In 1981, Hitoshi Nakada from Keio University, Yokohama, introduced for a parameter $\alpha\in [0,1]$ a family of continued fraction maps, and studied for $\alpha$ between $\frac12$ and 1 their natural extensions. These are the Nakada $\alpha$-expansions.

These $\alpha$-expansions, and in particular the fact that Hitoshi Nakada also obtained their natural extensions, played a crucial role in the 1983 proof of the so-called Doeblin-Lenstra conjecture by Wieb Bosma, Henk Jager and Freek Wiedijk. The results of Nakada were extended to values of $ \alpha$ between $\sqrt{2}-1$ and $\frac12$ by Marmi, Moussa and Yoccoz, who studied in 1997 various possible versions of the Brjuno functions. After this last paper there was a long silence, until 2008, when Luzzi and Marmi showed using simulations that the properties of $\alpha$-expansions for $\alpha$ between 0 and $\sqrt{2}-1$ are very mysterious.

In this talk I will use two operations on the digits of continued fraction expansions, one of which was already known to Perron, to shed some light on what is happening.