  We study the late time evolution of a class of exact anisotropic cosmological
solutions of Einstein's equations, namely spatially homogeneous cosmologies of
Bianchi type VII$_0$ with a perfect fluid source. We show that, in contrast to
models of Bianchi type VII$_h$ which are asymptotically self-similar at late
times, Bianchi VII$_0$ models undergo a complicated type of self-similarity
breaking. This symmetry breaking affects the late time isotropization that
occurs in these models in a significant way: if the equation of state parameter
$\gamma$ satisfies $\gamma \leq 4/3$ the models isotropize as regards the shear
but not as regards the Weyl curvature. Indeed these models exhibit a new
dynamical feature that we refer to as Weyl curvature dominance: the Weyl
curvature dominates the dynamics at late times. By viewing the evolution from a
dynamical systems perspective we show that, despite the special nature of the
class of models under consideration, this behaviour has implications for more
general models.
