 
Summary: Book Title
Book Editors
IOS Press, 2003
1
Pisot number system and its dual tiling
Shigeki Akiyama 1
Niigata Univ. JAPAN
Abstract. Number systems in Pisot number base are discussed in relation to arith
metic construction of quasicrystal model. One of the most important ideas is to
introduce a `dual tiling' of this system. This provides us a geometric way to under
stand the `algebraic structure' of the above model as well as dynamical understand
ing of arithmetics algorithms.
Keywords. Pisot Number, Number System, Symbolic Dynamical System, Tiling
1. Beta expansion and Pisot number system
For this section, the reader finds a nice survey by Frougny [41]. However we give a brief
review and concise proofs of fundamental results to make this note more selfcontained.
Let us fix > 1 and A = [0, ) Z. Denote by A
the set of finite words over A and
by AN
the set of right infinite words over A. By concatenation :
