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A new class of α-Farey maps and an application to normal numbers
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 15 September 2025, pp. 1-37
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- Article
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We define two types of the α-Farey maps Fα 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).
