k-integrals and k-Lie symmetries in discrete dynamical systems

We generalize the concept of symplectic maps to that of k- symplectic maps: maps whose kth iterates are symplectic. Similarly, k-symmetries and k-integrals are symmetries (resp. integrals) of the kth iterate of the map. It is shown that k-symmetries and k-integrals are related by the k-symplectic structure, as in the k = 1 continuous case (Noetherʹs theorem). Examples are given of k-integrals and their related k-symmetries for k = 1,…,4.