A note on symplecticity of step-transition mappings for multi-step methods

We prove that for a linear multi-step method ∑k=0mαkZk=τ∑k=0mβkf(Zk), even though the mappings Z0→Z1,…,Zm-2→Zm-1Z0→Z1,…,Zm-2→Zm-1 are chosen to be symplectic, Zm-1→ZmZm-1→Zm will be non-symplectic. Similarly, there is an interesting result for a sort of general linear methods.