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 Teichmuller 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.
| ISBN: | 9781470415228 |
| Publication date: | 30th July 2014 |
| Author: | Richard Evan Schwartz |
| Publisher: | American Mathematical Society |
| Format: | Hardback |
| Pagination: | 211 pages |
| Series: | Mathematical Surveys and Monographs |
| Genres: |
Mathematics Differential calculus and equations Geometry Topology |