Periodicity of β-expansions for certain Pisot units
ESAIM. Proceedings, Tome 36 (2012), pp. 48-60
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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} T β : [ 0 , 1 [ → [ 0 , 1 [ x → βx − ⌊ βx ⌋ . 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} x = ∑ i ≥ 0 e i β − i , where each ei can be superior to ⌊ β ⌋, its properties and the relation with Per (β).
@article{EP_2012_36_a5,
author = {Sandra Vaz and Pedro Martins Rodrigues},
title = {Periodicity of \ensuremath{\beta}-expansions for certain {Pisot} units},
journal = {ESAIM. Proceedings},
pages = {48--60},
year = {2012},
volume = {36},
doi = {10.1051/proc/201236005},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201236005/}
}
Sandra Vaz; Pedro Martins Rodrigues. Periodicity of β-expansions for certain Pisot units. ESAIM. Proceedings, Tome 36 (2012), pp. 48-60. doi: 10.1051/proc/201236005
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