In this paper we investigate asymptotic isotropization. We derive the
asymptotic dynamics of spatially inhomogeneous cosmological models with a
perfect fluid matter source and a positive cosmological constant near the de
Sitter equilibrium state at late times, and near the flat FL equilibrium state
at early times. Our results show that there exists an open set of solutions
approaching the de Sitter state at late times, consistent with the cosmic
no-hair conjecture. On the other hand, solutions that approach the flat FL
state at early times are special and admit a so-called isotropic initial
singularity. For both classes of models the asymptotic expansion of the line
element contains an arbitrary spatial metric at leading order, indicating
asymptotic spatial inhomogeneity. We show, however, that in the asymptotic
regimes this spatial inhomogeneity is significant only at super-horizon scales.