TY  - BOOK
AU  - Schwartz,Richard Evan
TI  - Octogonal PETs
T2  - Mathematical surveys and monographs
SN  - 9781470415228 (alk. paper)
U1  - 510MS 23
PY  - 2014///
CY  - Providence
PB  - American Mathematical Society
KW  - Polytopes
KW  - Geometry
KW  - Dynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Universality, renormalization
N1  - 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
N2  - 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
ER  -