Geodesic
Purely Periodic $\beta$-expansions in Cubic Salem Base in $\mathbb{F}_q((X^{-1}))$
Publications de l'Institut Mathématique, _N_S_100 (2016) no. 114, p. 279 .

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Let $\mathbb{F}_q$ be the finite field with $q$ elements and $\beta$ Salem series in $ \mathbb{F}_q((X^{-1}))$. It is proved in \cite{K} that, in this case, all elements in $\mathbb{F}_q(X,\beta)$ have purely periodic $\beta$-expansion. We characterize the formal power series $f$ in $\mathbb{F}_q(X,\beta)$ with purely periodic $\beta$-expansions by the conjugate vector $\widetilde{f}$ when $\beta$ is a cubic unit. No similar results exist in the real case.
DOI : 10.2298/PIM1614279M
Classification : 11R06, 37B50
Keywords: formal power series, finite fields, $\beta$-expansion, Salem element 
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     author = {Fa{\"\i}za Mahjoub},
     title = {Purely {Periodic} $\beta$-expansions in {Cubic} {Salem} {Base} in $\mathbb{F}_q((X^{-1}))$},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {279 },
     publisher = {mathdoc},
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     year = {2016},
     doi = {10.2298/PIM1614279M},
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Faïza Mahjoub. Purely Periodic $\beta$-expansions in Cubic Salem Base in $\mathbb{F}_q((X^{-1}))$. Publications de l'Institut Mathématique, _N_S_100 (2016) no. 114, p. 279 . doi : 10.2298/PIM1614279M. http://geodesic.mathdoc.fr/articles/10.2298/PIM1614279M/

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