Continuous homomorphisms between algebras of iterated Laurent series over a ring

We study continuous homomorphisms between algebras of iterated Laurent series over a commutative ring. We give a full description of such homomorphisms in terms of a discrete data determined by the images of parameters. In similar terms, we give a criterion of invertibility of an endomorphism and provide an explicit formula for the inverse endomorphism. We study the behavior of the higher-dimensional residue under continuous homomorphisms.