Abstract: | 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 of a Holder potential , is 1-sided Bernoulli, then f is expanding from the point of view of a pointwise section dimension of . If the measure of maximal entropy μ 0 is 1-sided Bernoulli, then f is shown to be distance expanding on Λ; and if is 1-sided Bernoulli for f expanding, then 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 is shown to have always exponential decay of correlations on Holder observables and to be mixing of any order. |