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The Octagonal PETs
Richard Evan Schwartz, Brown University, Providence, RI
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Mathematical Surveys and Monographs
2014; 212 pp; hardcover
Volume: 197
ISBN-10: 1-4704-1522-4
ISBN-13: 978-1-4704-1522-8
List Price: US$90 Member Price: US$72
Order Code: SURV/197

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An Introduction to Dynamical Systems: Continuous and Discrete, Second Edition - R Clark Robinson

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.