Cross sections for geodesic flows and α-continued fractions

Author: Arnoux Pierre   Schmidt Thomas A  

Publisher: IOP Publishing

ISSN: 0951-7715

Source: Nonlinearity, Vol.26, Iss.3, 2013-03, pp. : 711-726

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Abstract

We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the α-continued fractions, for each α ∈ (0, 1]. The argument applies in wide generality, as we illustrate with its application to the Rosen continued fractions and their recently introduced α-variants.