The Octagonal PETs
( Mathematical Surveys and Monographs )
Publication series ：Mathematical Surveys and Monographs
Author： Richard Evan Schwartz
Publisher： American Mathematical Society
Publication year： 2014
E-ISBN: 9781470417185
P-ISBN(Paperback): 9781470415228
Subject： O189 topology (geometry of situation)
Keyword： Geometry and Topology
Language： ENG
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The Octagonal PETs
Description
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.
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