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Octogonal PETs / Richard Evan Schwartz.

By: Material type: TextTextSeries: Mathematical surveys and monographs ; v 197Publication details: Providence : American Mathematical Society, c2014.Description: x, 212 p. : illustrations (some color) ; 26 cmISBN:
  • 9781470415228 (alk. paper)
Other title:
  • Octogonal polytope exchange transformations
Subject(s): DDC classification:
  • 510MS 23 Am512
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-- Bibliography.
Summary: 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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Holdings
Item type Current library Call number Status Date due Barcode Item holds
Books ISI Library, Kolkata 510MS Am512 (Browse shelf(Opens below)) Available 135870
Total holds: 0

Includes bibliographical references.

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--
Bibliography.

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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