A new class of <i>α</i>-Farey maps and an application to normal numbers

Abstract We define two types of the α -Farey maps F α and $F_{\alpha, \flat}$ for $0 \lt \alpha \lt \tfrac{1}{2}$ , which were previously defined only for $\tfrac{1}{2} \le \alpha \le 1$ by Natsui (2004). Then, for each $0 \lt \alpha \lt \tfrac{1}{2}$ , we construct the natural extension maps on the plane and show that the natural extension of $F_{\alpha, \flat}$ is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α -continued fractions does not vary by the choice of α , $0 \lt \alpha \lt 1$ . This extends the result by Kraaikamp and Nakada (2000).