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GENERALIZED RADIX REPRESENTATIONS AND DYNAMICAL SYSTEMS SHIGEKI AKIYAMA, HORST BRUNOTTE, ATTILA PETHO, AND JORG M. THUSWALDNER
 

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 well-known 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

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics