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On the Boundary of the Set of the Closure of Contractive Polynomials

Attila Pethő
  • Faculty of Informatics, University of Debrecen and Hungarian Academy of Sciences, Debrecen, Hungary. E-mail:
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Published Online: 2009-08-20 | DOI: https://doi.org/10.1515/INTEG.2009.026

Abstract

For and a = (a 1, . . . , ad ) T ∈ ℤ d , let τ r(a) = (a 2, . . . , ad , –⌊r T a⌋) T . Further, let . In this paper we prove that if some roots of the polynomial Xd + rdX d–1 + ⋯ + r 2 X + r 1 are t -th roots of unity and the others lie in the open unit disc, then |a k+tak | < c 1 with a constant c 1 which does not depend on k. Under some conditions this yields an algorithm to decide whether the sequence is, for all a, ultimately periodic, or becomes divergent for some a.

We study the boundary of the closure of 𝒟3 and show that large parts of it belong to 𝒟3, while others lie outside 𝒟3.

Keywords.: Contractive polynomials; shift radix systems

About the article

Received: 2008-07-24

Accepted: 2009-01-26

Published Online: 2009-08-20

Published in Print: 2009-08-01


Citation Information: Integers, Volume 9, Issue 3, Pages 311–325, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/INTEG.2009.026.

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