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Summary: GENERALIZED RADIX REPRESENTATIONS AND DYNAMICAL SYSTEMS
II
SHIGEKI AKIYAMA, HORST BRUNOTTE, ATTILA PETHO, AND JšORG M. THUSWALDNER
Abstract. For r Rd define r : Zd Zd by setting
r(a) = (a2, . . . , ad, - ra ) (a = (a1, . . . , ad)).
We call r a shift radix system if for each a Zd there exists an integer k > 0 with k
r (a) = 0.
Shift radix systems have been defined in the first part of this series of papers. It turns out that
they are intimately related to certain well known notions of number systems like -expansions
and canonical number systems.
It seems to be a hard problem to characterize all r Rd giving rise to a shift radix system. In
the present paper we give partial characterization results. After proving some general theorems
we are mainly concerned with the characterization of two dimensional shift radix systems.
1. Introduction
In the first part [4] of this series of papers we introduced the notion of shift radix system
and described its basic properties as well as its relations to -expansions and canonical number
systems1
. Specifically, let d 1 be an integer and r = (r1, . . . , rd) Rd
. To r we associate the
mapping r : Zd
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