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

TL;DR

This paper introduces new $eta$-Farey maps for $eta$ between 0 and 0.5, constructs their natural extensions, and demonstrates that the set of normal numbers associated with these maps remains invariant across different $eta$ values.

Contribution

It extends the definition of $eta$-Farey maps to a new parameter range and proves the invariance of normal numbers for these maps, generalizing previous results.

Findings

Natural extensions are metrically isomorphic for the new maps.

The set of normal numbers does not depend on the choice of $eta$ in (0,1).

Extension of previous invariance results to a broader class of maps.

Abstract

We define two types of the $\alpha$-Farey maps $F_{\alpha}$ and $F_{\alpha, \flat}$ for $0 < \alpha < \tfrac{1}{2}$, which were previously defined only for $\tfrac{1}{2} \le \alpha \le 1$ by R.~Natsui (2004). Then, for each $0 < \alpha < \tfrac{1}{2}$, 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 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).