Geodesic flows and slow downs of continued fraction maps

The connection between cutting sequences of geodesics on the modular surface $\mathrm{PSL}(2,\mathbb{Z})\mathrm{\backslash }\mathbb{H}$ and regular continued fractions was established by Series, and Heersink expanded the cross-section of the geodesic flow on the unit tangent bundle to the modular surface to describe the Farey tent-map as a slowdown of the Gauss map for the regular continued fractions. Boca and the author expanded the connection between cutting sequences of geodesics on the modular surface $\mathrm{\Theta }\mathrm{\backslash }\mathbb{H}$ and even continued fractions, which was previously established as a billiard flow by Bauer and Lopes. We will similarly expand the cross-section of the geodesic flow on this unit tangent bundle to describe the three-branch slowdown of the even Farey map.