# Periodicity of β-expansions for certain Pisot units*

Sandra Vaz; Pedro Martins Rodrigues

ESAIM: Proceedings (2012)

- Volume: 36, page 48-60
- ISSN: 1270-900X

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topVaz, Sandra, and Rodrigues, Pedro Martins. Fournier-Prunaret, D., Gardini, L., and Reich, L., eds. "Periodicity of β-expansions for certain Pisot units*." ESAIM: Proceedings 36 (2012): 48-60. <http://eudml.org/doc/251287>.

@article{Vaz2012,

abstract = {Given β > 1, let Tβ\begin\{eqnarray\} T\_\{\beta\}:[0,1[ & \rightarrow&
[0,1[ \nonumber\\ \hspace*\{0.5 cm\} x & \rightarrow & \beta x
-\lfloor \beta x \rfloor. \nonumber \end\{eqnarray\}The iteration of this transformation gives rise to the greedy β-expansion.
There has been extensive research on the properties of this expansion and its dependence
on the parameter β.In [17], K. Schmidt analyzed the set of periodic points of
Tβ, where β is a Pisot
number. In an attempt to generalize some of his results, we study, for certain Pisot
units, a different expansion that we call linear expansion\begin\{eqnarray\} x=\sum\_\{i \geq 0\} e\_i \beta^\{-i\},\nonumber
\end\{eqnarray\}where each ei can be superior to ⌊ β ⌋, its properties and the relation with
Per (β).},

author = {Vaz, Sandra, Rodrigues, Pedro Martins},

editor = {Fournier-Prunaret, D., Gardini, L., Reich, L.},

journal = {ESAIM: Proceedings},

keywords = {beta-expansions; beta-representations; Pisot numbers},

language = {eng},

month = {8},

pages = {48-60},

publisher = {EDP Sciences},

title = {Periodicity of β-expansions for certain Pisot units*},

url = {http://eudml.org/doc/251287},

volume = {36},

year = {2012},

}

TY - JOUR

AU - Vaz, Sandra

AU - Rodrigues, Pedro Martins

AU - Fournier-Prunaret, D.

AU - Gardini, L.

AU - Reich, L.

TI - Periodicity of β-expansions for certain Pisot units*

JO - ESAIM: Proceedings

DA - 2012/8//

PB - EDP Sciences

VL - 36

SP - 48

EP - 60

AB - Given β > 1, let Tβ\begin{eqnarray} T_{\beta}:[0,1[ & \rightarrow&
[0,1[ \nonumber\\ \hspace*{0.5 cm} x & \rightarrow & \beta x
-\lfloor \beta x \rfloor. \nonumber \end{eqnarray}The iteration of this transformation gives rise to the greedy β-expansion.
There has been extensive research on the properties of this expansion and its dependence
on the parameter β.In [17], K. Schmidt analyzed the set of periodic points of
Tβ, where β is a Pisot
number. In an attempt to generalize some of his results, we study, for certain Pisot
units, a different expansion that we call linear expansion\begin{eqnarray} x=\sum_{i \geq 0} e_i \beta^{-i},\nonumber
\end{eqnarray}where each ei can be superior to ⌊ β ⌋, its properties and the relation with
Per (β).

LA - eng

KW - beta-expansions; beta-representations; Pisot numbers

UR - http://eudml.org/doc/251287

ER -

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