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

Dajani, 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 -