Destroying ergodicity in geodesic flows on surfaces

On surfaces, metrics that contain focusing cap outside of which the curvature is non-positive give rise to ergodic, and indeed Bernouli, geodesic flow. There exist C-infinity small perturbations of these metrics that destroy the closed geodesic in the focusing cap, thereby producing what we call partially focusing systems. We prove that arbitrarily close to the ergodic focusing system are non-ergodic partially focusing systems. The proof involves perturbing near a homoclinic connection so as to produce a one-parameter family of horseshoe maps which leads to the existence of elliptic periodic orbits.