We consider a $C^\infty$ Lagrangian function $L: T^*M\rightarrow \mathbb{R}$ which is superlinear and convex in the fibers and has one antisymplectic symmetry. We prove that:

1. in every energy level strictly above the critical one, there exists a Mather set which is the union of some periodic orbits;

2. for $L$ generic, these orbits are hyperbolic;

3. on the torus, these orbits have one homoclinic orbit.