On some coding and mixing properties for a class of chaotic systems

We study certain ergodic properties of equilibrium measures of hyperbolic non-invertible maps f on basic sets with overlaps Λ. We prove that if the equilibrium measure \({\mu_\phi}\) of a Holder potential \({\phi}\) , is 1-sided Bernoulli, then f is expanding from the point of view of a pointwise section dimension of \({\mu_\phi}\) . If the measure of maximal entropy μ ₀ is 1-sided Bernoulli, then f is shown to be distance expanding on Λ; and if \({\mu_\phi}\) is 1-sided Bernoulli for f expanding, then \({\mu_\phi}\) must be the measure of maximal entropy. These properties are very different from the case of hyperbolic diffeomorphisms. Another result is about the non 1-sided Bernoullicity for certain equilibrium measures for hyperbolic toral endomorphisms. We also prove the non-existence of generating Rokhlin partitions for measure-preserving endomorphisms in several cases, among which the case of hyperbolic non-expanding toral endomorphisms with Haar measure. Nevertheless the system \({(\Lambda, f, \mu_\phi)}\) is shown to have always exponential decay of correlations on Holder observables and to be mixing of any order.