Cross sections for geodesic flows and \alpha-continued fractions

TL;DR

This paper develops a new coding of geodesic flows on the modular surface using \

Contribution

It introduces a modified coding approach that links geodesic flows with \

Findings

The coding applies to \

It extends to Rosen continued fractions and their \

The method provides a robust framework for analyzing \

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 \alpha-continued fractions, for each $\alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced \alpha-variants.