# Distribution of the periodic points of the Farey map

## TL;DR

This paper extends the connection between geodesic flow on the modular surface and Farey map periodic points, proving their equidistribution when ordered by geodesic length.

## Contribution

It introduces a new cross section for the geodesic flow that relates to the Farey map's natural extension, broadening the understanding of periodic points.

## Key findings

- Established a correspondence between closed geodesics and Farey map periodic points.
- Proved equidistribution of Farey map periodic points ordered by geodesic length.
- Extended previous results to include Farey map's periodic points.

## Abstract

We expand the cross section of the geodesic flow in the tangent bundle of the modular surface given by Series to produce another section whose return map under the geodesic flow is a double cover of the natural extension of the Farey map. We use this cross section to extend the correspondence between the closed geodesics on the modular surface and the periodic points of the Gauss map to include the periodic points of the Farey map. Then, analogous to the work of Pollicott, we prove an equidistribution result for the periodic points of the Farey map when they are ordered according to the length of their corresponding closed geodesics.