 
Summary: GENERALIZED RADIX REPRESENTATIONS AND DYNAMICAL SYSTEMS
III
SHIGEKI AKIYAMA, HORST BRUNOTTE, ATTILA PETHO, AND JšORG M. THUSWALDNER
Abstract. For r = (r1, . . . , rd) Rd the map r : Zd Zd given by
r(a1, . . . , ad) = (a2, . . . , ad,  r1a1 + · · · + rdad )
is called a shift radix system if for each a Zd there exists an integer k > 0 with k
r (a) = 0. As
shown in the first two parts of this series of papers shift radix systems are intimately related to
certain wellknown notions of number systems like expansions and canonical number systems.
In the present paper further structural relationships between shift radix systems and canon
ical number systems are investigated. Among other results we show that canonical number
systems related to polynomials
P(X) :=
d
i=0
piXi
Z[X]
of degree d with a large but fixed constant term p0 approximate the set of (d  1)dimensional
shift radix systems. The proofs make extensive use of the following tools: Firstly, vectors r Rd
which define shift radix systems are strongly connected to monic real polynomials all of whose
