  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.
