Geodesic flows and the mother of all continued fractions

Abstract

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