Totally ergodic generalised matrix equilibrium states have the Bernoulli property

TL;DR

This paper proves that totally ergodic generalized matrix equilibrium states exhibit Bernoulli properties, showing they are isomorphic to Bernoulli shifts and extending understanding of their stochastic structure.

Contribution

It establishes that all totally ergodic generalized matrix equilibrium states are psi-mixing and Bernoulli, resolving a specific case of a longstanding question.

Findings

They are psi-mixing with respect to natural partitions.

Their natural extensions are isomorphic to Bernoulli processes.

The results apply to self-affine repelling sets with generic translations.

Abstract

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.