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We show that every totally ergodic generalised matrix equilibrium state ispsi-mixing with respect to the natural partition into cylinders and hence ismeasurably isomorphic to a Bernoulli shift in its natural extension. Thisimplies that the natural extensions of ergodic generalised matrix equilibriumstates are measurably isomorphic to Bernoulli processes extended by finiterotations. This resolves a question of Gatzouras and Peres in the special caseof self-affine repelling sets with generic translations.

Totally ergodic generalised matrix equilibrium states have the Bernoulli property