Totally ergodic generalised matrix equilibrium states have the Bernoulli property (Q2231664)

Given a dynamical system, the repelling set will usually have zero Lebesgue measure. An important task is to look for invariant measures supported on the repelling set with the largest possible Hausdorff dimension. This raises natural fundamental questions: do such measures exist? are they unique? what are their ergodic properties? Questions and conjectures in this context have been raised on numerous occasions by various authors. \textit{D. Gatzouras} and \textit{Y. Peres} [Ergodic Theory Dyn. Syst. 17, No. 1, 147--167 (1997; Zbl 0876.58039)] gave the following conjecture: {Conjecture}. Let \(f: M \rightarrow M\) be an expanding map and \(K \subseteq M\) be a compact invariant set which satisfies specification. Then \(K\) supports a unique ergodic \(f\)-invariant measure with the same Hausdorff dimension as \(K\). This measure is mixing for \(f\) and, perhaps, its natural extension is measurably isomorphic to a Bernoulli shift. For the case of conformal expanding maps, the above conjecture has been substantially understood. In this case all Lyapunov exponents of a given invariant measure are guaranteed to be equal. Except this special case the problem is more difficult and one has to consider equilibrium states of a potential which is defined in terms of several distinct Lyapunov exponents and cannot be reduced to the classical thermodynamic formalism of continuous potentials. In this paper, the author shows 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 answers a question of \textit{D. Gatzouras} and \textit{Y. Peres} [loc. cit.] in the special case of self-affine repelling sets with generic translations.