A natural extension for the greedy \(\beta\)-transformation with three arbitrary digits
type: Item
instance of: scholarly article
title: A natural extension for the greedy \(\beta\)-transformation with three arbitrary digits
zbMATH Open document ID: 1212.37001
published in: Acta Mathematica Hungarica
publication date: 2010-08-13T00:00:00Z
full work available at URL: https://dspace.library.uu.nl/handle/1874/197282
review text: For a real number \(\beta>1\) and finite set of real numbers \(A\) (the digits), an expression of the form \(x=\sum_{n=1}^{\infty}a_n\beta^{-n}\) with \(a_n\in A\) for all \(n\geq1\) is called a \(\beta\)-expansion for \(x\). \textit{M. Pedicini} [Theor. Comput. Sci. 332, No. 1-3, 313--336 (2005; Zbl 1080.11009)] gave conditions on \(A\) to give a range of values of \(x\) that are guaranteed to have a \(\beta\)-expansion, and an algorithm to generate the so-called greedy expansion. Here a planar version of the natural invertible extension of the piecewise-linear map that generates this greedy expansion in the case \(| A|=3\). Using this, a closed formula is found for the density of the unique invariant measure absolutely continuous with respect to Lebesgue measure, and this measure is shown to be exact and weakly Bernoulli.
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 11K55
zbMATH DE Number: 5770413
zbMATH Keywords: greedy expansion
zbMATH Keywords: natural expansion
zbMATH Keywords: absolutely continuous invariant measure
author: K. Dajani
author: C. Kalle
MaRDI profile type: Publication
OpenAlex ID: W2098106291
cites work: $\beta$-transformation, natural extension and invariant measure
cites work: Local dimensions for the random beta-transformation
cites work: Random \(\beta\)-expansions with deleted digits
cites work: The natural extension of the \(\beta\)-transformation
cites work: Invariant densities for random \(\beta\)-expansions
cites work: On the Morphology of γ-Expansions with Deleted Digits
cites work: Invariant measures for piecewise linear transformations of the interval
cites work: On theβ-expansions of real numbers
cites work: Greedy expansions and sets with deleted digits
cites work: The Hausdorff Dimension of λ-Expansions with Deleted Digits
cites work: Representations for real numbers and their ergodic properties
cites work: Bounded variation and invariant measures
cites work: Almost Every Number Has a Continuum of b-Expansions
cites work: Ergodic properties of a class of piecewise linear transformations
DOI: 10.1007/S10474-009-8212-0
Recommended article: Local dimensions for the random beta-transformation
Recommended article: On \(\alpha \)-greedy expansions of numbers
Recommended article: The natural extension of the random beta-transformation
Recommended article: A generalization of the \(G\)-transformation and the related algorithms
Recommended article: The natural extension of the \(\beta\)-transformation
Recommended article: Mathematical Foundations of Computer Science 2005
Recommended article: Approximation and Online Algorithms
Recommended article: On sorting by 3-bounded transpositions
Recommended article: On Sorting by 3-Bounded Transpositions
Recommended article: A note on the distribution of digits in triadic expansions
description: scientific article; zbMATH DE number 5770413