Digital expansions with negative real bases

Wolfgang Steiner (1)

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Similarly to Parry's characterization of $\beta$-expansions of real numbers in real bases $\beta > 1$, Ito and Sadahiro characterized digital expansions in negative bases, by the expansions of the endpoints of the fundamental interval. Parry also described the possible expansions of $1$ in base $\beta > 1$. In the same vein, we characterize the sequences that occur as $(-\beta)$-expansion of $\frac{-\beta}{\beta+1}$ for some $\beta > 1$. These sequences also describe the itineraries of $1$ by linear mod one transformations with negative slope.

Domaines: Systèmes dynamiques [math.DS]; Théorie des nombres [math.NT]

Wolfgang Steiner. Digital expansions with negative real bases. Acta Mathematica Hungarica, 2013, 139 (1-2), pp.106-119. ⟨10.1007/s10474-012-0252-1⟩. ⟨hal-00648536⟩