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GENERALIZED RADIX REPRESENTATIONS AND DYNAMICAL SYSTEMS SHIGEKI AKIYAMA, TIBOR BORBELY, HORST BRUNOTTE, ATTILA PETHO,
 

Summary: GENERALIZED RADIX REPRESENTATIONS AND DYNAMICAL SYSTEMS
I
SHIGEKI AKIYAMA, TIBOR BORBŽELY, HORST BRUNOTTE, ATTILA PETHO,
AND JšORG M. THUSWALDNER
Abstract. In this paper we are concerned with families of dynamical systems which are related
to generalized radix representations. The properties of these dynamical systems lead to new
results on the characterization of bases of Pisot Number Systems as well as Canonical Number
Systems.
1. Introduction
The present paper is devoted to the study of a certain dynamics on the set of integer vectors Zd
,
which is intimately related to generalized radix representations. In particular, this dynamics on Zd
will be useful for characterizing base numbers with certain finiteness properties. Furthermore, it
reveals unexpected relations between different notions of radix representations. Although we will
make no use of the theory of dynamical systems, the reader will find that this simple dynamical
system on Zd
plays an essential role in understanding the global structure of various numeration
systems.
The first kind of radix representations we are dealing with are the so-called -expansions which
go back to RŽenyi [34] and Parry [29]. Let > 1 be a non-integral real number and let A =

  

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

 

Collections: Mathematics