# Invariant densities for random $\beta $-expansions

Karma Dajani; Martijn de Vries

Journal of the European Mathematical Society (2007)

- Volume: 009, Issue: 1, page 157-176
- ISSN: 1435-9855

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topDajani, Karma, and de Vries, Martijn. "Invariant densities for random $\beta $-expansions." Journal of the European Mathematical Society 009.1 (2007): 157-176. <http://eudml.org/doc/277444>.

@article{Dajani2007,

abstract = {Let $\beta >1$ be a non-integer. We consider 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 existence and uniqueness of a $K_\beta $-invariant probability measure, absolutely continuous with respect to $m_p\otimes \lambda $, where $m_p$ is the Bernoulli measure on $\lbrace 0,1\rbrace ^\{\mathbb \{N\}\}$ with parameter $p$ ($0<p<1$) and $\lambda $ is the normalized Lebesgue measure on $[0,\lfloor \beta \rfloor (\beta −1)]$. Furthermore, this measure is of the form $m_p\otimes \mu _\{\beta ,p\}$, where $\mu _\{\beta ,p\}$ is equivalent to $\lambda $. We prove that the measure of maximal entropy and $m_p\otimes \lambda $ are mutually singular. In case the number 1 has a finite greedy expansion with positive
coefficients, the measure $m_p\otimes \mu _\{\beta ,p\}$ is Markov. In the last section we answer a question concerning
the number of universal expansions, a notion introduced in [EK].},

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

journal = {Journal of the European Mathematical Society},

keywords = {greedy expansions; lazy expansions; absolutely continuous invariant measures; measures of maximal entropy; Markov chains; universal expansions; greedy expansions; lazy expansions; absolutely continuous invariant measures; measures of maximal entropy; Markov chains; universal expansions},

language = {eng},

number = {1},

pages = {157-176},

publisher = {European Mathematical Society Publishing House},

title = {Invariant densities for random $\beta $-expansions},

url = {http://eudml.org/doc/277444},

volume = {009},

year = {2007},

}

TY - JOUR

AU - Dajani, Karma

AU - de Vries, Martijn

TI - Invariant densities for random $\beta $-expansions

JO - Journal of the European Mathematical Society

PY - 2007

PB - European Mathematical Society Publishing House

VL - 009

IS - 1

SP - 157

EP - 176

AB - Let $\beta >1$ be a non-integer. We consider 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 existence and uniqueness of a $K_\beta $-invariant probability measure, absolutely continuous with respect to $m_p\otimes \lambda $, where $m_p$ is the Bernoulli measure on $\lbrace 0,1\rbrace ^{\mathbb {N}}$ with parameter $p$ ($0<p<1$) and $\lambda $ is the normalized Lebesgue measure on $[0,\lfloor \beta \rfloor (\beta −1)]$. Furthermore, this measure is of the form $m_p\otimes \mu _{\beta ,p}$, where $\mu _{\beta ,p}$ is equivalent to $\lambda $. We prove that the measure of maximal entropy and $m_p\otimes \lambda $ are mutually singular. In case the number 1 has a finite greedy expansion with positive
coefficients, the measure $m_p\otimes \mu _{\beta ,p}$ is Markov. In the last section we answer a question concerning
the number of universal expansions, a notion introduced in [EK].

LA - eng

KW - greedy expansions; lazy expansions; absolutely continuous invariant measures; measures of maximal entropy; Markov chains; universal expansions; greedy expansions; lazy expansions; absolutely continuous invariant measures; measures of maximal entropy; Markov chains; universal expansions

UR - http://eudml.org/doc/277444

ER -

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