The Octagonal PETs Hardback

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.