The octagonal PETs

by Schwartz, Richard Evan

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Title Statements

Full title :The octagonal PETs / Richard Evan Schwartz.

Summary

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.

Edition Statements

Publication Date:2014.

Publication Statements

Publisher(s):American mathematical Society
Place of Publication:Providence, Rhode Island
Language:eng
Publication Date:2014.

The Physical Object

Description:x, 212 pages. : : illustrations (some color) ; : 26 cm.

Edition Identifiers

ISBN:9781470415228, 1470415224
oclc:ocn909625556
LC Classification:QA691
Dewey Decimal:515/.48

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.