Metric properties and exceptional sets of \(\beta \)-expansions over formal Laurent series
type: Item
instance of: scholarly article
title: Metric properties and exceptional sets of \(\beta \)-expansions over formal Laurent series
zbMATH Open document ID: 1153.11043
author: Bing Li
author: Jun Wu
author: Jian Xu
published in: Monatshefte für Mathematik
publication date: 2008-11-25T00:00:00Z
review text: Let \(\mathbb{F}\) be a finite field with \(q\) elements, and consider the field of formal Laurent series \(\mathbb{F}((X^{-1}))\). This is a complete metric space under the norm \(\| x\|=q^{-v(x)}\), where \(v\) is the standard exponential valuation. Let \(I=\{x:\| x\|<1\}\); as an abelian group, we have the direct sum decomposition \(\mathbb{F}((X^{-1}))=\mathbb{F}[X]\oplus I\), with projection functions \([-]\) and \(\{-\}\) onto the first and second summand, respectively. Fix now \(\beta\in\mathbb{F}((X^{-1}))\) with \(\|\beta\|>1\), and define the \(\beta\)-transformation \(T_\beta:I\to I\) by \(T_\beta(x)=\{\beta x\}\). The mapping \(T_\beta\) is a direct analogue of the well known \(\beta\)-transformation on the real unit interval \(R_\beta:[0,1)\to[0,1)\); here \(\beta>1\) is a fixed irrational number, and \(R_\beta(x)\) is the fractional part of \(\beta x\). The transformations \(T_\beta\) and \(R_\beta\) have obvious analogies, but also remarkable differences, the Laurent series case being somehow smoother. For example, the sets of all \(x\in I\) whose \(T_\beta\)-orbit is finite, or eventually periodic, have been characterized by \textit{M. Hbaib} and \textit{M. Mkaouar} [Int. J. Number Theory 2, No. 3, 365--378 (2006; Zbl 1157.11004)] and by \textit{K. Scheicher} [Finite Fields Appl. 13, No. 2, 394--410 (2007; Zbl 1152.11037)], while the real case is not yet fully understood. Also, it is known that not every sequence \(\varepsilon\in\{0,1,\dots,\lfloor\beta\rfloor\}^\mathbb{N}\) is the \(R_\beta\)-expansion of some \(x\in[0,1)\) (\(\varepsilon\) is the \(R_\beta\)-expansion of \(x\) if \(\varepsilon_n=\lfloor \beta R_\beta^{n-1}(x) \rfloor\), for every \(n\geq1\)). On the other hand, every sequence in \(\{\alpha\in\mathbb{F}[X]:\|\alpha\|<\|\beta\|\}^\mathbb{N}\) is the \(T_\beta\)-expansion of some \(x\in I\). In the present paper it is proved that the Haar measure on the compact abelian group \(I\) is \(T_\beta\)-invariant, and that the resulting dynam…
reviewed by: Giovanni Panti
Mathematics Subject Classification ID: 11K55
zbMATH DE Number: 5373508
zbMATH Keywords: \(\beta\)-expansion
zbMATH Keywords: Laurent series
zbMATH Keywords: Haar measure
zbMATH Keywords: finite field
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
full work available at URL: s00605-008-0531-7
OpenAlex ID: W2060299996
cites work: Q4510760
cites work: Q4135662
cites work: Q5515914
cites work: \(\beta\)-expansions and symbolic dynamics
cites work: Q3995301
cites work: Q4529547
cites work: Finite beta-expansions
cites work: Q3268372
cites work: SUR LE BÊTA-DÉVELOPPEMENT DE 1 DANS LE CORPS DES SÉRIES FORMELLES
cites work: Beta-expansion and continued fraction expansion
cites work: On theβ-expansions of real numbers
cites work: Representations for real numbers and their ergodic properties
cites work: \(\beta\)-expansions in algebraic function fields over finite fields
cites work: Symbolic dynamics for $\beta$-shifts and self-normal numbers
cites work: On Periodic Expansions of Pisot Numbers and Salem Numbers
DOI: 10.1007/S00605-008-0531-7
Recommended article: Metric properties and exceptional sets of the Oppenheim expansions over the field of Laurent series
Recommended article: Metric properties and exceptional sets of beta-continued fractions of Laurent series
Recommended article: Some metric properties of Lüroth expansions over the field of Laurent series
Recommended article: Metrical properties for continued fractions of formal Laurent series
Recommended article: Metric properties of some special p-adic series expansions
Recommended article: Exceptional sets of the Oppenheim expansions over the field of formal Laurent series
Recommended article: Q4953559
Recommended article: Metric properties for p-adic Oppenheim series expansions
Recommended article: Q5128193
Recommended article: Beta-expansion and continued fraction expansion over formal Laurent series
description: scientific article; zbMATH DE number 5373508