Besicovitch-Federer projection theorem and geodesic flows on Riemann   surfaces
  We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports such that their projections to the base manifold are 2-dimensional but the supports of the projections are Lebesgue negligible. In particular, the union of complete geodesics has Hausdorff dimension 2 and is Lebesgue negligible. 
Risto Hovila, Esa J\"arvenp\"a\"a, Maarit J\"arvenp\"a\"a and   Fran\c{c}ois Ledrappier
math.CA math.DG math.DS