chapter 7. Elementary properties 6576
- Chapter 8. Orbit stability and combinatorics 7182
- Chapter 9. Bilateral symmetry 7788
- Chapter 10. Proof of the main theorem 8192
- Chapter 11. The renormalization map 8798
- Chapter 12. Properties of the tiling 93104
- Chapter 13. The filling lemma 101112
- Chapter 14. The covering lemma 105116
- Chapter 15. Further geometric results 111122
- Chapter 16. Properties of the limit set 115126
- Chapter 17. Hausdorff convergence 121132
- Chapter 18. Recurrence relations 127138
- Chapter 19. Hausdorff dimension bounds 131142
- Chapter 20. Controlling the limit set 141152
- Chapter 21. The arc case 149160
- Chapter 22. Further symmetries of the tiling 157168
- Chapter 23. The forest case 163174
- Chapter 24. The Cantor set case 167178
- Chapter 25. Dynamics in the arc case 175186
- Chapter 26. Computational methods 185196
- Chapter 27. The calculations 193204
- Chapter 28. The raw data 203214
- Bibliography 211222
- Back Cover Back Cover1226

#### Readership

Graduate students and research mathematicians interested in dynamical systems.