A new class of α-Farey maps and an application to normal numbers

2024/05/17 by Dajani, Karma, Kraaikamp, Cornelis, Nakada, Hitoshi +1
#11J70 #11K50 #37A10 #37A44 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.10921

Abstract

We define two types of the α-Farey maps F_α and F_{α, \flat} for 0 < α< \tfrac12, which were previously defined only for \tfrac12 ≤ α≤ 1 by R.~Natsui (2004). Then, for each 0 < α< \tfrac12, 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).

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