Summary: PENTAGONAL DOMAIN EXCHANGE
SHIGEKI AKIYAMA AND EDMUND HARRISS
Abstract. Self-inducing structure of pentagonal piecewise isometry is
applied to show detailed description of periodic and aperiodic orbits,
and further dynamical properties. A Pisot number appears as a scaling
constant and plays a crucial role in the proof. Further generalization is
discussed in the last section.
Adler-Kitchens-Tresser  and Goetz  initiated the study of piecewise
isometries. This class of maps shows the way to possible generalizations of
results on interval exchanges to higher dimensions [16, 30]. In this paper
we examine the detailed properties of the map shown in Figure 1 from an
algebraic point of view.
Figure 1. A piecewise rotation T on two pieces. The tri-
angle is rotated 2/5 around a and the trapezium is ro-
tated 2/5 around b. Periodic points with short periods are
shown below, in two colours to illustrate that they cluster
into groups, each forming a pentagon.
The goal of this paper is to see how this map is applied to show number
theoretical results. First we reprove that almost all orbits in the sense of
The first author is supported by the Japanese Society for the Promotion of Science