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 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.
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Partially supported by CNPq, FAPERJ, TWAS
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Ruggiero, R.O. The set of smooth metrics in the torus without continuous invariant graphs is open and dense in the C 1 topology. Bull Braz Math Soc, New Series 35, 377–385 (2004). https://doi.org/10.1007/s00574-004-0020-0
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DOI: https://doi.org/10.1007/s00574-004-0020-0