  We show that every totally ergodic generalised matrix equilibrium state is
psi-mixing with respect to the natural partition into cylinders and hence is
measurably isomorphic to a Bernoulli shift in its natural extension. This
implies that the natural extensions of ergodic generalised matrix equilibrium
states are measurably isomorphic to Bernoulli processes extended by finite
rotations. This resolves a question of Gatzouras and Peres in the special case
of self-affine repelling sets with generic translations.
