Growth rate of periodic orbits for geodesic flows on surfaces with regions of positive curvature

We construct a Margulis measure on the unit tangent bundle of compact surfaces that have regions of positive curvature. This measure is then used to obtain a lower bound Cesaro estimate on P epsilon(t), which is the number of periodic orbits of period t +/- epsilon for the geodesic flow. This is the first step toward obtaining precise asymptotics and suggests that Pepsilon(t) grows like ektt , for some constant k.