The modular surface and continued fractions. (English) Zbl 0545.30001

Series, Caroline

J. Lond. Math. Soc., II. Ser. 31, 69-80 (1985).

This note clarifies the somewhat elusive but well known connection between geodesics on the modular surface \(\mathbb H/\text{SL}(2,\mathbb Z)\) and continued fractions. Use of the Farey tesselation \(F\) corresponding to the congruence subgroup mod 2 allows simultaneous coding of geodesics by their cutting sequences across F and determination of the continued fraction expansions of their endpoints. The geodesic flow is a special flow over a shift which is essentially the natural extension of the continued fraction transformation, under a height function given by crossing times across regions in \(F\).

Reviewer: C. Series

MSC:

37C10	Dynamics induced by flows and semiflows
11J06	Markov and Lagrange spectra and generalizations
30F35	Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)

Keywords:

modular group; Farey tesselation; geodesic flow