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The Octagonal PETs

Richard Evan Schwartz
Format
Book
Published
Providence, Rhode Island : American Mathematical Society, [2014]
Language
English
Variant Title
Octagonal polytope exchange transformations
Series
Mathematical Surveys and Monographs
ISBN
1470415224 (alk. paper), 9781470415228 (alk. paper)
Abstract
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.
Contents
  • 1. Introduction
  • 2. Background
  • 3. Multigraph PETs
  • 4. The alternating grid system
  • 5. Outer billiards on semiregular octagons
  • 6. Quarter turn compositions
  • 7. Elementary properties
  • 8. Orbit stability and combinatorics
  • 9. Bilateral symmetry
  • 10. Proof of the main theorem
  • 11. The renormalization map
  • 12. Properties of the tiling
  • 13. The filling lemma
  • 14. The covering lemma
  • 15. Further geometric results
  • 16. Properties of the limit set
  • 17. Hausdorff convergence
  • 18. Recurrence relations
  • 19. Hausdorff dimension bounds
  • 20. Controlling the limit set
  • 21. The arc case - - 22. Further symmetries of the tiling
  • 23. The forest case
  • 24. The cantor set case
  • 25. Dynamics in the arc case
  • 26. Computational methods
  • 27. The calculations
  • 28. The raw data.
Description
x, 212 pages : illustrations (some color) ; 26 cm.
Notes
Includes bibliographical references (pages 211-212).
Series Statement
Mathematical surveys and monographs ; volume 197
Mathematical surveys and monographs ; no. 197
Technical Details
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    a| 1. Introduction -- 2. Background -- 3. Multigraph PETs -- 4. The alternating grid system -- 5. Outer billiards on semiregular octagons -- 6. Quarter turn compositions -- 7. Elementary properties -- 8. Orbit stability and combinatorics -- 9. Bilateral symmetry -- 10. Proof of the main theorem -- 11. The renormalization map -- 12. Properties of the tiling -- 13. The filling lemma -- 14. The covering lemma -- 15. Further geometric results -- 16. Properties of the limit set -- 17. Hausdorff convergence -- 18. Recurrence relations -- 19. Hausdorff dimension bounds -- 20. Controlling the limit set -- 21. The arc case - - 22. Further symmetries of the tiling -- 23. The forest case -- 24. The cantor set case -- 25. Dynamics in the arc case -- 26. Computational methods -- 27. The calculations -- 28. The raw data.
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    a| 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.
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