We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component. Considering the extension of an endomorphism to the completion we count the number of orbits for the action of the subgroup of fixed points (resp. periodic) points on the set of infinite fixed (resp. periodic) points. Finally, we study the dynamics of infinite points: for type II endomorphisms we prove that every infinite point is either periodic or wandering, which implies that the dynamics is asymptotically periodic. We also prove the latter for the case of automorphisms.
Year of publication: 2020