Geodesic flows and slow downs of continued fraction maps

TL;DR

This paper explores the relationship between geodesic flows on modular surfaces and continued fraction maps, extending previous work to describe a three-branch slowdown of the even Farey map.

Contribution

It introduces an expanded cross-section of the geodesic flow on the modular surface to model the three-branch slowdown of the even Farey map, building on prior connections with continued fractions.

Findings

Established a connection between geodesic flows and even continued fractions.

Described the three-branch slowdown of the even Farey map.

Extended the cross-section framework for geodesic flows.

Abstract

The connection between cutting sequences of geodesics on the modular surface $\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$ 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 $\Theta\backslash\mathbb{H}$ 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.