Title: 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.
Subjects: Polytopes, Geometry, Dynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Universality, renormalization, Dynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Maps of the interval (piecewise continuous, continuous, smooth), Dynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Combinatorial dynamics (types of periodic orbits)