A new class of $α$-Farey maps and an application to normal numbers
We define two types of the $α$-Farey maps $F_α$ and $F_{α, \flat}$ for $0 < α< \tfrac{1}{2}$, which were previously defined only for $\tfrac{1}{2} \le α\le 1$ by R.~Natsui (2004). Then, for each $0 < α< \tfrac{1}{2}$, we construct the natural extension maps on the plane and show that the natural extension of $F_{α, \flat}$ 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 $α$-continued fractions does not vary by the choice of $α$, $0 < α< 1$. This extends the result by C.~Kraaikamp and H.~Nakada (2000).
math.DS↗