On a class of dynamical spacetimes which are asymptotic as t\to\infty to a stationary spacetime containing a horizon \mathcal{H}_0, we show the existence of a unique null hypersurface \mathcal{H} which is asymptotic to \mathcal{H}_0. This is a special case of a general unstable manifold theorem for perturbations of flows which translate in time and have a normal sink at an invariant manifold in space. Examples of horizons \mathcal{H}_0 to which our result applies include event horizons of subextremal Kerr and Kerr-Newman black holes as well as event and cosmological horizons of subextremal Kerr-Newman-de Sitter black holes. In the Kerr(-Newman) case, we show that \mathcal{H} is equal to the boundary of the black hole region of the dynamical spacetime.
