### 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 |
---|---|

Pages (from-to) | 333-348 |

Number of pages | 16 |

Journal | Indagationes Mathematicae |

Volume | 19 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 2008 |

### Fingerprint

### 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

**Generalized radix representations and dynamical systems. IV.** / Akiyama, Shigeki; Brunotte, Horst; Pethő, A.; Thuswaldner, Jörg M.

Research output: Contribution to journal › Article

*Indagationes Mathematicae*, vol. 19, no. 3, pp. 333-348. https://doi.org/10.1016/S0019-3577(09)00006-8

}

TY - JOUR

T1 - Generalized radix representations and dynamical systems. IV

AU - Akiyama, Shigeki

AU - Brunotte, Horst

AU - Pethő, A.

AU - Thuswaldner, Jörg M.

PY - 2008/9

Y1 - 2008/9

N2 - For r = (r1,..., rd) ∈ ℝd the mapping τr:ℤd →ℤd given by. τr(a1,...,ad) = (a2, ..., ad,-⌊r1a1+...+ rdad⌋). where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ℤd there exists an integer k > 0 with τr 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).

AB - For r = (r1,..., rd) ∈ ℝd the mapping τr:ℤd →ℤd given by. τr(a1,...,ad) = (a2, ..., ad,-⌊r1a1+...+ rdad⌋). where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ℤd there exists an integer k > 0 with τr 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).

KW - Beta expansion

KW - Canonical number system

KW - Contracting polynomial

KW - Periodic point

KW - Pisot number

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UR - http://www.scopus.com/inward/citedby.url?scp=67449110973&partnerID=8YFLogxK

U2 - 10.1016/S0019-3577(09)00006-8

DO - 10.1016/S0019-3577(09)00006-8

M3 - Article

AN - SCOPUS:67449110973

VL - 19

SP - 333

EP - 348

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

SN - 0019-3577

IS - 3

ER -