Transformations generating negative \(\beta\)-expansions (Q2855522)

Piecewise linear transformations of the form \(x \mapsto - \beta x - d\) generate expansions in base \(-\beta < -1\). The authors consider mainly the transformations defined by \(d = 0\) if \(x < \alpha\) and \(d = 1\) if \(x \geq \alpha\), with \(1 < \beta \leq 2\) and \(\frac{-1}{\beta(\beta^2-1)} \leq \alpha \leq \frac{1}{\beta^2-1} - \frac{1}{\beta}\). For certain parameters, the support of the unique absolutely continuous invariant measure is determined. Using Hofbauer's trick, a formula for the invariant density is given. For \(\frac{-1}{\beta(\beta+1)} < \alpha < \frac{1-\beta}{\beta(\beta+1)}\), (the restriction to the attractor of) the transformation is conjugate to the map \(-\beta x - \beta (\beta+1) \alpha \bmod 1\).NEWLINENEWLINEThe set of corresponding digit sequences is characterised in terms of expansions of \(\alpha\). The greedy expansion of a number is defined by the authors as the one that is maximal with respect to the alternating order. The set of these expansions cannot be generated by a piecewise linear transformation, but they can be described in terms of random negative \(\beta\)-expansions. Some results on unique expansions for positive bases are adapted to negative bases, but the authors fail to notice that \(x\) has a unique negative \(\beta\)-expansion (with digits \(0\) and \(1\)) if and only if \(\frac{1}{\beta^2-1} - x\) has a unique \(\beta\)-expansion.