A \((-\beta)\)-expansion associated to Sturmian sequences (Q2911027)

The authors describe a construction of Sturmian sequences and corresponding divisions of the unit interval that give rise to expansions in negative base \(-\beta\), when \(\beta > 1\) is a quadratic number satisfying \(\beta^2 = k \beta + 1\), \(k \geq 1\). These expansions can also be generated by the transformation \(T(x) = -\beta x + (1 + \lfloor \beta^2 x \rfloor)/\beta\) on \([-1/\beta^2,k/\beta^2)\) and \(T(x) = -\beta x + k/\beta\) on \([k/\beta^2,1/\beta]\). They differ from those defined in [\textit{S. Ito} and \textit{T. Sadahiro}, Integers 9, No. 3, 239--259, A22 (2009; Zbl 1191.11005)] and belong to those considered in [\textit{K. Dajani} and \textit{C. Kalle}, Integers 11B, A05, 18 p. (2011; Zbl 1283.11014)]. The corresponding shift space is of finite type; the set of forbidden words is \(\{00,10,\dots,(k{-}1)0\}\). For general \(\beta\), one obtains a variant of the Ostrowski numeration system.