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

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)
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