State dependent symplecticity of symmetric methods

Despite symmetric one-step methods applied to Hamiltonian dynamical systems fail in general to be symplectic, we show that symmetry implies, however, a relation which is close to symplecticity and that we called state dependent symplecticity. We introduce such definition for general maps and analyze it from an analytical viewpoint in one simpler case. Some numerical tests are instead reported as a support of this feature in relation with the good long time behaviour of the solutions generated by symmetric methods.