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.