Geodesic flows and the mother of all continued fractions

Abstract

We extend the Series' connection between the modular surface $\mathcal{M}=\mathrm{PSL}(2,\mathbb{Z})\backslash \mathbb{H}$, cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits $(1,+1)$ and $(2,-1)$ 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 $\frac{dxdy}{(1+xy{)}^2}$ is invariant under the natural extension map.

12 pages, 4 figures

Cited by in corpus (1)

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