A new class of $\alpha$-Farey maps and an application to normal numbers

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