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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?

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The geodesic flow on (possibly noncompact) hyperbolic surfaces of finite type is ergodic. The original reference for this is E. Hopf: „Fuchsian groups and ergodic theory“ Trans. AMS 39, 299-314 (1936).

In negative curvature, there is only one closed geodesic in each free homotopy class of curves. So you can not have more than one closed geodesic in a cylinder. (This is not related to ergodicity, I guess.)

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  • $\begingroup$ Thank you and +1 for your answer. As you said, the uniqueness of closed geodesic is not related to curvature. it is a result of Gauss Bonnet Theorem. But in my question I search for this possible obstruction: Is ergodicity of geodesic flow a possible obstruction to have a negatively curved cylinder with a foliation by geodesic with AT LEAST(so exactly) one closed geodesic. $\endgroup$ – Ali Taghavi Jul 8 '18 at 6:35
  • $\begingroup$ BTW, as an independednt question, is ergodicity of geodesic flow special to negative curvature?I mean that is there a (compact) manifold whose geodesic flow is ergodic but its curvature is not negative? $\endgroup$ – Ali Taghavi Jul 8 '18 at 6:37

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