The set of smooth metrics in the torus without continuous invariant graphs is open and dense in the C ¹ 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.

Authors and Affiliations

1. Pontifícia Universidade Católica do Rio de Janeiro, PUC-Rio, Departamento de Matemática, Rua Marquês de São Vicente, 225, Gávea, Rio de Janeiro, BRASIL

   Rafael O. Ruggiero

Corresponding author

Correspondence to Rafael O. Ruggiero.

Partially supported by CNPq, FAPERJ, TWAS