Skip to main content
Article
Licensed
Unlicensed Requires Authentication

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

Published/Copyright: August 20, 2009
Become an author with De Gruyter Brill
Author Information Explore this Subject
Integers
From the journal Volume 9 Issue 3

Abstract

For and a = (a1, . . . , ad)T ∈ ℤd, let τr(a) = (a2, . . . , ad, –⌊rTa⌋)T. Further, let . In this paper we prove that if some roots of the polynomial Xd + rdXd–1 + ⋯ + r2X + r1 are t -th roots of unity and the others lie in the open unit disc, then |ak+t – ak| < c1 with a constant c1 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
Received: 2008-07-24
Accepted: 2009-01-26
Published Online: 2009-08-20
Published in Print: 2009-August

© de Gruyter 2009

Articles in the same Issue

  1. Preface
  2. Symmetric CNS Trinomials
  3. On the β-Expansion of an Algebraic Number in an Algebraic Base β
  4. A Planar Integral Self-Affine Tile with Cantor Set Intersections with Its Neighbors
  5. Beta-Expansions with Negative Bases
  6. Convergence in Möbius Number Systems
  7. Factor Complexity of Infinite Words Associated with Non-Simple Parry Numbers
  8. On the Boundary of the Set of the Closure of Contractive Polynomials
  9. Christoffel Words and Markoff Triples
  10. Characterizations of Words with Many Periods
https://doi.org/10.1515/INTEG.2009.026
Keywords for this article
Contractive polynomials; shift radix systems

Articles in the same Issue

  1. Preface
  2. Symmetric CNS Trinomials
  3. On the β-Expansion of an Algebraic Number in an Algebraic Base β
  4. A Planar Integral Self-Affine Tile with Cantor Set Intersections with Its Neighbors
  5. Beta-Expansions with Negative Bases
  6. Convergence in Möbius Number Systems
  7. Factor Complexity of Infinite Words Associated with Non-Simple Parry Numbers
  8. On the Boundary of the Set of the Closure of Contractive Polynomials
  9. Christoffel Words and Markoff Triples
  10. Characterizations of Words with Many Periods
Downloaded on 18.4.2026 from https://www.degruyterbrill.com/document/doi/10.1515/INTEG.2009.026/html
Scroll to top button