Measures of maximal entropy for random \(\beta\)-expansions
type: Item
instance of: scholarly article
title: Measures of maximal entropy for random \(\beta\)-expansions
zbMATH Open document ID: 1074.28008
DOI: 10.4171/JEMS/21
author: K. Dajani
author: Martijn de Vries
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2005-03-21T00:00:00Z
review text: Summary: Let \(\beta>1\) be a non-integer. We consider \(\beta\)-expansions of the form \(\sum^\infty_{i=1}d_i/\beta^i\), where the digits \((d_i)_{i\geq 1}\) are generated by means of a Borel map \(K_\beta\) defined on \(\{0,1\}^\mathbb{N}\times [0,\lfloor\beta\rfloor/(\beta-1)]\). We show that \(K_\beta\) has a unique mixing measure \(\nu_\beta\) of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure \(\nu_\beta\) the digits \((d_i)_{i\geq 1}\) form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of \(\beta\)-expansions.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37A45
zbMATH DE Number: 2148276
zbMATH Keywords: greedy expansions
zbMATH Keywords: lazy expansions
zbMATH Keywords: Markov chains
zbMATH Keywords: measures of maximal entropy
zbMATH Keywords: Parry measure
MaRDI profile type: Publication
cites work: From greedy to lazy expansions and their driving dynamics
cites work: Random \beta-expansions
cites work: On a Family of Symmetric Bernoulli Convolutions
cites work: Characterization of the unique expansions $1=\sum^{\infty}_{i=1}q^{-n_ i}$ and related problems
cites work: Developments in non-integer bases
cites work: Q4345785
description: scientific article; zbMATH DE number 2148276