### Abstract

For r = (r_{1},..., r_{d}) ∈ ℝ^{d} the mapping τ_{r}:ℤ^{d} →ℤ^{d} given by. τ_{r}(a_{1},...,a_{d}) = (a_{2}, ..., a_{d},-⌊r_{1}a_{1}+...+ r_{d}a_{d}⌋). where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ℤ^{d} there exists an integer k > 0 with τ_{r}^{k}(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).

Original language | English |
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Pages (from-to) | 333-348 |

Number of pages | 16 |

Journal | Indagationes Mathematicae |

Volume | 19 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 2008 |

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### Keywords

- Beta expansion
- Canonical number system
- Contracting polynomial
- Periodic point
- Pisot number

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indagationes Mathematicae*,

*19*(3), 333-348. https://doi.org/10.1016/S0019-3577(09)00006-8