Random \(N\)-continued fraction expansions
type: Item
instance of: scholarly article
title: Random \(N\)-continued fraction expansions
zbMATH Open document ID: 1395.11112
author: K. Dajani
author: Margriet Oomen
published in: Journal of Approximation Theory
publication date: 2018-02-20T00:00:00Z
review text: \textit{E. B. Burger} et al. [J. Number Theory 128, No. 1, 144--153 (2008; Zbl 1210.11078)] introduced the \(N\)-continued expansion, showing that for any \(N>1\) any \(x\in\mathbb{R}\) has a representation of the form \(x=d_0+\frac{N}{d_1+\frac{N}{d_2+\cdots}}\), in which the usual continued fraction has the repeated numerator~\(1\) replaced by \(N\). ~~\textit{K. Dajani} et al. [J. Number Theory 133, No. 9, 3183--3204 (2013; Zbl 1325.11066)] showed how this expansion may be associated to a dynamical system, and used this dynamical approach to recover an earlier result of \textit{M. Anselm} and \textit{S. H. Weintraub} [J. Number Theory 131, No. 12, 2442--2460 (2011; Zbl 1263.11007)] that every \(x\in\mathbb{R}\) has infinitely many such expansions and found invariant measures for several dynamical systems generating \(N\)-continued fraction expansions. Here \(N\)-continued fraction expansions generated by a random dynamical system that randomizes across all such expansions is studied for \(N=2\) and \(3\). The approach is to use ergodic theory to study the random system as a measure-theoretic dynamical system, in particular using work of \textit{T. Inoue} [Stud. Math. 208, No. 1, 11--29 (2012; Zbl 1267.37054)] to prove the existence a measure of product type. Dynamical properties of the system are established, allowing asymptotic behaviour of the expansions to be studied.
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 11K50
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37C40
zbMATH DE Number: 6840976
zbMATH Keywords: \(N\)-continued fractions
zbMATH Keywords: random dynamical systems
zbMATH Keywords: ergodicity
zbMATH Keywords: finite invariant measures
MaRDI profile type: Publication
full work available at URL: j.jat.2017.11.003
OpenAlex ID: W2563770607
cites work: Annealed and quenched limit theorems for random expanding dynamical systems
cites work: A generalization of continued fractions
cites work: Q4357052
cites work: Shrinking the period lengths of continued fractions while still capturing convergents
cites work: The Möbius function and the residue theorem
cites work: Q4548818
cites work: Ergodicity of \(N\)-continued fraction expansions
cites work: Measures of maximal entropy for random \(\beta\)-expansions
cites work: Invariant measures for position dependent random maps with continuous random parameters
cites work: The random continued fraction transformation
cites work: Q4744504
DOI: 10.1016/J.JAT.2017.11.003
description: scientific article; zbMATH DE number 6840976