Ergodicity of \(N\)-continued fraction expansions
type: Item
instance of: scholarly article
title: Ergodicity of \(N\)-continued fraction expansions
zbMATH Open document ID: 1325.11066
author: Niels van der Wekken
author: K. Dajani
author: Cor Kraaikamp
published in: Journal of Number Theory
publication date: 2014-09-09T00:00:00Z
review text: The authors of this paper study a special kind of continued fraction expansions, firstly introduced by \textit{E. B. Burger} and his co-authors in the paper [J. Number Theory 128, No. 1, 144--153 (2008; Zbl 1210.11078)]. Applying the map \(T_{N}:[0,1)\rightarrow [0,1)\), defined by \(T_{N}(x)=\frac{N}{x}-\lfloor\frac{N}{x}\rfloor\) with \(x\neq0\) and \(T_{N}(0)=0\), every \(x\in \mathbb{R}\) can be written in the form \[ x=n_{0}+\frac{N}{n_{1}+\frac{N}{n_{2}+\ddots +\frac{N}{n_{k}+\ddots}}}=[n_{0};n_{1},n_{2},\ldots,n_{k}\ldots]_{N}, \] where \(N\in\mathbb{R}\) and \(N \neq 0\). The first theorem (Theorem 3.1) is: Let \(N\in \mathbb{N}\). Then the dynamical system \(([0,1),\mathcal{B},\mu_{N},T_{N})\) is exact and therefore ergodic. Here \(\mathcal{B}\) is the Borel \(\sigma\)-algebra on \([0,1)\), and \(\mu_{N}\) is a \(T_{N}\)-invariant probability measure on \(([0,1), \mathcal{B})\) with density \(\frac{1}{\log\frac{N+1}{N}}\frac{1}{N+x}1_{[0,1)}(x)\). Other theorem with the same conditions (Theorem 3.3) asserts that the dynamical system \(([0,1),\mathcal{B},\mu_{-N},T_{-N})\) is also ergodic. In particular case \(N \geq 2\), \(\mu_{-N}\) is a probability measure with density \(\frac{1}{\log\frac{N}{N-1}}\frac{1}{N-x}1_{[0,1)}(x)\). Similar theorems with other conditions are proved and many examples are discussed. Special attention is turned to the periodicity of digits and conjectures on the base of calculation and their histograms.
reviewed by: Tonko Tonkov
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11K60
Mathematics Subject Classification ID: 37A45
zbMATH DE Number: 6341801
zbMATH Keywords: \(\sigma\)-finite
zbMATH Keywords: infinite invariant measure
MaRDI profile type: Publication
full work available at URL: j.jnt.2013.02.017
OpenAlex ID: W2036454529
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DOI: 10.1016/J.JNT.2013.02.017
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description: scientific article; zbMATH DE number 6341801