  We present results on the non-linear dynamics of inhomogeneous cosmological
models with irrotational dust and a positive cosmological constant,
considering, in particular, a wide class with vanishing magnetic Weyl tensor.
For those patches of the universe that do not recollapse to a singularity we
find a unique attractor, representing a de Sitter vacuum phase. For the
(re-)collapsing regions we find a family of (Kasner) attractors, so that
generically these regions fall in spindle-like singularities. These results
give substantial support to the idea that the universe can be very
inhomogeneous on ultra-large, super-horizon scales, whit observers living in
those (almost) isotropic regions that emerge from an inflationary phase.
