Singularity of projections of 2-dimensional measures invariant under the   geodesic flow
  We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure. 
Risto Hovila, Esa J\"arvenp\"a\"a, Maarit J\"arvenp\"a\"a and   Fran\c{c}ois Ledrappier
math.DS math-ph math.DG math.MP