Two-dimensional superintegrable mappings and integrable hierarchies in the (2|2) superspace

The formalism of integrable mappings is applied to the problem of constructing hierarchies of (1 + 2) dimensional integrable systems in the (2|2) superspace. We find new supersymmetric integrable mappings and corresponding to them new hierarchies of integrable systems which, at the reduction to the (1|2) superspace, possess N = 2 supersymmetry. The general formulae obtained for the hierarchies are used to explicitly derive their first nontrivial equations possessing a manifest N = 2 supersymmetry. New bosonic substitutions and hierarchies are obtained from the supersymmetric counterparts in the bosonic limit.