Spheres with positive curvature and nearly dense orbits for the geodesic flow

For any ε > 0, we construct an explicit smooth Riemannian metric on the sphere Sⁿ, n ≥ 3, that is within ε of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is ε-dense in the unit tangent bundle. Moreover, for any ε > 0, we construct a smooth Riemannian metric on Sⁿ, n ≥ 3, that is within ε of the round metric and has a geodesic for which the complement of the closure of the corresponding orbit of the geodesic flow has Liouville measure less than ε.