Natural extensions and entropy of \(\alpha \)-continued fractions (Q2905059)

The paper under review deals with Nakada's \(\alpha\)-continued fractions. The underlying dynamical system \(T_\alpha: [\alpha - 1, \alpha] \rightarrow [\alpha - 1, \alpha]\) is introduced, and natural (invertible) extensions are constructed for these maps with \(\alpha \in (0,1]\) with a rather complicated domain \(\Omega_\alpha \subset {\mathbb R}^2\). Establishing a conjecture of \textit{L. Luzzi} and \textit{S. Marmi} [Discrete Contin. Dyn. Syst. 20, No. 3, 673--711 (2008; Zbl 1179.11022)], it is shown that \(h(T_\alpha) \mu( \Omega_\alpha) = \pi^2/6\), where \(h(\cdot)\) denotes the entropy and \(\mu\) is a measure on \({\mathbb R}^2\) with density \(d\mu = (1+xy)^{-2} dx dy\). Following this result, the dependence of \(\mu(\Omega_\alpha)\) on \(\alpha\) is studied, and a number of results are derived. Among them is the continuous dependence on \(\alpha\) of \(h(T_\alpha)\) claimed by Luzzi and Marmi [loc. cit.]; and it is shown that the maximal interval where the entropy is constant is \([g^2, g]\), where \(g = (\sqrt{5} - 1)/2\).NEWLINENEWLINEThe paper ends with a nice list of open questions.