Associated to any normal unital completely positive map (“UCP map”) on a von Neumann algebra is an abstractly defined W∗ -dynamical system, its “asymptotic lift.” This generalizes the passage from an injective ∗-endomorphism α : M → M to its restriction to an automorphism of its “tail algebra” T k≥0 α k (M). We explicitly calculate the asymptotic lift of an arbitrary endomorphism, and show how this computes the asymptotic lift of any UCP map in terms of its dilation theory. This is joint work with W. Arveson.