We will study transitive sets (typically, homoclinic classes) which are partially hyperbolic with one dimensional center direction. We are specially interested in the case where this direction is genuinely non-hyperbolic (i.e., there are some hyperbolic periodic points which are expanding in the central direction and other periodic points which are contracting).

In this setting, the space of ergodic measures splits into three parts according to the exponent corresponding to the central direction: positive (expanding), negative (contracting), and zero (neutral). In many cases, in very rough terms, the expanding and contracting measures are glued throughout the neutral ones. But this is not always the case, and in some case special configurations arise. A key ingredient in those discussions are the so-called exposed pieces of dynamics.