Abstract
We show that the set of C∞ metrics in the two dimensional torus with no continuous invariant graphs of the geodesic flow is open and dense in the C1 topology. The generic nonexistence of invariant graphs with rational rotation numbers was known in the C∞ topology for metrics, and in general the generic nonexistence in the C∞ topology of invariant graphs with Liouville rotation numbers is known for twist maps and Hamiltonian flows in the torus. The main idea of the proof is that small C1 bumps are enough to prevent the existence of invariant graphs.
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