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Summary: GENERALIZED RADIX REPRESENTATIONS AND DYNAMICAL SYSTEMS
IV
SHIGEKI AKIYAMA, HORST BRUNOTTE, ATTILA PETHO, AND J¨ORG M. THUSWALDNER
Abstract. For r = (r1, . . . , rd) Rd the mapping r : Zd Zd given by
r(a1, . . . , ad) = (a2, . . . , ad, - r1a1 + · · · + rdad )
where · denotes the floor function, 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 Part I of this series of papers, shift
radix systems are intimately related to certain well-known notions of number systems like -
expansions and canonical number systems. After characterization results on shift radix systems
in Part II of this series of papers and the thorough investigation of the relations between shift
radix systems and canonical number systems in Part III, the present part is devoted to further
structural relationships between shift radix systems and -expansions. In particular we establish
the distribution of Pisot polynomials with and without the finiteness property (F).
1. Introduction
This is the fourth part of a series of papers that is devoted to the systematic study of so-
called shift radix systems. Shift radix systems are dynamical systems that are strongly related to
well-known notions of number systems. First of all, let us recall their exact definition.
Definition 1.1. Let d 1 be an integer and r = (r1, . . . , rd) Rd
. To r we associate the mapping
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