The characteristic DEL recursion/reduction axioms can be thought of as establishing a system of recursive equations. Using modal frame correspondence theory, one can see that these characterize updates induced by epistemic events as a generalized p-morphism family: a set of ‘generalized’ partial p-morphisms, whose domains are modally definable. This raises the issue of finding generalizations of the standard modal preservation/invariance results under p-morphisms. We show that invariance now turns into ‘backwards translation’: this is reflected in the known results establishing co-expressivity of a dynamic-epistemic language with a suitably chosen static-epistemic basis, which are once again based on the reduction/recursion axioms. But the perspective proposed here in terms of systems of (co-)recursive equations becomes even more useful, when we move to languages with common knowledge or other fixed-point constructions. This leads to a new view on the ‘heavy’ system of epistemic PDL which has been used in this setting, that gives new insights and a greatly simplified presentation of its recursion laws.