 
Summary: A CERTAIN FINITENESS PROPERTY OF PISOT
NUMBER SYSTEMS
SHIGEKI AKIYAMA, HUI RAO AND WOLFGANG STEINER
Abstract. In the study of substitutative dynamical systems and
Pisot number systems, an algebraic condition, which we call `weak
niteness', plays a fundamental role. It is expected that all Pisot
numbers would have this property. In this paper, we prove some
basic facts about `weak niteness'. We show that this property is
valid for cubic Pisot units and for Pisot numbers of higher degree
under a dominant condition.
1. Introduction
Let > 1 be a real number. The transformation is a piecewise
linear transformation on [0; 1) dened by
T : x ! x bxc;
where bc is the largest integer not exceeding . By iterating this map
and considering its trajectory
x x 1
! T (x) x 2
! T 2
(x) x 3
