# Measures of maximal entropy for random $\beta $-expansions

Karma Dajani; Martijn de Vries

Journal of the European Mathematical Society (2005)

- Volume: 007, Issue: 1, page 51-68
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topDajani, Karma, and de Vries, Martijn. "Measures of maximal entropy for random $\beta $-expansions." Journal of the European Mathematical Society 007.1 (2005): 51-68. <http://eudml.org/doc/277322>.

@article{Dajani2005,

abstract = {Let $\beta >1$ be a non-integer. We consider $\beta $-expansions of the form
$\sum _\{i=1\}^\infty d_i/\beta ^i$, where
the digits $(d_i)_\{i\ge 1\}$ are generated by means of a Borel map $K_\beta $ defined on $\lbrace 0,1\rbrace ^\{\mathbb \{N\}\}\times [0,\lfloor \beta \rfloor /(\beta −1)]$. 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 $(d_i)_\{i\ge 1\}$ 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.},

author = {Dajani, Karma, de Vries, Martijn},

journal = {Journal of the European Mathematical Society},

keywords = {greedy expansions; lazy expansions; Markov chains; measures of maximal entropy; Parry measure; greedy expansions; lazy expansions; Markov chains; measures of maximal entropy; Parry measure},

language = {eng},

number = {1},

pages = {51-68},

publisher = {European Mathematical Society Publishing House},

title = {Measures of maximal entropy for random $\beta $-expansions},

url = {http://eudml.org/doc/277322},

volume = {007},

year = {2005},

}

TY - JOUR

AU - Dajani, Karma

AU - de Vries, Martijn

TI - Measures of maximal entropy for random $\beta $-expansions

JO - Journal of the European Mathematical Society

PY - 2005

PB - European Mathematical Society Publishing House

VL - 007

IS - 1

SP - 51

EP - 68

AB - Let $\beta >1$ be a non-integer. We consider $\beta $-expansions of the form
$\sum _{i=1}^\infty d_i/\beta ^i$, where
the digits $(d_i)_{i\ge 1}$ are generated by means of a Borel map $K_\beta $ defined on $\lbrace 0,1\rbrace ^{\mathbb {N}}\times [0,\lfloor \beta \rfloor /(\beta −1)]$. 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 $(d_i)_{i\ge 1}$ 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.

LA - eng

KW - greedy expansions; lazy expansions; Markov chains; measures of maximal entropy; Parry measure; greedy expansions; lazy expansions; Markov chains; measures of maximal entropy; Parry measure

UR - http://eudml.org/doc/277322

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.