Does the ergodicity of geodesic flow of compact surfaces with negative curvature stile hold for non compact case?

Is not the ergocity theorems of geodesic flow an obstruction to have a cylinder with negative curvature which is foliated by geodesics such that there is a unique closed curve for this foliation?

**Motivation:** More than 15 years ago I heard from some one who said me you can not reach the following aim because of ergodicity of geodesic flow in negative curvature

Limit cycles as closed geodesics(in negatively or positively curved space)

But I do not see why this is realy an obstruction?