Number systems with negative bases

Dammak, Salma; Hbaib, M.

doi:10.1007/s10474-013-0383-z

Cited by 4 publications

(3 citation statements)

References 4 publications

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“…We say that β > 1 has the negative finiteness property, or Property (−F), if each element of Z[β −1 ] has a finite (−β)-expansion. By Dammak and Hbaib [10], we know that β must be a Pisot number, as in the positive case. It was shown in [20] that the Pisot roots of x 2 − mx + n, with positive integers m, n, m ≥ n + 2, satisfy the Property (−F).…”

Section: Introductionmentioning

confidence: 98%

Finite beta-expansions with negative bases

2017

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International audienceThe finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $\beta$ having the negative finiteness property, that is the set of finite $(-\beta)$-expansions is equal to $\mathbb{Z}[\beta^{-1}]$. For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of $(-\beta)$-integers. We also give conditions excluding the negative finiteness property

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Section: Introductionmentioning

confidence: 98%

Finite beta-expansions with negative bases

2017

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“…Hence it is worth investigating problems considered for β-shifts in the case of (−β)-shifts. Recently, dynamical properties of (−β)-shifts are also studied by many authors from viewpoints of ergodic theory and number theory [9,10,11,12,14].…”

Section: Introductionmentioning

confidence: 99%

“…Hence it is worth considering the problems of (−β)-shifts, which are also considered in β-shifts. Dynamical properties of (−β)-shifts have been also studied in recent years by many authors from viewpoints of ergodic theory and number theory [9][10][11][12]13].…”

Section: Introductionmentioning

confidence: 99%

Intrinsic ergodicity for factors of $(-\beta)$ -shifts

Shinoda

Yamamoto

2019

Nonlinearity

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We show that every subshift factor of a (−β)-shift is intrinsically ergodic, when β ≥ 1+ √ 5 2 and the (−β)-expansion of 1 is not periodic with odd period. Moreover, the unique measure of maximal entropy satisfies a certain Gibbs property. This is an application of the technique established by Climenhaga and Thompson to prove intrinsic ergodicity beyond specification. We also prove that there exists a subshift factor of a (−β)-shift which is not intrinsically ergodic in the cases other than the above.

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Finiteness in real real cubic fields

Masáková

Tinková

2017

Acta Math. Hungar.

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Number systems with negative bases

Cited by 4 publications

References 4 publications

Finite beta-expansions with negative bases

Finite beta-expansions with negative bases

Intrinsic ergodicity for factors of $(-\beta)$ -shifts

Finiteness in real real cubic fields

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