Geodesic flows and the mother of all continued fractions

We extend the Series [The modular surface and continued fractions, J. London Math. Soc. (2) 31(1) (1985) 69–80] connection between [Formula: see text], cutting sequences, regular fractions to slow converging Lehner Farey with digits text] in notation used for fractions. also introduce an alternative insertion singularization algorithm expansions other non-semiregular dual expansion so that is invariant under natural extension map.