The set of smooth metrics in the torus without continuous invariant graphs is open and dense in the C1 topology

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 C ¹ 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 C ¹ bumps are enough to prevent the existence of invariant graphs.Partially supported by CNPq, FAPERJ, TWAS