Totally Ergodic Generalised Matrix Equilibrium States have the Bernoulli Property

Abstract

We show that every totally ergodic generalised matrix equilibrium state is $$\psi $$ ψ -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.